Computer-implemented method, computer-readable medium, computer program product, and corresponding system for generating an aerial image of a photolithographic mask

Abstract

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

Description

[Technical Field]

[0001] Related Applications This application claims the benefit of German Patent Application No. 10 2022 135019.3, filed December 29, 2022, which is incorporated by reference in its entirety.

[0002] 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 photolithography mask. The method, computer-readable medium, computer program product, and system may be utilized for quantitative metrology, photolithography mask defect detection, photolithography mask defect conformance assessment, photolithography mask improvement, system simulation, or process control, process monitoring, or process improvement. [Background technology]

[0003] Wafers, made from thin slices of silicon, serve as substrates for microelectronic devices, which contain semiconductor structures built within and on the wafer. The semiconductor structures are built layer by layer using iterative processing steps involving repeated chemical, mechanical, thermal, and optical processes. The dimensions, shape, and placement of the semiconductor structures and patterns are influenced by a variety of factors. One of the most critical steps is the photolithography process.

[0004] Photolithography is a process used to fabricate patterns on wafer substrates. The patterns printed on the surface of the substrate are typically generated by computer-aided design (CAD). From the design, a photolithography mask is generated layer by layer. The photolithography mask contains a magnified image of the computer-generated pattern to be etched into the substrate. The photolithography mask can be further adapted, for example, by optical proximity correction techniques. During the printing process, the illuminated image projected from the photolithography mask is focused onto a thin photoresist film formed on the substrate. A semiconductor chip that powers a cell phone or tablet, for example, contains approximately 80 to 120 patterned layers. Traditionally, when photolithography required less precision, the circuit layout corresponded to the mask pattern, and the mask pattern corresponded to the wafer pattern.

[0005] Due to increasing integration density in the semiconductor industry, photolithography masks must image smaller and smaller features onto wafers. The aspect ratio and number of layers in integrated circuits are constantly increasing, and structures are growing in three dimensions (vertically). Current memory stack heights exceed 12 μm. Meanwhile, feature sizes are shrinking. Minimum feature sizes, or critical dimensions, are less than 10 nm, e.g., 7 nm or 5 nm, and are approaching feature sizes of less than 3 nm in the near future. While the complexity and dimensions of semiconductor structures are growing in three dimensions, the lateral dimensions of integrated semiconductor structures are becoming smaller. To fabricate the smaller feature dimensions imaged onto wafers, photolithography masks or templates for nanoimprint photolithography with smaller features or pattern elements are required. Therefore, the fabrication process for photolithography masks and templates for nanoimprint photolithography is becoming increasingly complex and, as a result, more time-consuming and ultimately more costly. With the advent of EUV photolithography scanners, the nature of masks has changed from transmission patterning to reflection patterning.

[0006] Currently, the minimum feature size on masks has reached sub-wavelength dimensions. This leads to the so-called optical proximity effect, which is caused by non-uniform energy intensity due to optical diffraction during the exposure process. As a result, the image formed on the substrate does not faithfully reproduce the pattern on the photolithography mask.

[0007] Therefore, many applications require an aerial image of a photolithography mask, which simulates the radiation intensity distribution at the substrate level. In this way, the aerial image enables analysis of the semiconductor structure being printed on the substrate during the printing process. However, generating an aerial image is time-consuming and costly. Therefore, methods for generating an aerial image based on a model of the photolithography mask have become important.

[0008] Among these methods, there are time-consuming rigorous simulations such as finite-difference time-domain (FDTD) or rigorous coupled-wave analysis (RWCA), and fast approximations such as thin element approximation (TEA). Rigorous simulations are generally not used in commercial computational photolithography software due to the heavy computational load in full-chip applications. Thin element approximation (TEA) assumes that the thickness of structures on a photolithography mask is very small compared to the wavelength and that the width of structures on a photolithography mask is very large compared to the wavelength. However, as lithographic processes use radiation with shorter wavelengths and structures on patterning devices become increasingly smaller with larger vertical dimensions, these assumptions no longer hold. The interaction of incoming radiation with absorber structures leads to mask 3D effects, which must be taken into account by simulation. Therefore, TEA produces inaccurate aerial images for short-wavelength radiation.

[0009] A typical mask 3D effect is, for example, mask shadowing. The chief ray angle specifies the angle between the optical axis and the normal vector of the mask surface. Current EUV projection systems, for example, use a CRA of 6°. Mask shadowing occurs due to the height of the absorber structures and non-telecentric illumination at mask level, modulating the intensity captured from the shadowed mask area through the reflective optics onto the wafer. At wafer level, this results in asymmetric shadowing, image shifts and size deviations depending on feature orientation, and shifts in the process window.

[0010] Another mask 3D effect is the phase shift caused by diffraction in the absorber structures. These phase effects produce imaging effects that are very similar to the phase deformations caused by wavefront aberrations in projection systems.

[0011] Another mask 3D effect can be attributed to the reflective properties of EUV photolithography masks. The dominant portion of the reflected light comes from the multilayers, which are designed to provide high reflectivity over a sufficiently large range of angles of incidence. However, there is also some light reflected from the top of the absorber, which causes double images.

[0012] These mask 3D effects should not be ignored during the lithography process. However, rigorous simulation methods such as finite-difference time-domain (FDTD) or rigorous coupled-wave analysis (RWCA) that take mask 3D effects into account are not computationally feasible.

[0013] Therefore, there is a need for an accurate and fast simulation method for the aerial image of a photolithographic mask that takes into account at least some of the 3D effects of the mask.

[0014] A known method for generating an aerial image of a photolithography mask is disclosed in WO 2019 / 214909. The method includes generating one or more electromagnetic field determination equations based on Maxwell's equations and the quantum Schrödinger equation. The method includes determining the electromagnetic field by propagating an electromagnetic wave through a mask stack region of interest based on Maxwell's equations and the quantum Schrödinger equation.

[0015] It is an object of the present invention to obtain an alternative method for generating an aerial image. A further object of the present invention is to generate an aerial image with high accuracy. In particular, it is an object of the present invention to account for mask 3D effects during the generation of the aerial image. A further object of the present invention is to generate an aerial image that requires less computation time. Another object of the present invention is to obtain an aerial image generation method that is applicable to both transmission and reflection photolithography masks. Another object of the present invention is to make the resolution of the aerial image independent of the smallest design feature. Another object of the present invention is to enable sub-pixel design feature resolution. Another object of the present invention is to enable more flexible representation of photolithography masks. A further object of the present invention is to improve the design of photolithography masks without the need to actually print them on a wafer. Another object of the present invention is to detect defects or placement errors in photolithography masks or measure structures on photolithography masks with high accuracy and short computation time. Another object of the present invention is to assess the suitability of defects detected in photolithography masks with high accuracy and short computation time.

[0016] The object is achieved by the invention as specified in the independent claims. Advantageous embodiments and further developments of the invention are specified in the dependent claims. Summary of the Invention

[0017] Embodiments of the present invention relate to computer-implemented methods, computer-readable media, computer program products, and corresponding systems for generating an aerial image of a photolithography mask or for detecting defects and assessing the suitability of the defects in a photolithography mask.

[0018] An embodiment of the present invention involves a computer-implemented method for generating an aerial image of a model of a photolithographic mask under illumination with incident electromagnetic waves, the method including: a) approximately simulating propagation of the incident electromagnetic waves within a first section of the photolithographic mask including a plurality of structures; b) analytically or numerically simulating propagation of the simulated electromagnetic waves from step a) within a second section of the photolithographic mask; c) simulating a representation of the electromagnetic near field within the model of the photolithographic mask by propagating the simulated electromagnetic waves from step b) to a near-field plane; and d) generating the aerial image of the photolithographic mask by applying a simulation of an imaging process of a photolithography or metrology system to the representation of the electromagnetic near field.

[0019] The methods may be used for various purposes, for example, to improve a model of a photolithography mask, e.g., a design pattern, to repair a photolithography mask, to determine the quality of a photolithography mask, to perform measurements of a photolithography mask, to detect or assess defects in a photolithography mask, to select illumination settings for a photolithography system, for source mask optimization or reverse photolithography.

[0020] According to an embodiment of the present invention, particularly with respect to a transmissive photolithography mask, e.g., a DUV photolithography mask, a simulated electromagnetic wave is incident on a reference plane of the photolithography mask, passes through a second section of the photolithography mask, and then propagates through a first section of the photolithography mask to a near-field plane.

[0021] According to an embodiment of the present invention, particularly for a reflective photolithography mask, e.g., an EUV photolithography mask, the second section includes a multilayer in the form of a stack of optical thin films for reflecting electromagnetic waves, where the simulated electromagnetic waves are incident on the structure plane, propagate through the first section of the photolithography mask, are reflected in the multilayer of the second section of the photolithography mask, propagate back through the first section of the photolithography mask, and propagate to the structure plane and the near-field plane. Thus, according to an embodiment of the present invention, the electromagnetic waves propagate through the first section, are then reflected by the second section, and propagate again through the first section of the photolithography mask to the near-field plane.

[0022] According to an embodiment of the present invention, for reflection-type measurements of a DUV photolithography mask, the second section includes a mask carrier including a glass substrate, and a simulated electromagnetic wave is incident on a structure plane that separates the first section from the outside, and the electromagnetic wave is partially reflected by the structure plane, partially propagates through the first section of the photolithography mask, and partially reflected by the mask carrier of the photolithography mask and propagates through the first section of the photolithography mask.

[0023] In each of the above embodiments, additional reflection and interference effects at other material interfaces may also be considered.

[0024] The term "photolithographic mask" refers to a general-purpose patterning device that can be used to impart a patterned cross-section to an incoming radiation beam that corresponds to the pattern to be created in a target portion of a substrate.

[0025] The photolithography mask may 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 photolithography mask may be approximately rectangular. The photolithography mask may preferably be 5 to 7 inches long and wide, and most preferably 6 inches long and wide. Alternatively, the photolithography mask may be 5 to 7 inches long and 10 to 14 inches wide, and preferably 6 inches long and 12 inches wide.

[0026] A "model" of a photolithography mask refers to a representation of a photolithography mask or a section thereof. A model may include, for example, a computer-readable file such as a CAD file or a GDS file, or a technical drawing, a set of polygons representing a structure of a photolithography mask or a section thereof. A model of a photolithography mask may include material information, such as the complex refractive index, permittivity, permeability, or derived representation of a material included in the photolithography mask. A model of a photolithography mask may include parameters describing the dimensions of a structure of the photolithography mask, such as the thickness of a layer within a multilayer of an EUV mask or the thickness of an absorber layer, or the dimensions of an absorber structure. A model of a photolithography mask may include parameters describing the position of a structure in the photolithography mask, such as the position of an absorber structure or a layer within a multilayer. A model of a photolithography mask may include parameters describing the shape of a structure in the photolithography mask, such as the sidewall angle or corner rounding. A model of a photolithographic mask may include an image representing properties of the photolithographic mask, for example, a 2D image or a 3D image (e.g., voxels of a volume or a number of 2D slices of a volume). The image may include one, two, or more channels. The image may include image elements, for example, pixels or voxels. The properties of the photolithographic mask may include material properties, for example, refractive index, permittivity, permeability, or derived expressions. The model of a photolithographic mask may include descriptions of structures within the photolithographic mask, for example, in the form of curves, contours, polygons, splines, NURBS, Bezier curves, etc.

[0027] The model of the photolithographic mask preferably describes the photolithographic mask at least in part in a dimension orthogonal to a reference plane of the photolithographic mask. The model of the photolithographic mask may include one or more different sections or portions thereof, such as a first section and / or a second section, of the photolithographic mask. The one or more different sections may be located at different depths relative to a normal to the surface.

[0028] The first section of the photolithography mask includes a plurality of structures. These structures may be arranged in a design pattern or model that determines the pattern imprinted on the wafer during the printing process. The design pattern or model may include structures and non-structures, particularly absorber and non-absorber structures. The second section of the photolithography mask may include a mask carrier, which may include one or more layers of one or more materials. The structures and non-structures may be deposited on the mask carrier. The mask carrier may include a substrate layer. The second section may be configured to transmit incident electromagnetic waves (in the case of a transmission-type photolithography mask) or to reflect incident electromagnetic waves (in the case of a reflection-type photolithography mask). The first section may be directly adjacent to the second section of the photolithography mask. Thus, the first and second sections may have a common boundary, e.g., a boundary plane. The mask carrier of the photolithography mask may be separated by a boundary plane and a reference plane. The boundary plane may be a surface plane of the mask carrier. The reference plane is preferably parallel to the boundary plane. The reference plane may delimit the second section from the outside. It may form an interface between the mask carrier and the outside of the photolithography mask through which the electromagnetic waves propagate. The structure of the first section of the photolithography mask may be delimited by the boundary plane and the structure plane. The structure plane may delimit the first section of the photolithography mask from the outside. The structure plane may include a portion of the surface of the structure and face away from the boundary plane. Preferably, the structure plane is parallel to the boundary plane. The first section of the photolithography mask may extend between the structure plane and the boundary plane and be delimited by these planes. The second section of the photolithography mask may extend between the boundary plane and the reference plane. The second section may be delimited by the boundary plane and the reference plane. The second section may include a stack of homogeneous parallel layers. Homogeneous means that material properties do not vary within the layers. Other configurations of photolithographic masks including first and second sections may also be used.

[0029] The electromagnetic near-field refers to the distribution of electromagnetic waves in a near-field plane. The near-field plane can be located next to a structure plane of the photolithography mask that separates the first section of the photolithography mask from the outside. Preferably, the near-field plane is parallel to the structure plane of the photolithography mask. The near-field plane can be located anywhere between the structure plane and the wafer plane. For example, the near-field plane can be located at a distance of 0 to 1000 nm from the structure plane, preferably at a distance of 0 to 100 nm, more preferably at a distance of 0 to 50 nm, even more preferably at a distance of 0 to 20 nm, and most preferably at a distance of 0 to 10 nm. In a preferred embodiment of the present invention, the near-field plane and the structure plane are identical. In principle, the near-field plane can be located within the first section, within the second section, on the structure plane, on the reference plane, or outside the photolithography mask next to the reference plane of the photolithography mask, for example, if the electromagnetic wave propagates through the first section and then re-propagates into the photolithography mask.

[0030] The expression for the electromagnetic (near) field is the (complex) electric field E or the (complex) scattered electric field E SC =EE inc You can also refer to E inc denotes the incident electric field. Complex electromagnetic fields can be expressed, for example, in terms of real and imaginary parts, or amplitude and phase. The expression of an electromagnetic field is the (complex) magnetic field H or the (complex) scattered magnetic field H SC =HH inc It can also refer to H inc denotes the incident magnetic field. Expressions of the electromagnetic field may be expressed as the total or scattered electric field, or the envelope of the total or scattered magnetic field, e.g., the total electric field envelope

[0031]

number

number

[0032] The aerial image shows the radiation intensity distribution at the wafer plane and is generated from the representation of the electromagnetic near field by applying a simulation of the imaging process of a photolithography or metrology system to the representation of the electromagnetic near field.

[0033] An optical metrology system refers to a system that measures an aerial image of at least a portion of a photolithographic mask or a quantity that can be derived from the aerial image, such as critical dimension (CD), normalized image log slope (NILS), edge placement, defects, etc.

[0034] In a photolithography system, the wafer plane refers to the plane in the resist on the wafer when it is placed in the photolithography system. In an optical metrology system, the wafer plane refers to the plane where the camera sensor is located.

[0035] The electromagnetic near field is calculated differently in the first and second sections of the photolithography mask. Within the first section, several assumptions, as described below, may be made in the photolithography setup, enabling simplified and fast calculation of the electromagnetic wave propagation within the first section. The electromagnetic wave propagation within the first section is calculated using a wave propagation method, which takes into account the inhomogeneity of the medium within the first section of the photolithography mask. Within the second section, the electromagnetic wave propagation is calculated analytically or numerically. In this way, a highly accurate approximation of the electromagnetic wave propagation within the photolithography mask is obtained, requiring computational time that is somewhat less than that required for rigorous simulation methods. Therefore, simulation of electromagnetic near fields and aerial images within industrial applications becomes feasible.

[0036] According to a first example of the embodiment, in step a), the propagation of the incident electromagnetic wave in the first section of the photolithographic mask is approximately simulated using the Helmholtz equation, in this way the approximation is simplified and therefore the complexity and calculation time are reduced.

[0037] According to a second example of the embodiment, in step a), the propagation of the incident electromagnetic wave in the first section of the photolithographic mask is approximately simulated using a machine learning model. By using the machine learning model, after training, a single fast forward pass is sufficient to calculate the propagation of the incident electromagnetic wave, so that the calculation time can be greatly reduced.

[0038] According to a first embodiment aspect, the Helmholtz equation is approximated using the forward Helmholtz equation, in this way the approximation is simplified and therefore the complexity and computation time are reduced.

[0039] The forward Helmholtz equation can be solved using the beam propagation method, thus simplifying the approximation and therefore reducing the complexity and computation time.

[0040] In a preferred embodiment, the forward Helmholtz equation is solved using a wave propagation method that approximately describes the propagation of electromagnetic waves through inhomogeneous media. By using the wave propagation method, the forward Helmholtz equation is solved quickly, thereby reducing the computational time of the method. Furthermore, by taking into account the inhomogeneity of the first section of the photolithography mask, for example, due to different materials of the absorber and non-absorber structures, the wave propagation is simulated with high accuracy.

[0041] According to an aspect of a preferred embodiment, a first section of a photolithographic mask is decomposed into different materials by defining characteristic functions for each material that indicate the presence of that material in different locations of the first section of the photolithographic mask, at least one characteristic function being non-binary. This allows for a more generalized representation of the material distribution within the first section of the photolithographic mask, whereby intermediate material properties can be expressed as a weighted average of the individual materials. Such intermediate values ​​may arise, for example, when a material representation that is effective for approximating sharp contrasts and the interaction of electromagnetic waves is used. This can be used as a mathematical tool to describe material distribution, leading to improved simulation results and, therefore, a more accurate aerial image.

[0042] In an embodiment, the characteristic functions form an affine combination at each location of the first section of the photolithographic mask. An affine combination of functions is a linear combination, so that the sum of all functions at each location amounts to 1. This mathematically guarantees that the sum over all materials at each location amounts to 1.

[0043] In an embodiment, the characteristic function is band-limited, which makes it possible to choose a sampling grid with lower resolution than that required for a binary characteristic function, thus reducing the computation time of the method.

[0044] In an embodiment, a low pass filter is applied to the characteristic function, which allows for fast calculation of the band-limited characteristic function.

[0045] Applying the low-pass filter may involve applying a spatial analytical Fourier transform to the characteristic function followed by an inverse Fourier transform. In this way, the low-pass filter may be applied quickly in a reduced manner and with reduced computation time.

[0046] In an embodiment, the wave propagation method approximates the analytical Fourier transform by a fast Fourier transform and / or the inverse analytical Fourier transform by an inverse fast Fourier transform, thereby reducing the computation time of the method.

[0047] According to an aspect, the wave propagation method approximates the analytical Fourier transform by a fast Fourier transform, and the wave propagation method takes into account the angle of the incident electromagnetic wave by assuming quasi-periodic boundary conditions for the fast Fourier transform at one or more pairs of opposing boundaries perpendicular to a reference plane of the photolithographic mask, thereby improving the accuracy of the approximation of the electromagnetic wave propagation.

[0048] In an embodiment, the electromagnetic wave in the first section has a dispersion relation that depends on the angle of the incident electromagnetic wave. Preferably, the dispersion relation in the first section is modified by a phase shift in a coordinate parallel to the reference plane of the photolithography mask. In this way, the approximation accuracy of the electromagnetic wave propagation is improved.

[0049] The photolithographic mask may be a transmissive photolithographic mask or a reflective photolithographic mask.

[0050] In the case of a reflective photolithography mask, the second section includes a multilayer body in the form of a stack of optical thin films for reflecting electromagnetic waves. According to an embodiment, simulating the reflection of the electromagnetic wave within the multilayer body includes analytical or numerical calculation of a reflection coefficient at the boundary between the second section and the first section of the photolithography mask, the reflection coefficient describing the propagation of the electromagnetic wave within the stack of optical thin films of the multilayer body. Instead of repeatedly propagating the electromagnetic wave through each layer of the multilayer body, the reflection coefficient at the boundary only needs to be calculated once to simulate the reflection within the multilayer body. In this way, a highly accurate simulation of the electromagnetic wave propagation within the multilayer body is obtained with greatly reduced calculation time.

[0051] According to an embodiment, the reflection coefficients at the boundaries are calculated separately inside and outside the structures of the first section of the photolithography mask, thereby improving the accuracy of the electromagnetic wave propagation.

[0052] In an embodiment, simulating propagation of the simulated electromagnetic wave within the second section of the photolithographic mask includes applying a reflection coefficient to the electromagnetic wave incident on the boundary, thereby obtaining a fast and accurate simulation within the second section.

[0053] According to a preferred embodiment of the present invention, the method for generating an aerial image of a model of a photolithographic mask further includes adjusting at least one parameter of the method to minimize differences between one or more reference aerial images of the photolithographic mask and one or more corresponding generated aerial images of the model of the photolithographic mask, the at least one parameter being from a group including mask parameters and optical parameters. The reference aerial image can be, for example, an aerial image acquired using some type of aerial image acquisition system, or a simulated aerial image, e.g., a target aerial image, that exhibits the desired results of the aerial image generation method. The mask parameters describe characteristics of the photolithographic mask, such as bias, corner rounding, sidewall angle, layer height, refractive index, etc., regarding the size of structures on the photolithographic mask relative to their size in the model or design. The optical parameters describe the characteristics of the optical system of the system used to acquire the reference aerial image, such as the illumination settings (NA, pupil shape, intensity distribution, coherence, etc.), imaging settings (NA, pupil shape, obscuration, aberrations, apodization, defocus, distortion, magnification, etc.), sensor model (intensity integral over the active sensor area, pixel size, dark current, shot noise, stray light, etc.), motion blur due to scanning, field-dependent effects (aberrations in both illumination and imaging), etc. By adjusting at least one parameter of a method based on the reference aerial image, the accuracy of the generated aerial image is improved. This calibration step further enables various applications requiring highly accurate simulation of aerial images that can be compared with acquired aerial images, such as defect detection in die-to-database methods.

[0054] In a preferred embodiment (which may be combined with the previous preferred embodiment), the method for generating an aerial image of a model of a photolithographic mask further includes registering one or more reference aerial images of the photolithographic mask to corresponding generated aerial images of the model of the photolithographic mask and reporting at least one registration parameter. The registration parameters include any (isomorphic) transformations, such as translation, shift, rotation, scaling, shear, nonlinear transformations, etc., to align the reference aerial images with the generated aerial images. This allows accurate comparison of the reference aerial images with simulated aerial images for various applications, such as defect detection in die-to-database methods.

[0055] According to an aspect of a preferred embodiment, the one or more reference aerial images include a focus stack of a photolithographic mask. In this manner, calibration and registration parameters that accommodate different defocus values ​​can be found.

[0056] A computer-implemented method for improving a model of a photolithography mask, e.g., a design pattern, repairing a photolithography mask, determining the quality of a photolithography mask, performing measurements on a photolithography mask, detecting or assessing defects in a photolithography mask, or selecting illumination settings in a photolithography system according to an embodiment includes generating an aerial image of the model of the photolithography mask using a method according to any of the embodiments, examples, or aspects described above, analyzing the generated aerial image accordingly, and using the analysis results to improve the design of the photolithography mask, repair the photolithography mask, determine the quality of the photolithography mask, detect or assess defects in the photolithography mask, or select illumination settings for the photolithography system.

[0057] According to an embodiment, a computer-implemented method for training a machine learning model that maps a model of a photolithographic mask to an aerial image of the photolithographic mask includes generating a plurality of aerial images of the model of the photolithographic mask using the method of any one of the above-described embodiments, examples, or aspects, and training the machine learning model using training data that includes the generated aerial images. In this way, particularly fast and efficient training of the machine learning model can be achieved, since a large portion of the training data can be obtained by simulation instead of time-consuming acquisition.

[0058] According to an embodiment, a computer-implemented method for training a machine learning model for defect detection in acquired aerial images of photolithography masks includes generating model pairs for a plurality of photolithography masks, each model pair including a defect-free model of the photolithography mask, i.e., a model without defects, and a defect model of the same photolithography mask, i.e., a model including one or more defects; generating aerial image pairs from the model pairs by applying a method of any one of the embodiments, examples, or aspects described above to the defect-free model and the defect model of each model pair; and training a machine learning model using training data including the aerial image pairs.

[0059] A computer-readable medium according to an embodiment of the present invention has stored thereon a computer program executable by a computing device, the computer program including code for performing a method for generating an aerial image of a photolithography mask according to any of the above-described embodiments, examples, or aspects.

[0060] A computer program product according to an embodiment of the present invention includes instructions that, when executed by a computer, cause the computer to perform a method for generating an aerial image of a photolithography mask according to any of the above-described embodiments, examples, or aspects.

[0061] A system for generating an aerial image of a model of a photolithography mask according to an embodiment of the present invention includes a data analysis device including at least one memory and at least one processor configured to perform steps of a computer-implemented method for generating an aerial image of a photolithography mask according to any of the embodiments, examples, or aspects described above.

[0062] A system for improving a model of a photolithography mask, repairing a photolithography mask, determining the quality of a photolithography mask, performing measurements on a photolithography mask, detecting or assessing defects in a photolithography mask, or selecting illumination settings for a photolithography system according to embodiments of the present invention includes a data analysis device including at least one memory and at least one processor configured to perform steps of a computer-implemented method for generating an aerial image of a photolithography mask according to any of the embodiments, examples, or aspects described above. The system may also include a subsystem for acquiring an aerial image of the photolithography mask, which may be used in analyzing the generated aerial image of the photolithography mask, e.g., for comparison.

[0063] A system for detecting defects in a photolithography mask according to an embodiment of the present invention includes a subsystem for acquiring an aerial image of the photolithography mask, and a data analysis device including at least one memory and at least one processor configured to perform steps of a computer-implemented method for generating an aerial image of a photolithography mask according to any of the embodiments, examples, or aspects described above.

[0064] A system for assessing the suitability of defects in a photolithography mask according to a tenth embodiment of the present invention includes a subsystem for acquiring a charged particle beam image of the photolithography mask, and a data analysis device including at least one memory and at least one processor configured to perform the steps of the computer-implemented method according to the fourth embodiment of the present invention.

[0065] The invention described by the examples and embodiments is not limited to the embodiments and examples, but can be implemented by those skilled in the art through various combinations or modifications thereof. [Brief explanation of the drawings]

[0066] [Figure 1] 1 illustrates an exemplary transmissive photolithography system, such as a deep ultraviolet (DUV) photolithography system. [Figure 2] 1 illustrates the propagation of an incoming electromagnetic wave through a transmission photolithographic mask. [Figure 3] 1 illustrates an exemplary reflective photolithography system, such as an extreme ultraviolet (EUV) photolithography system. [Figure 4] 1 illustrates the propagation of an incoming electromagnetic wave through a reflective photolithographic mask. [Figure 5a] 1 shows the simulated electromagnetic near-field amplitude of a photolithography mask generated using the Rigorous Coupled-Wave Analysis (RCWA) method. [Figure 5b] 1 shows the simulated electromagnetic near-field amplitude of the same photolithography mask using the thin element approximation (TEA) method. [Figure 6] 1 shows a flowchart of a computer-implemented method according to an embodiment of the present invention. [Figure 7] 1 shows a flowchart of a computer-implemented method for generating an aerial image of a model of a photolithographic mask, with the example being a transmission photolithographic mask. [Figure 8] 10 shows a flowchart of a computer-implemented method for generating an aerial image of a model of a photolithographic mask, according to another example of a reflective photolithographic mask. [Figure 9] 1 shows a flowchart of a computer-implemented method for generating an aerial image of a model of a photolithographic mask according to an embodiment. [Figure 10a]The effect of the sampling grid resolution during sampling of the characteristic function on the approximation error is shown. [Figure 10b] The effect of the sampling grid resolution during sampling of the characteristic function on the approximation error is shown. [Figure 11] A comparison of the convergence rates of the wave propagation method implemented using a discretized binary characteristic function or a discretized band-limited characteristic function is presented. [Figure 12] 1 shows the dependence of the phase shift vector on the angle φ of the incoming electromagnetic wave. [Figure 13a] 3 illustrates steps of a computer-implemented method for generating an aerial image of a model of a photolithographic mask according to an embodiment of the present invention. [Figure 13b] 3 illustrates steps of a computer-implemented method for generating an aerial image of a model of a photolithographic mask according to an embodiment of the present invention. [Figure 13c] 3 illustrates steps of a computer-implemented method for generating an aerial image of a model of a photolithographic mask according to an embodiment of the present invention. [Figure 13d] 3 illustrates steps of a computer-implemented method for generating an aerial image of a model of a photolithographic mask according to an embodiment of the present invention. [Figure 14a] 1 shows a comparison of aerial images of a model of a photolithography mask obtained by three different simulation methods. [Figure 14b] 1 shows a comparison of aerial images of a model of a photolithography mask obtained by three different simulation methods. [Figure 14c] 1 shows a comparison of aerial images of a model of a photolithography mask obtained by three different simulation methods. [Figure 15] 1 illustrates the use of a calibration step within a defect detection method for detecting defects in an aerial image of a photolithographic mask. [Figure 16] 16 illustrates the calibration step of FIG. 15, including optimization of mask and / or optical parameters. [Figure 17] 16 illustrates the effect of the calibration step of FIG. [Figure 18] 16 illustrates the calibration step of FIG. 15 including registration. [Figure 19] 16 illustrates the calibration step of FIG. 15, which includes a combination of a prior mask parameter and / or optical parameter optimization followed by registration. [Figure 20] 16 illustrates the calibration steps of FIG. 15, including optimization and registration of joint mask and / or optical parameters. [Figure 21] 1 illustrates a training method for a machine learning model to generate an aerial image of a model of a photolithography mask. [Figure 22] 22 shows a sample model used as training data to train the machine learning model of FIG. 21. [Figure 23] We present a training method for a machine learning model for defect detection in photolithography masks. [Figure 24] FIG. 23 illustrates the generation of training data for training a machine learning model for defect detection. [Figure 25] A method is presented for improving a model of a photolithography mask, repairing a photolithography mask, determining the quality of a photolithography mask, performing measurements on a photolithography mask, detecting or assessing defects in a photolithography mask, or selecting illumination settings for a photolithography system. [Figure 26] 1 illustrates a computer-implemented method for detecting defects in a photolithography mask, according to an embodiment of the present invention. [Figure 27] 1 illustrates a computer-implemented method for assessing the suitability of defects in a photolithography mask according to an embodiment of the present invention. [Figure 28] 1 illustrates a system for generating an aerial image of a model of a photolithographic mask according to an embodiment of the present invention. [Figure 29] 1 illustrates a system for detecting defects in a photolithography mask according to an embodiment of the present invention. [Figure 30]1 illustrates a system for assessing the suitability of defects in a photolithography mask according to an embodiment of the present invention. DETAILED DESCRIPTION OF THE INVENTION

[0067] In the following, advantageous exemplary embodiments of the present invention are described and illustrated schematically in the drawings, wherein the same reference numerals are used throughout the drawings and description to denote the same features or components.

[0068] The methods and systems herein may be used in a variety of photolithography systems, for example, a transmissive photolithography system 10 or a reflective photolithography system 10'.

[0069] 1 shows an exemplary transmissive photolithography system 10, e.g., a DUV photolithography system. The main components are a radiation source 12, which may be a deep ultraviolet (DUV) excimer laser source; imaging optics, which may include, e.g., optics for defining partial coherence and shaping the radiation from the radiation source 12; a photolithography mask 14; illumination optics 16, which illuminate the photolithography mask 14; and projection optics 17, which projects an image of a photolithography mask model 92, e.g., a design pattern, onto a wafer plane 18. An adjustable filter or aperture in the pupil plane of the projection optics 17 may limit the range of beam angles that impinge on the wafer plane 18; the maximum possible angle defines 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 exiting the projection optics 17 that can also impinge on the wafer plane 18.

[0070] In this document, the terms "radiation" or "beam" are used to encompass all types of electromagnetic radiation, including ultraviolet (e.g., having wavelengths of 365, 248, 193, 157, or 126 nm) and EUV (extreme ultraviolet, e.g., having wavelengths in the range of about 3-100 nm).

[0071] Illumination optics 16 may include optical components for shaping, conditioning, and / or projecting radiation from radiation source 12 before the radiation passes through photolithography mask 14. Projection optics 17 may include optical components for shaping, conditioning, and / or projecting radiation after the radiation passes through photolithography mask 14. Illumination optics 16 excludes light source 12, and projection optics excludes photolithography mask 14.

[0072] Illumination optics 16 and projection optics 17 may include various types of optics, including, for example, refractive optics, reflective optics, apertures, and catadioptric optics, and illumination optics 16 and projection optics 17 may also include components operating according to any of these design types to collectively or individually direct, shape, or control the projection beam of radiation.

[0073] 2, the simulated electromagnetic wave is incident on reference plane 34, propagates from reference plane 34 to boundary plane 32 within second section 27 of photolithographic mask 14, and propagates from boundary plane 32 to structure plane 30 within first section 25 of photolithographic mask 14. In this manner, the computer-implemented method for generating aerial image 64 may be applied to transmission photolithographic masks, such as DUV photolithographic masks.

[0074] FIG. 2 illustrates the propagation of an incoming electromagnetic wave 22 through a transmissive photolithography mask 14, e.g., a DUV photolithography mask. The photolithography mask 14 includes a first section 25 and a second section 27. The first section 25 includes a diffraction grating 24, and the second section 27 includes a mask carrier 48. The diffraction grating 24 is formed by combining an absorber structure 26 and a non-absorber structure 28. The absorber structure 26 is made of one or more materials that absorb the electromagnetic wave 22, such as titanium nitride or tantalum nitride. The non-absorber structure 28 is made of one or more materials that absorb the electromagnetic wave 22 to a lesser extent than the absorber material. For example, the non-absorber structure 28 may include a vacuum. Thus, the diffraction grating 24 is an inhomogeneous medium. The absorber structure 26 and the non-absorber structure 28 are deposited on a mask carrier 48. The mask carrier 48 may include a substrate layer 46. The mask carrier 48 of the photolithography mask 14 is bounded by a boundary plane 32 and a reference 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 reference plane 34 is a boundary plane through which the electromagnetic wave 22 enters the diffraction grating 24. The incoming electromagnetic wave 22 impinges on the reference plane 34. The reference plane 34 forms an interface between the mask carrier 48 and the outside of the photolithography mask 14 through which the electromagnetic wave 22 propagates. The absorber structure 26 in the diffraction grating 24 of the photolithography mask 14 is bounded by the boundary plane 32 and a structure plane 30. The structure plane 30 is a boundary plane that includes a portion of the surface of the absorber structure 26 that faces away from the boundary plane 32. Preferably, the structure plane 30 is parallel to the boundary plane 32. The first section 25 of the photolithography mask 14 extends between the structure plane 30 and the boundary plane 32 and is bounded by these planes. A second section 27 of the photolithographic mask 14 extends between and is bounded by the boundary plane 32 and the reference plane 34 .

[0075] In the case of a transmission photolithography mask 14, by way of example, the simulated electromagnetic wave 22 is incident on the reference plane 34, propagates from the reference plane 34 to the boundary plane 32 within the second section 27 of the photolithography mask 14, and propagates from the boundary plane 32 to the structure plane 30 within the first section 25 of the photolithography mask 14.

[0076] 3 shows an exemplary reflective photolithography system 10′, e.g., an extreme ultraviolet (EUV) lithography system. The main components are a radiation source 12, which may be a laser-plasma light source; illumination optics 16, which may include, e.g., optics for defining partial coherence and shaping the radiation from the radiation source 12; a photolithography mask 14; a photolithography mask model 92; and projection optics 17, which projects an image of the design pattern onto a wafer plane 18. An adjustable filter or aperture in the pupil plane of the projection optics 17 may limit the range of beam angles that impinge on the wafer plane 19; the maximum possible angle defines 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 exiting the projection optics 17 that can also impinge on the wafer plane 18.

[0077] 4, mask carrier 48 includes a multilayer 38 in the form of a stack of optical thin films for reflecting electromagnetic waves, the simulated electromagnetic waves being incident on structure plane 30, propagating from structure plane 30 to boundary plane 32 within first section 25 of photolithography mask 14, reflected within multilayer 38 of second section 27 of photolithography mask 14, and propagating from boundary plane 32 to structure plane 30 within first section 25 of photolithography mask 14. In this way, the computer-implemented method for generating aerial image 64 can be applied to reflective photolithography masks, for example, EUV photolithography masks.

[0078] FIG. 4 illustrates the propagation of an incoming electromagnetic wave 22 through a reflective photolithography mask 14, e.g., an EUV photolithography mask. The photolithography mask 14 includes a first section 25 and a second section 27. The first section 25 includes a diffraction grating 24, and the second section 27 includes a mask carrier 48. The diffraction grating 24 includes an absorber structure 26 and a non-absorber structure 28 that form a model 92 on at least a portion of the mask carrier 48 printed on a wafer. The absorber structure 26 is made of one or more materials that absorb the electromagnetic wave 22, such as titanium nitride or tantalum nitride. The non-absorber structure 28 is made of one or more materials that absorb the electromagnetic wave 22 to a lesser extent than the absorber material. For example, the non-absorber structure 28 can include a vacuum. Thus, the absorber structure 26 and the non-absorber structure 28 form an inhomogeneous medium. The absorber structures 26 and non-absorber structures 28 are deposited on a mask carrier 48. The mask carrier 48 includes a multilayer structure 38 in the form of a stack of optical thin films 40 for reflecting the electromagnetic wave 22. The mask carrier 48 may include a capping layer 42 and / or a substrate layer 46. Reflection of the electromagnetic wave 22 by the stack of optical thin films 40 corresponds to reflection of the electromagnetic wave 22 at the effective mirror surface 44. The mask carrier 48 of the photolithography mask 14 is bounded by a boundary plane 32 and a reference plane 34 that is preferably parallel to the boundary plane 32. The boundary plane 32 is the surface plane of the mask carrier 48. The absorber structures 28 in the diffraction grating 24 of the photolithography mask 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 the surface of the absorber structures 26 that faces away from the boundary plane 32. Preferably, the structure plane 30 is parallel to the boundary plane 32.

[0079] The structure plane 30 is a boundary plane through which the electromagnetic wave 22 enters a first section 25, e.g., a diffraction grating 24. The incoming electromagnetic wave 22 impinges on the structure plane 30. The structure plane 30 forms an interface between the mask 14 and the exterior of the photolithography mask 14 through which the electromagnetic wave 22 propagates. The first section 25 of the photolithography mask 14 extends between and is bounded by the structure plane 30 and the boundary plane 32. The second section 27 of the photolithography mask 14 extends between and is bounded by the boundary plane 32 and the reference plane 34.

[0080] In the case of a reflective photolithography mask 14, by way of example, the mask carrier 48 includes a multilayer body 38 in the form of a stack of optical thin films 40 for reflecting the electromagnetic wave 22, the simulated electromagnetic wave 22 being incident on the structure plane 30, propagating from the structure plane 30 to the boundary plane 32 within the first section 25 of the photolithography mask 14, being reflected within the multilayer body 38 of the second section 27 of the photolithography mask 14, and propagating from the boundary plane 32 to the structure plane 30 within the first section 25 of the photolithography mask 14.

[0081] The electromagnetic near-field 20 represents the distribution of electromagnetic waves 22 in a near-field plane 52. The near-field plane may 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 reference plane 34 of the photolithography mask 14. The near-field plane 52 may generally be located anywhere between the structure plane 30 and the wafer plane 18, e.g., at a distance of 0 to 1000 nm from the structure plane 30, preferably at a distance of 0 to 100 nm, more preferably at a distance of 0 to 50 nm, even more preferably at a distance of 0 to 20 nm, and most preferably at a distance of 0 to 10 nm. In a preferred embodiment of the present invention, the near-field plane 52 and the structure plane 30 are the same.

[0082] Known methods for simulating the electromagnetic near field 20 or aerial image 64 often require too much computation time or are not accurate enough.

[0083] To simulate the interaction of electromagnetic wave 22 with photolithographic mask 14, one must consider the propagation of electromagnetic wave 22 within different layers of photolithographic mask 14, which may comprise different materials having different refractive indices.

[0084] To simulate the electromagnetic near-field 20 or 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, FIG. 5a shows the amplitude of the simulated electromagnetic near-field 20 of a photolithography mask 14 using the rigorous coupled-wave analysis (RCWA) method. However, these methods are computationally prohibitively expensive, making them unsuitable for full-chip applications. A full-mask simulation can require multiple years.

[0085] Figure 5b shows the amplitude of the simulated electromagnetic near field 20 for the same photolithography mask 14 as Figure 5a using the thin element approximation (TEA) method. The thin element approximation method is an efficient method for analyzing diffractive optical elements. The TEA method assumes that the thickness of structures on the photolithography mask 14 is very small compared to the wavelength of the incident light and that the width of structures on the photolithography mask 14 is very large compared to the wavelength. However, as photolithographic processes use radiation with increasingly shorter wavelengths and the structures on the photolithography mask 14 become smaller and smaller, the assumptions of the TEA method can break down. In this case, the photolithography mask 14 can no longer be approximated by a planar photolithography mask. Instead, the interaction of radiation at wavelengths less than the height of the structures on the photolithography mask 14 must be considered, leading to the so-called mask 3D effect. Therefore, a method for generating an aerial image 64 of the photolithography mask 14 that is fast and accurate even for short wavelengths is needed.

[0086] To achieve these objectives, a computer-implemented method for generating an aerial image of a model of a photolithographic mask according to an embodiment of the present invention will be described with reference to FIG.

[0087] According to an embodiment, a computer-implemented method 54 for generating an aerial image of a model of a photolithography mask under illumination by incident electromagnetic waves includes: a) in a first section simulation step 56, approximately simulating the propagation of incident electromagnetic waves within a first section of a photolithography mask including a plurality of structures; b) in a second section simulation step 58, analytically or numerically simulating the propagation of the simulated electromagnetic waves from step a) within a second section of the photolithography mask; c) in a near-field generation step 60, simulating a representation of the electromagnetic near field within the model of the photolithography mask by propagating the simulated electromagnetic waves from step b) to a near-field plane; and d) in an aerial image generation step 63, generating an aerial image of the photolithography mask by applying a simulation of the imaging process of a photolithography system or metrology system to the representation of the electromagnetic near field.

[0088] According to an embodiment, the photolithography mask 14 includes a mask carrier 48 and a diffraction grating 24, the diffraction grating 24 including absorber structures 26 and non-absorber structures 28 that form a model 92 or design pattern on at least a portion of the mask carrier 48. The photolithography mask 14 includes a first section 25 extending between a structure plane 30 and a boundary plane 32 of the photolithography mask 14, and a second section 27 extending between the boundary plane 32 and a reference plane 34 of the photolithography mask 14. The first section 25 includes the diffraction grating 24, and the second section 27 includes the mask carrier 48.

[0089] According to an example, simulating the imaging process includes resampling the simulated electromagnetic near-field 20. Thus, the resolution of the simulated aerial image 64 can be increased without significantly increasing the computation time. In this way, an aerial image 64 can be obtained with a resolution comparable to that obtained by a rigorous simulation method.

[0090] The computer-implemented method 54 according to the embodiment may be applied to a transmissive photolithographic mask 14 and a reflective photolithographic mask 14 .

[0091] 7 shows a flowchart of a computer-implemented method 54′ for generating an aerial image 64 of a model of a photolithographic mask 14 for the transmission photolithographic mask 14 shown in FIG. 2. In the computer-implemented method 54′, a simulated electromagnetic wave 22 is incident on the photolithographic mask, e.g., on the reference plane 34, and propagates within a second section 27 of the photolithographic mask, e.g., from the reference plane 34 to the boundary plane 32, in a second section simulation step 58, and propagates within a first section 25 of the photolithographic mask 14, e.g., from the boundary plane 32 to the structure plane 30, in a first section simulation step 56. A representation of the electromagnetic near field 20 of the model of the photolithographic mask 14 in the near-field plane 52 is then obtained in a near-field generation step 60. Finally, an aerial image 64 is generated from the representation of the near field 20 in an aerial image generation step 63 by applying a simulation of the imaging process of a photolithography or metrology system to the representation of the electromagnetic near field.

[0092] 8 shows a flowchart of a computer-implemented method 54″ for generating an aerial image 64 of a model of a photolithography mask 14 for a reflective photolithography mask 14 such as that shown in FIG. 4. In the computer-implemented method 54″, a mask carrier 48 includes a multilayer body 38 in the form of a stack of optical thin films 40 for reflecting electromagnetic waves 22, the simulated electromagnetic waves 22 being incident on the photolithography mask, e.g., on the structure plane 30, propagating within a first section 25 of the photolithography mask 14, e.g., from the structure plane 30 to the boundary plane 32, in a first section simulation step 56, being reflected within the multilayer body 38 of a second section 27 of the photolithography mask 14, in a second section simulation step 58, and propagating within the first section 25 of the photolithography mask 14, e.g., from the boundary plane 32 to the structure plane 30, in a second first section simulation step 56. A representation of the electromagnetic near field 20 of the model of the photolithographic mask 14 at the near-field plane 52 is then 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 a photolithography or metrology system to the representation of the electromagnetic near field in an aerial image generation step 63.

[0093] Instead of solving Maxwell's equations directly in the first section 25, different approximations can be used to reduce the computation time of the method. According to an example, in step a), the propagation of the incident electromagnetic wave in the first section 25 of the photolithographic mask 14 is approximately simulated using the Helmholtz equation, in particular the forward Helmholtz equation.

[0094] In a photolithography setup, the following assumptions can be made: 1) the refractive index is similar for different materials of the photolithography mask 14, e.g., the refractive index of the structures 26, especially the absorber structures, is close to the refractive index outside the structures 26, especially the non-absorbing structures, e.g., a vacuum; 2) the refractive index profile in the first sections 25 is piecewise constant without the need for modeled transitions; 3) the main propagation direction 50 of the incoming electromagnetic wave 22 is approximately perpendicular to the main surface of the photolithography mask, especially the reference plane 34. These assumptions allow for a simplified approximation of the propagation of the electromagnetic wave 22 within the first sections 25.

[0095] Based on Maxwell's equations, the following equation can be derived for the electric field E of the electromagnetic wave 22:

number

[0096]

number

number

[0097] The Helmholtz equation can be further simplified using the following relationship between the magnitude of the wavenumber |k| and its wavelength λ:

number

number

number

[0098]

number

number

[0099] This equation is:

number

[0100] Here, the square root Helmholtz operator is introduced, which is formally defined as a power series. Furthermore, the commutator

number

number

[0101] Ordinary partial differential equations can be solved using multiplication with integral factors.

number

[0102] The exponential operator can be approximated by an integral operator as shown in Appendix A of the doctoral thesis "Efficient wave-optical simulations for the modeling of micro-optical elements" by Soeren Schmidt of the University of Jena. Full reference is made to the aforementioned doctoral thesis, and the disclosure thereof is incorporated into the description of the present invention. From the integral operator approximation,

[0103]

number

[0104] This technique is called the angular spectrum of a plane wave decomposition (ASPW), as shown in equation 1.8 of the aforementioned doctoral thesis. Assume that the electromagnetic wave propagates in a homogeneous medium with a refractive index n. However, this is not true for the first section 25 of the photolithographic mask 14, which contains structures 26 and non-structures.

[0105] Therefore, an extension of the ASPW to inhomogeneous media is required to describe the propagation of the electromagnetic wave 22 within the first section 25 of the photolithographic mask 14.

[0106] To account for inhomogeneous media, the propagation constant at a plane z following a given plane z0 is calculated according to the refractive index profile as described in section 1.4 of the aforementioned doctoral thesis.

number

[0107] Thus, by way of example, the forward Helmholtz equation can be solved using the wave propagation method, which is a generalization of the ASPW to inhomogeneous media,

number

number

number

[0108] In an embodiment, the first section 25 of the photolithography mask 14 includes structures 26 and non-structures 28 that form an inhomogeneous medium, e.g., a diffraction grating 24 includes absorber and non-absorber structures. The simulation of the propagation of the electromagnetic wave 22 within the first section 25 takes into account the inhomogeneity of the material within the first section 25. At the same time, some simplifying assumptions may be utilized in the photolithography setup. Additionally, the simulation of the propagation of the electromagnetic wave 22 within the second section 27 is calculated analytically or numerically. In this manner, an accurate and fast simulation of the propagation of the electromagnetic wave 22 within the photolithography mask 14 is obtained.

[0109] Alternatively, the forward Helmholtz equation can be solved using the beam propagation method, which is described, for example, in chapter 1.3 of the above-mentioned doctoral thesis by Soeren Schmidt, "Efficient wave-optical simulations for the modeling of micro-optical elements."

[0110] In an example, in step a), the propagation of the incident electromagnetic wave within the first section of the photolithography mask is approximately simulated using a machine learning model. The machine learning model may include, for example, a neural network, such as a deep learning model. For example, the machine learning model may include a U-Net or a neural network using at least one attention mechanism, such as a Transformer machine learning model. The machine learning model may use a model of the photolithography mask, such as a design pattern, as input and map the input to an electromagnetic field as output. The machine learning model may be trained, for example, using training data obtained from the simulation described above. By using the machine learning model, after training, a single fast forward pass is sufficient to calculate the propagation of the incident electromagnetic wave, thereby greatly reducing computation time.

[0111] Due to the dependence of the dispersion relation in (3) on the spatial variables (x, y), the wave propagation method in (2) cannot be implemented using the fast Fourier transform (FFT). To use the FFT and reduce the computation time, the wave propagation method in (2) can be reformulated using characteristic functions.

[0112] In an example, the first section 25 of the photolithography mask 14 is decomposed into different materials by defining characteristic functions for each material that indicate the presence of the material in different locations of the first section 25 of the photolithography mask 14, and at least one characteristic function is non-binary.

[0113] The first section 25 of the photolithographic mask 14 is made up of a finite number M of mutually prime, homogeneous regions of refractive index n mThe refractive index profile n(x,y,z) in a given layer z can then be rewritten using the characteristic function for material m:

number

number

number

number

number

[0114] FIG. 9 shows a flowchart of a computer-implemented method 54''' for generating an aerial image 64 of a model of photolithographic mask 14 according to an example, including an additional characteristic function step 61.

[0115] The additional characteristic function step 61 involves creating a model 92 of the photolithography mask 14, e.g., identifying the material of the number M of structures 26 in the first section 25 that form the design pattern, and a subset of the x / y plane at z=z0.

number

number

number

[0116] However, discretizing commonly used binary characteristic functions is problematic. Because binary characteristic functions are discontinuous, the Shannon-Nyquist theorem requires a very high sampling frequency (at least twice the maximum frequency of the signal), thereby requiring a very high-resolution sampling grid. In particular, if the edges of the structure 26 are not aligned with the sampling grid, the sampling will be inaccurate. In addition, the resolution of the sampling grid depends on the size of the smallest feature. A high resolution of the sampling grid also leads to long computation times for generating the aerial image 64.

[0117] Thus, according to an example embodiment, the characteristic function is band-limited. A band-limited characteristic function is one for which there exists a finite frequency ω such that F(ω)=0 if |ω|>ω.

[0118] According to the Shannon-Nyquist theorem, on the one hand, the required sampling frequency of the discretization of a band-limited characteristic function depends on its maximum frequency, and on the other hand, a given sampling frequency of the discretization of a band-limited characteristic function directly implies its maximum frequency.

[0119] By using a band-limited characteristic function, the maximum frequency of the characteristic function can be limited. In this way, according to the Shannon-Nyquist theorem, the required sampling frequency is reduced, and therefore a lower-resolution sampling grid (than in the case of a binary characteristic function) can be used to discretize the characteristic function. In this way, the computation time required to generate the aerial image 64 can be reduced. In addition, the resolution of the sampling grid is independent of the feature size of the features in the model of the photolithography mask, e.g., the design pattern. In contrast, for a binary characteristic function, the sampling grid resolution depends on the smallest feature in the model of the photolithography mask.

[0120] The reason for using a discretized band-limited characteristic function is given below: If we assume that the electromagnetic field E contains only energy at long wavelengths in the x / y plane perpendicular to the reference plane 34 of the photolithographic mask 14, then a linear space-invariant low-pass filter P will have no effect when applied to the electromagnetic field E, i.e., P(E)≈E.

[0121] Equivalently, P can be written as a convolution in the time domain, which is P(E)=∫p(t')E(t-t')dt' ≒∫p(t')E(t)dt' =E(t)∫p(t')dt' =E This suggests that:

[0122] If a filter P is applied to the product of E and a function Θ that has energy at shorter wavelengths, then: P(E·Θ)=∫p(t')E(t-t')Θ(t-t')dt' ≒∫p(t')E(t)Θ(t-t')dt' =E(t)∫p(t')Θ(t-t')dt' =E P(Θ)

[0123] Thus, if a low pass filter is applied to the product of a slowly varying function E and a fast varying function Θ, the result is approximately the product of the slowly varying function E and a filtered fast varying function P(Θ).

[0124] Applying this result to the propagation function of the wave propagation method (4),

number

number

number

number

[0125] Therefore, the propagation function of the low frequency part of the field E in the wave propagation method of (4) is obtained by applying the filter P to the characteristic function.

[0126] FIG. 10a illustrates the effect of sampling grid resolution on the approximation error 66 during sampling of a binary characteristic function 62. Each column shows a different sample spacing, ranging from 30.18 nm to 2 nm. In the top row, sampled binary characteristic functions 62 for a model 92 of a photolithography mask 14, e.g., a design pattern, are shown for different sample spacings (pixel size in nm). Using the binary characteristic functions 62 shown, an aerial image 64 is generated by a computer-implemented method 54''' for generating an aerial image 64 of a model of a photolithography mask 14, including the characteristic function step 61 described above. In the middle row, a generated aerial image 64 corresponding to each binary characteristic function 62 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 precisely simulated aerial image 64 for the same underlying model 92 of a photolithography mask 14, e.g., a design pattern. From the results it can be concluded that a sample spacing of less than 2 nm is required for an approximation error 66 of less than 1% for a binary characteristic function 62. Therefore, a high-resolution sampling grid is required for a binary characteristic function 62, which leads to long calculation times.

[0127] FIG. 10b illustrates the effect of sampling grid resolution on the approximation error 66 during sampling of the band-limited characteristic function 68. Each column shows a different sample spacing, ranging from 30.18 nm to 2 nm. The top row shows sampled band-limited characteristic functions 68 for a model 92 of a photolithography mask 14, e.g., a design pattern, for different sample spacings (pixel size in nm). Using the illustrated band-limited characteristic functions 68, an aerial image 64 is generated by a computer-implemented method 54''' for generating an aerial image 64 of a model of a photolithography mask 14, including the characteristic function step 61 described above. The middle row shows the generated aerial image 64 corresponding to each band-limited characteristic function 68 in the top row. The bottom row shows the approximation error 66 as the difference between the generated aerial image 64 and a precisely simulated aerial image 64 for the same underlying model 92 of a photolithography mask 14, e.g., a design pattern. From the results, it can be concluded that a sample spacing of less than 8 nm is required for an approximation error of less than 1% for the band-limited characteristic function 68. Therefore, even a coarse resolution is sufficient to obtain an accurate aerial image 64 with reduced computation time for the band-limited characteristic function 68.

[0128] FIG. 11 shows a comparison of the convergence rates of the wave propagation method of Equation (4) implemented using a discretized binary characteristic function 62 or a discretized band-limited characteristic function 68. On the upper horizontal axis 78, the sample spacing is shown. On the lower horizontal axis 80, the corresponding number of pixels in one dimension of the sampling grid is shown. On the vertical axis 82, the approximation error 66 is shown. The plot shows the root-mean-square error 70 and maximum error 72 of the binary characteristic function 62, and the root-mean-square error 74 and maximum error 76 of the band-limited characteristic function 68. From the plot, it can be concluded that the convergence rate of the band-limited characteristic function 68 is exponential compared to the convergence rate of the binary characteristic function 62.

[0129] By generalizing the concept of the characteristic function to a binary characteristic function, the characteristics of the sub-pixel design can be resolved and a speedup of about 100 times can be achieved.

[0130] Besides the band-limited characteristic function 68, it may also be advantageous to use other non-binary characteristic functions to describe the presence of particular materials at different locations (x,y)∈X×Y of the photolithographic mask 14 at z=z0.

[0131] For example, it may be advantageous to use continuous or complex-valued characteristic functions, in which case the material distribution within the photolithographic mask may be described in a more flexible way, leading to a more accurate approximation.

[0132] According to an example embodiment, the value range D of the at least one characteristic function is

number

[0133] According to an example embodiment, the characteristic function forms an affine combination at each location of the photolithographic mask at z=z0.

number

[0134] According to an example embodiment, obtaining the characteristic function includes decomposing a model 92 of the photolithography mask 14, e.g., a design pattern, into elements 94 (e.g., using a mathematical function describing the contour or area of ​​the structure 26, such as a polygon, spline, or curve element), representing the elements 94 by a characteristic function, particularly a binary characteristic function, and applying a low-pass filter to the characteristic function. The elements 94 may be represented, for example, by a characteristic function that takes a non-zero value, e.g., 1, inside the element 94 and 0 outside the element 94. For example, each element 94 may be decomposed into one or more triangles, and the triangles may be represented by a characteristic function. The Fourier transform of the polygon may be obtained as described in Appendix A of the doctoral thesis "Photolithography Simulation by Heinrich Kirchauer at the Technical University of Wien." Reference is made in full to the aforementioned doctoral thesis, the disclosure of which is incorporated herein by reference. By applying a low-pass filter to the characteristic function, a band-limited characteristic function 68 is obtained. Therefore, the wave propagation method in (4) can be simulated using the coarse sampling grid mentioned above, thereby reducing the computation time.

[0135] In particular, applying a low-pass filter to the characteristic function can include applying a spatial analytical Fourier transform to the characteristic function, followed by an inverse fast Fourier transform. The analytical Fourier transform can be calculated only for spatial frequencies in the discretized domain of the inverse FFT. This subsampling of the spatial domain limits the maximum frequency of the characteristic function in accordance with the Shannon-Nyquist theorem. Thus, the discretization corresponds to a low-pass filtering of the characteristic function. The result is a band-limited characteristic function representing a model of the photolithography mask, e.g., a design pattern, which can be discretized using a sampling grid with significantly lower resolution than the binary characteristic function, thereby reducing computation time.

[0136] By way of example, the analytical Fourier transform used in the wave propagation method of equation (4) is approximated by a fast Fourier transform (FFT). In this way, the computation time is reduced.

[0137] The FFT suggests periodic boundary conditions. However, due to the arbitrary angle of the incident electromagnetic wave, this assumption no longer holds. This inaccuracy is often ignored by approximation methods. Even if the mask model 92, e.g., the design pattern, is assumed to be periodic, an arbitrary illumination angle of the incident electromagnetic wave 22, e.g., with respect to the normal 84 of the structure plane 30, suggests that the solution of equation (4) is merely quasi-periodic according to Floquet's theorem, which means that it is periodic with an additional phase shift α. E(x+nΔx)=E(x)exp inα Thus, by way of example, the wave propagation method takes into account the angle φ of the incident electromagnetic wave 22, e.g., the angle with respect to the normal 84 of the structure plane 30, by assuming quasi-periodic boundary conditions in the propagator step of equation (4) at one or more pairs of opposing boundaries in the x / y plane that are perpendicular to the reference plane 34 of the photolithographic mask 14. Assuming quasi-periodic boundary conditions improves the accuracy of the simulated electromagnetic near field.

[0138] Let E(x,y,z0) be quasi-periodic in the x and y coordinates. Then, by Floquet's theorem, E can be expressed as a non-periodic phase shift α=(α x ,α y ) can be rewritten as the part E' periodic in x and y multiplied by

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[0139] this is,

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[0140] From this it can be concluded that the phase shift α of an input field that is linear in the x and y coordinates can be accommodated by reformulating the dispersion relation of equation (3) as follows:

number

[0141] In particular, the dispersion relation of the electromagnetic wave 22 in the first section 25 depends on the angle φ of the incident electromagnetic wave 22 .

[0142] 12 shows the dependence of the phase shift vector α on the angle φ of the incoming electromagnetic wave 22. The angle φ can be measured relative to the normal 84 of the structure plane z0. The electromagnetic wave 22 propagates in the direction of the wave vector 86. Let x0 and x1 refer to the boundaries of the unit cell in the x-direction, which is the smallest aperiodic subset of the periodic model 92, e.g., the design pattern.

[0143]

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[0144] In the case of a reflective photolithographic mask 14, analytically or numerically simulating the propagation of the simulated electromagnetic wave 22 from step a) within the second section 27 of the photolithographic mask 14 includes 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 body 38.

[0145] Thus, by way of example, simulating the reflection of the simulated electromagnetic wave 22 from step a) within the multilayer body 38 comprises an analytical calculation of a reflection coefficient at the boundary between the second section 27 and the first section 25 of the photolithographic mask 14, e.g., at the boundary plane 32, which reflection coefficient describes the propagation of the electromagnetic wave 22 within the stack of optical thin films 40 of the multilayer body 38. The propagation within the stack of optical thin films 40 of the multilayer body 38 corresponds to a reflection at an effective mirror surface 44 that is at a particular distance from the boundary plane 32.

[0146] In particular, the reflection coefficient at boundary 32 may be calculated separately within and outside structures 26 of first section 25 of photolithographic mask 14. For example, the reflection coefficient may be calculated separately for the respective media of absorbing and non-absorbing structures of diffraction grating 24 at boundary plane 32. In this way, the accuracy of generated aerial image 64 is improved.

[0147] In an example, simulating the propagation of the simulated electromagnetic wave 22 within the second section 27 of the photolithographic mask 14 includes applying a reflection coefficient to the electromagnetic wave 22 incident on the boundary 32.

[0148] In particular, simulating the reflection of the electromagnetic wave 22 within the multilayer body 38 involves the phase term in (4)

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[0149] The analytical reflection coefficient r for each of the N optical thin films 40 in the multilayer 38 is given by equations 33-41 in the paper by S. Kim, Y. Kim, and I. Park, "Optical properties of a thin-film stack illuminated by a focused field," Journal of the Optical Society of America A, Vol. 17, No. 8, August 2000. m can be calculated for s-polarized and p-polarized waves as follows:

[0150]

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[0151] where ε0 denotes the vacuum permittivity, μ0 denotes the vacuum permeability, and n j+1 represents the refractive index of the j+1th optical thin film 40, and d j+1 denotes the thickness of the j+1th optical thin film 40. Full reference is made to the aforementioned paper, the contents of which are included in the description of the present invention.

[0152] In another example, the reflection of an electromagnetic wave by the multilayer body 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 the paper "Domain Decomposition Method for Maxwell's Equations: Scattering off Periodic Structures," by Achim Schadle, Lin Zschiedrich, Sven Burger, Roland Klose, and Frank Schmidt, in arXiv:math / 0602179v1). In a third step, the superposition of the reflected Fourier modes results in the reflected electromagnetic wave. Alternatively, a machine learning model can be trained to numerically simulate the propagation of the electromagnetic wave within the second section of the photolithography mask.

[0153] 13a-13d illustrate 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 an embodiment of the present invention. A model 92 of the photolithographic mask 14 includes elements 94 composed of polygons in the form of rectangles, as shown in FIG. 13a. In a characteristic function step 61, the elements 94 are represented by a characteristic function obtained by any of the methods described above, for example, by a binary characteristic function 62. For example, the elements 94 are represented by a binary characteristic function 62 having values ​​1 inside the elements 94 and values ​​0 outside. A spatial analytic Fourier transform is then applied to the characteristic function, followed by an inverse FFT for the inverse transform, resulting in a band-limited characteristic function 68. Here, the analytic Fourier transform is calculated only for spatial frequencies in the discretized domain of the inverse FFT. This subsampling of the spatial domain limits the maximum frequency of the characteristic function according to the Shannon-Nyquist theorem. Thus, the discretization corresponds to a low-pass filtering of the characteristic function. The result is a band-limited, discretized representation of model 92 of photolithography mask 14, i.e., band-limited characteristic function 68 sampled on the low-resolution sampling grid shown in FIG. 13b. Based on band-limited characteristic function 68, a representation of electromagnetic near-field 20 in the form of its amplitude is shown in FIG. 13c, which is simulated by propagating a simulated electromagnetic wave into the near-field plane. Finally, aerial image 64, shown in FIG. 13d, is calculated by applying a simulation of the imaging process of photolithography system 10, 10′ within projection section 19 between near-field plane 52 and wafer plane 18 to the representation of electromagnetic near-field 20. The imaging process may include resampling of electromagnetic near-field 20 to a higher-resolution grid. By calculating aerial image 64 by applying characteristic function step 61 and aerial image generation step 63, an accurate aerial image 64 can be simulated for model 92 of photolithography mask 14 with low computation time due to the low resolution of the sampling grid.Therefore, the computational time to obtain the aerial image 64 is reduced compared to simulating the aerial image 64 by applying a rigorous simulation method (such as RCWA) to a model 92 of the photolithographic mask 14, e.g., a design pattern, in a rigorous simulation step 95 that requires a high-resolution sampling grid.

[0154] FIGS. 14a-14c show a comparison of aerial images 64 of a model of photolithographic mask 14 obtained by three different simulation methods. All three figures show the intensity distribution of the obtained aerial images 64. FIG. 14a shows an aerial image 64 simulated using the TEA method described above. FIG. 14b shows an aerial image 64 generated using the computer-implemented method for generating an aerial image 64 of a model of photolithographic mask 14 described above. FIG. 14c shows an aerial image 64 simulated by a rigorous simulation method, the RCWA method described above. The results show that the computer-implemented methods 54, 54', 54'', and 54''' for generating an aerial image 64 of a model of photolithographic mask 14 approximate the true aerial image 64 with a very low error rate and are therefore more accurate than the TEA method.

[0155] The computation time of the above-described method can be further reduced using acceleration methods known to those skilled in the art, such as Graphics Processing Units (GPUs), distributed GPUs, Field Programmable Gate Arrays (FPGAs), etc.

[0156] The computation time for the TEA method in FIG. 14a is 0.4 seconds. The computation time for the computer-implemented method 54, 54', 54'', and 54''' for generating the aerial image 64 in FIG. 14b is 0.01 to 0.4 seconds, depending on the grid structure. When 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, typically, the model 92, e.g., the design pattern, is provided with 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 approximately 0.4 seconds of computation time. The computation time for the rigorous simulation method, the RCWA method in FIG. 14c, is 10 to 1000 seconds, depending on the minimum feature size, e.g., the defect size. Thus, the computer-implemented method according to an embodiment of the present invention provides an approximation of the aerial image 64 with a very low error rate in a computation time comparable to common aerial image approximation methods. Compared to rigorous simulation methods, methods according to embodiments of the present invention are approximately 2-4 orders of magnitude faster with error levels of only a few percent.

[0157] For many applications, a comparison of the generated aerial image with a reference aerial image is performed. The reference aerial image may, for example, be an acquired aerial image (so-called die-to-die mode) or a simulated aerial image (so-called die-to-database mode). To obtain meaningful results from such a comparison, the generated aerial image must be highly accurate and accurately reproduce the reference aerial image, e.g., the reference aerial image acquisition conditions. To this end, calibration of the generated aerial image is advantageous.

[0158] In the example shown in FIG. 15 , defects 93 are detected by comparing an acquired aerial image 83, here acquired using several metrology systems 85, with a simulated aerial image 64. The acquired aerial image 83 includes defects 93. The simulated aerial image 64 is generated using the method 54, 54′, 54″, 54′″ for generating an aerial image 64 of a model 92 of a photolithography mask according to an embodiment, implementation, or aspect of the invention described above. In this manner, the simulated aerial image 64 is generated accurately and with low computational time. The acquired aerial image 83 is compared with the simulated aerial image 64, for example, by image subtraction, thereby generating a comparison result 90. From the comparison result 90, a defect map 91 is generated indicating the locations of the defects 93, for example, by thresholding, adaptive thresholding, or applying machine learning methods for defect detection.

[0159] In order for the acquired aerial image 83 acquired by the metrology system 85 to be comparable to the aerial image 64 simulated using the method 54, 54′, 54″, 54′″ in accordance with the present invention, it is important to consider the characteristics of the metrology system 85 and the photolithographic mask when generating the aerial image 64 from the model 92 of the photolithographic mask. Characteristics of the metrology system 85 to consider include, for example, illumination settings, imaging settings, sensor model, motion blur due to scanning, field-dependent effects, defocus, distortion, aberrations, apodization, etc. Characteristics of the mask to consider include, for example, bias, corner rounding, sidewall angle, layer height, refractive index, etc., which are described in more detail above. To account for these properties, the method for generating the aerial image of the model of the photolithographic mask may include adjusting the mask parameters and / or optical parameters described above.

[0160] Thus, in a preferred embodiment, the method 54, 54', 54", 54'" of generating an aerial image 64 from a model 92 of a photolithographic mask further includes adjusting at least one parameter of the method to minimize differences between one or more reference aerial images 88 of the one or more photolithographic masks and the corresponding generated aerial images 64 of the corresponding models 92 of the one or more photolithographic masks, the at least one parameter being from a group including mask parameters and optical parameters. These parameter adjustments may be performed in a calibration step 89 shown in detail in FIG. 16.

[0161] As shown in FIG. 16 , calibration step 89 involves adjusting parameter p of method 54 for generating an aerial image of a model of a photolithographic mask by minimizing the difference between one or more reference aerial images 88 and corresponding generated aerial images 64. One or more reference aerial images 88 may include, for example, acquired aerial images acquired by metrology system 85 of one or more photolithographic masks. Corresponding generated aerial images 64 may be generated, for example, using an embodiment of the method for generating an aerial image using one or more corresponding models 92 of the photolithographic mask. Other methods for generating an aerial image from a model of a photolithographic mask may also be used. However, the method according to the present invention has the advantage of being fast and accurate, allowing for the calculation of gradients that greatly simplify the optimization of method parameters. One or more reference images 88 may include acquired aerial images of model 92 of a photolithographic mask to be inspected for defects 93.

[0162] The parameters p of the method 54, including mask parameters and / or optical parameters, may be optimized in an iterative method 79, including, for example, single or multiple iterations as shown in Figure 16. To this end, the optimization problem may be solved as follows: p opt =argmin p χ[I acq ,Igen (p),p] p opt is the acquired aerial image I acq and the generated aerial image I gen is a target parameter vector that minimizes the difference criterion χ between p and p. χ is an objective or loss function that defines the optimality condition. It may be linked to a noise model of the acquired aerial image; for example, in the case of Gaussian iid noise, the L2 norm may be used. The objective function may also include an additional regularization term as a function of p to obtain a particularly well-posed objective function.

[0163] Partially coherent imaging and the resulting aerial image I generated using the incoming electromagnetic near-field corresponding to different illumination angles in step d) of methods such as the Hopkins method, the Abbe method and the local Hopkins method. gen Different techniques are known for calculating the intensity of

[0164] The Hopkins method relies on the observation that for small changes in the angle of incidence of a light wave, only very small deviations in the intensity, phase, and polarization of the light wave can be expected. Therefore, a change in the illumination angle results approximately only in a frequency shift in the respective diffraction spectrum of the photolithographic mask. Therefore, the incoming electromagnetic near field E in The same mask spectrum F{E in (x,y)} is used for all illumination angles with a shift due to the illumination angle.

[0165]

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[0166] This approach is simple and fast. For simulations using thin masks or Kirchhoff methods such as TEA, this assumption is always met. However, when the thickness of structures on photolithographic masks can no longer be ignored and a rigorous electromagnetic field simulation of mask diffraction for various illumination angles is required, the Hopkins method is not accurate enough.

[0167] In this case, the Abbe method uses the illumination angle-dependent diffraction spectrum F{E in,i}, i=1,...N Abbe Since we assume that .gtoreq..times ...

number

[0168] However, because the electromagnetic near field must be simulated for every single illumination angle, the Abbe method is very computationally expensive and may therefore not be suitable for use in, for example, full-chip optical proximity correction (OPC) simulation or verification systems.

[0169] To obtain a fast and accurate simulation method for the aerial image of a photolithographic mask, a local Hopkins method, such as that disclosed in U.S. Patent Application Publication No. 2007 / 0253637, can be used. The local Hopkins method is a combination of the Hopkins method and the Abbe method, which is based on the local assumption of a constant diffraction spectrum of the photolithographic mask. For this purpose, the source map is divided into several segments. For each segment, the diffraction spectrum is assumed to be constant, thereby simulating only a single diffraction spectrum per segment. Therefore, using the local Hopkins method, a small number of spectra F{E} for a selected subset of illumination angles are obtained. in,j}, j=1,...N Hop , N Hop < <N Abbe is simulated. For the remaining illumination angles, the simulated spectrum is shifted according to the illumination angle.

[0170]

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[0171] The local Hopkins method requires careful selection of segments and illumination angles within the segments, for example, as described in US Patent Application Publication No. 2007 / 0253637, to simulate a diffraction spectrum.

[0172] The Hopkins, Abbe, or local Hopkins method allows for the calculation of a gradient with respect to the parameter vector p. Thus, by using one of these techniques to calculate the aerial image Igen generated from the electromagnetic near field in the above objective function χ, the parameter p can be optimized in an iterative manner, for example, by gradient descent. Note that optimizing optical parameters (e.g., Zernike aberrations) that can be modeled by changes in the pupil function, as opposed to mask parameters (e.g., bias, corner rounding, etc.), simply requires a re-evaluation of the Hopkins, Abbe, or local Hopkins method without requiring a new simulation of the electromagnetic near field, thereby simplifying and speeding up the optimization of the parameter vector p.

[0173] FIG. 17 illustrates the benefits of adjusting mask and optical parameters of a method for generating an aerial image of a model of a photolithographic mask. In the center of the top row, an aerial image 83 acquired for a given photolithographic mask is shown. The acquired aerial image 83 may be an aerial image acquired using a metrology system. On the left side of the top row, a generated aerial image without calibration 87 is shown, which is obtained by applying a method for generating an aerial image of a model of a photolithographic mask according to the present invention to the photolithographic mask model without performing a calibration step 89. On the right side of the top row, a generated aerial image 64 including calibration is shown, which is obtained by applying a method for generating an aerial image of a model of a photolithographic mask according to the present invention to the photolithographic mask model and performing a calibration step 89. In the bottom row, a comparison result 90, i.e., a difference image, is presented, which clearly illustrates the benefit of the additional calibration step 89.

[0174] For mask analysis, metrology systems often compare the positions of features in a reference aerial image, e.g., an acquired aerial image, with the corresponding positions in the generated aerial image. For this purpose, accuracy of the reference aerial image, e.g., subpixel accuracy of less than about 1 nm, is required. However, due to the low-pass filtering effect of the numerical aperture and mask 3D effects or optical proximity effects that lead to displacement of features in the aerial image, it is not possible to directly estimate the structure positions from the reference aerial image. In addition, the reference aerial image of a metrology system often suffers from aberrations and additional image error sources, such as distortion, apodization, and noise (e.g., shot noise, readout noise).

[0175] Thus, according to the example shown in FIG. 18 , the method for generating an aerial image of a model 92 of a photolithographic mask further includes registering 99 one or more acquired aerial images 83 of the photolithographic mask, e.g., the acquired aerial images, to a corresponding generated aerial image 64 of the model 92 of the photolithographic mask and reporting at least one registration parameter. Registration 99 may also be understood as or as part of calibration step 89. In this manner, structural displacements are reduced and defect detection is improved. In FIG. 18 , acquired aerial image 83 on the left is an acquired aerial image of a photolithographic mask using a metrology system. The acquired aerial image is registered 99 to generated aerial image 64 generated from the underlying model 92 of the photolithographic mask. Registration results 97 are shown on the right.

[0176] Various registration methods can be used to register aerial images. For example, the (subpixel) shift of one of the aerial images can be optimized by minimizing an error norm, such as the L2 norm or the Huber loss, or by maximizing a similarity measure, such as the cross-correlation of the aerial images. Note that the subpixel shift for a Nyquist-sampled, simulated (noise-free) reference image is accurate when using sinc interpolation (Fourier shift) and ignoring boundary effects. Nonlinear registration methods can also be used. For example, continuous optimization methods, such as the calculus of variations, can be used for registration. Additional constraints can be imposed on the registration results. Different registration methods with subpixel accuracy are described, for example, in "Efficient subpixel image registration algorithms," Manuel Guizar-Sicairos, Samuel T. Thurman, and James R. Fienup, Opt. Lett. 33, pp. 156-158, 2008.

[0177] Besides registration parameters, critical dimension (CD) parameters can also be estimated by optimization methods. CD can be a global parameter (constant across the entire model) or a local parameter (locally varying across the model) of the mask model and a loss function.

[0178] According to an example, the one or more acquired aerial images 83 include a focus stack of the photolithography mask. The focus stack includes multiple aerial images of the same measurement site acquired at different focus settings. The focus stack can be used in various 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 photolithography mask during registration. Alternatively, all aerial images in the focus stack can be considered during registration, for example, by calculating the norm of the complete difference image stack during optimization.

[0179] In an example, calibration step 89 includes both optimization 81 of mask and optical parameters and registration 99. In one example shown in FIG. 19, optimization 81 may be performed before registration 99. An aerial image 64 is then generated using the optimized mask and optical parameters. In another example shown in FIG. 20, mask parameters and / or optical parameters and / or registration parameters are jointly optimized in an iterative 79 manner using single or multiple iterations. In each optimization step, mask and optical parameters of the generated aerial image 64 are optimized 81, followed by registration 99 of the optimized aerial image to the acquired aerial image 83. Finally, the optimized mask and optical parameters p opt , as well as the complete registration, are reported. Optimizing the parameters in a collaborative manner yields more accurate results. For example, parameters are optimized based on the most recent measurements, and in this way parameter drift is taken into account.

[0180] The generated aerial images of models of photolithographic masks obtained according to embodiments, examples or aspects of the present invention as described above may be advantageously used to train machine learning models.

[0181] For example, in the case of an EUV photolithography mask, the 7 λ×10 7Due to the large size of a photolithography mask compared to the wavelength of λ, generating an aerial image of a complete photolithography mask is very time-consuming, even using the method according to the present invention described above. Therefore, according to an embodiment of the present invention, a machine learning model is trained to mimic the mapping from the model of the photolithography mask to the aerial image. The machine learning model is even faster than the method according to the present invention. In the example shown in FIG. 21 , a computer-implemented method for training a machine learning model 128 that maps a photolithography mask model 92 to an aerial image 64 of the photolithography mask includes generating aerial images 64 of multiple photolithography mask models 92 using the method for generating an aerial image of a model of a photolithography mask 54 according to any one of the embodiments, examples, or aspects described above, and training the machine learning model 128 using training data including the generated aerial images 64. By using the method for generating an aerial image of a model of a photolithography mask 54 according to any one of the embodiments, examples, or aspects described above, aerial images 64 for various photolithography mask models 92 can be generated accurately and quickly. Using the model 92 and the generated aerial image 64 as training data, a machine learning model 128 can be efficiently trained to mimic the method of generating an aerial image of a model of a photolithographic mask. The trained machine learning model 128 can then be used to rapidly generate an aerial image 64 of the model 92 of a photolithographic mask. The machine learning model can include, for example, a neural network, e.g., a deep learning model. Because only a single forward pass is required during inference, the trained machine learning model 128 requires very little computation time. FIG. 22 shows a sample model 92 of a photolithographic mask that can be used to generate training data. In addition to the generated training data, other training data, such as acquired aerial images or simulated aerial images obtained by a different method, can be used. For example, after pre-training on the generated training data, additional training data can be considered using transfer learning.

[0182] Machine learning models are also particularly promising for other photolithography applications, such as defect detection in photolithography masks, due to their high-quality results and inference speed. However, these methods require large amounts of realistic, annotated training data, which is typically not available based on acquired aerial images.

[0183] Thus, the method for generating an aerial image of a model of a photolithography mask according to the above-described embodiments, examples, or aspects of the invention may also be used to help train machine learning models for other applications, such as for defect detection as shown in FIG. 23. The machine learning model for defect detection 136 receives as input the reference aerial image 132 and the acquired aerial image 134, which are mapped to a defect map 138 with marked defects 93 as output. The reference aerial image 132 may be, for example, an acquired aerial image (die-to-die defect detection) or a simulated aerial image (die-to-database defect detection).

[0184] As shown in FIG. 24, a computer-implemented method for training a machine learning model 136 for defect detection in acquired aerial images 134 of photolithography masks includes generating model pairs 140 for a plurality of photolithography masks, each model pair 140 including a defect-free model 142 of the photolithography mask, i.e., a model without defects, and a defect model 144 of the same photolithography mask, i.e., a model including one or more defects 93; generating aerial image pairs 146 including a defect-free aerial image 148 and a defect aerial image 150 of the photolithography mask from the model pairs 140 by applying a method for generating aerial images of models of photolithography masks according to the above-described 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 including the aerial image pairs 146.

[0185] Noise 152 may be added to aerial image pair 146, thereby generating noisy aerial image pair 154, which includes a noisy defect-free aerial image 156 and a noisy defect aerial image 158. The noise may include all randomly varying parameter realizations (e.g., shot noise, read noise, aberrations, line edge roughness, etc.). By adding noise to aerial image pair 146, the training data becomes more realistic, especially for die-to-die defect detection, thereby improving the training results.

[0186] In addition to the generated training data, other training data may be used, e.g., acquired or simulated aerial images obtained by different methods, e.g., defect-free acquired or simulated aerial images. The additional training data may be considered using, e.g., transfer learning after pre-training on the generated training data.

[0187] The generated aerial image of the model of the photolithographic mask can be used in different ways.

[0188] For example, based on the accurate aerial image, a model of the corresponding photolithography mask, e.g., a design pattern, may be improved, and the mask 3D effect may be mitigated, for example, by modifying the model of the photolithography mask. Alternatively, the material or thickness in the first section, for example, the absorber material and / or absorber thickness in the diffraction grating of the photolithography mask, may be modified. Alternatively, optical proximity correction techniques may be applied to the photolithography mask, for example, by adding sub-resolution assist functions.

[0189] For example, the generated aerial image may be used for defect detection. Given an acquired or simulated aerial image of a photolithography mask, the photolithography mask may be checked for defects by generating an aerial image of the photolithography mask based on a model of the photolithography mask and comparing the acquired or simulated aerial image with the generated aerial image.

[0190] For example, the suitability of defects in an acquired charged particle beam image of a photolithography mask can be assessed by generating an aerial image using the acquired charged particle beam image of the photolithography mask as a model of the photolithography mask, and 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 photolithography mask.

[0191] For example, the generated aerial image may be used to generate a digital twin of a machine that uses the acquired aerial image of a photolithography mask. The digital twin of a machine is a digital simulation of a machine that uses a method for generating an aerial image of a model of a photolithography mask to simulate the acquisition of the aerial image within the machine. The digital twin of a machine may be used for many different purposes, such as to specify the functionality and requirements of a machine, to present the functionality of a machine to a customer before the machine is assembled or shipped, or to accelerate the development of parts of a machine, such as a user interface.

[0192] In these applications, instead of acquiring an aerial image of the photolithographic mask, a generated aerial image of a model of the photolithographic mask is used, thereby significantly reducing computation time.

[0193] According to the embodiment shown in FIG. 25 , a computer-implemented method 103 for generating an aerial image 64 of a model 92 of a photolithography mask 14, e.g., for improving a design pattern, repairing the photolithography mask 14, determining the quality of the photolithography mask 14, performing measurements on the photolithography mask 14, detecting or assessing the suitability of defects in the photolithography mask 14, or selecting illumination settings for a photolithography system, includes generating an aerial image 64 of the model of the photolithography mask 14 using the method 54 for generating an aerial image of a model of a photolithography mask according to any of the embodiments, examples, or aspects described above, appropriately analyzing the generated aerial image 64 in an analysis step 105, and using the analysis results in an application step 107 to improve the design of the photolithography mask 14, repairing the photolithography mask 14, determining the quality of the photolithography mask 14, detecting or assessing the suitability of defects in the photolithography mask 14, or selecting illumination settings for a photolithography system. The method may use further information of the photolithographic mask acquired in further information step 101, for example an acquired aerial image, which may be acquired using some aerial image acquisition system or metrology system or by simulation, or an SEM image of the photolithographic mask, information about layer dimensions, type of mask, etc. The method may apply iterations of the method steps mentioned above, for example source mask optimization, inverse photolithography, or selection of illumination settings. Iterative optimization, for example gradient-based optimization, of mask or illumination parameters may be performed, requiring iterative generation of aerial images and parameter adjustment.

[0194] To improve the model of the photolithography mask, the analysis step 105 may include comparing the generated aerial image 64 of the model of the photolithography mask with an acquired aerial image of the photolithography mask, e.g., an acquired aerial image or a simulated aerial image (e.g., a simulated target aerial image), to, for example, detect defects or measure critical dimensions. Based on the analysis results, the model of the photolithography mask may be modified, e.g., within the context of source mask optimization or inverse photolithography. Based on the analysis results, a repair shape indicating the locations of defects and their corrections may be generated and used to repair the photolithography mask. Based on the analysis results, the quality of the photolithography mask may be determined, for example, by using several types of quality measures, such as the number of defects, the average number of defects per area, the maximum severity of the defects, the types of defects and their frequency, etc. Based on the analysis results, defects may be detected. Based on the analysis results, the suitability of the defects can be assessed by checking whether the defects detected in the generated aerial image of the photolithography mask are actually printed in the acquired aerial image of the photolithography mask.

[0195] Using the acquired aerial image for comparison with the generated aerial image in analysis step 105 is optional. For example, defects may also be detected by analyzing only the generated aerial image. Measurements of the photolithography mask may be performed using only the generated aerial image. Illumination settings for the photolithography system may be selected by generating different aerial images using different illumination settings in analysis step 105 and selecting the illumination settings that result in the highest quality aerial image.

[0196] The analyzing step 105 may be performed once or iteratively, for example, during iterative optimization. For example, during source mask optimization, a photolithographic mask may be optimized by iteratively generating an aerial image of a current model of the photolithographic mask and modifying the model based on the aerial image.

[0197] 26 illustrates a computer-implemented method 96 for detecting defects in a photolithography mask 14, which includes obtaining an aerial image of the photolithography mask 14 in an aerial image step 98, generating an aerial image 64 of a model of the photolithography mask 14 using a computer-implemented method 54, 54′, 54″, 54′″ according to an embodiment of the present invention, and detecting defects in the photolithography mask 14 in a defect detection step 100 by comparing the acquired aerial image with the generated aerial image 64. The aerial image of the photolithography mask may be obtained, for example, by acquiring an aerial image of the photolithography mask using an acquisition tool or by simulating or generating an aerial image using several methods, such as those described above.

[0198] FIG. 27 illustrates a computer-implemented method 102 for assessing suitability of defects in a photolithography mask 14 according to an embodiment of the present invention, the computer-implemented method 102 including providing a charged particle beam image of a photolithography mask including one or more defects in an imaging step 104, generating an aerial image 64 of a model of the photolithography mask using a computer-implemented method 54, 54′, 54″, 54′″, where the charged particle beam image is used as a model of the photolithography mask 14, and assessing suitability of one or more defects in the photolithography mask 14 using the generated aerial image 64 in an assessment step 106. A defect is assessed as suitable if it prints on a wafer during the printing process. In contrast, a defect that does not print on the wafer is assessed as unsuitable. The charged particle beam image is acquired by a charged particle beam device, such as a helium ion microscope (HIM), a cross-beam device including a FIB and a SEM, or any charged particle imaging device. The assessment step 106 may include a comparison of the generated aerial image 64 with the charged particle beam image. For example, one or more locations of one or more defects in the charged particle beam image may be compared with one or more corresponding locations in the generated aerial image 64. If a defect is not visible in the generated aerial image 64, it may be concluded that it was not printed on the wafer and is therefore not compatible. If a defect is visible in the generated aerial image 64, it may be concluded that it was printed on the wafer and is therefore compatible. The generated aerial image 64 may also be compared to a reference image, such as another generated, simulated, or acquired aerial image 64 of the photolithography mask 14, to assess the compatibility of one or more defects. For example, if the generated aerial image 64 is very similar to the reference image of the locations of the defects, the defects may be assessed as not compatible. If the simulated aerial image 64 differs from the reference image of the locations of the defects, the defects may be assessed as not compatible. The assessment step 106 may additionally or alternatively include a calculation of critical dimensions (CDs). The calculated CD can be compared to a predefined CD.For example, if the calculated CD is lower than a predefined CD at one or more locations, those locations may be assessed as compatible defects.

[0199] 28 illustrates a system 108 for generating an aerial image 64 of a model of a photolithographic mask 14 according to an embodiment of the present invention, the system 108 including a data analysis device 110 including at least one memory 114 and at least one processor 112 configured to perform steps of the computer-implemented method for generating an aerial image according to an embodiment of the present invention described above. The processor 112 may be implemented as a CPU or a GPU.

[0200] 28 also illustrates a system 108 for improving a model of a photolithography mask, repairing a photolithography mask, determining the quality of a photolithography mask, performing measurements of a photolithography mask, detecting or assessing defects in a photolithography mask, or selecting illumination settings for a photolithography system. It includes a data analysis device 110 including at least one memory 114 and at least one processor 112 configured to perform steps of a computer-implemented method for generating an aerial image of a photolithography mask according to any of the embodiments, examples, or aspects described above. The system may also include a subsystem for acquiring an aerial image of the photolithography mask, which can be used to analyze the generated aerial image of the photolithography mask, e.g., for comparison, as shown in FIG. 29 .

[0201] 29 illustrates a system 116 for detecting defects in a photolithographic mask 14 according to an embodiment of the present invention, the system 116 including a subsystem 118 for obtaining an aerial image 64 of the photolithographic mask 14 and a data analysis device 110 including at least one memory 114 and at least one processor 112 configured to execute steps of a computer-implemented method 96 for detecting defects according to an embodiment of the present invention. The subsystem 118 for obtaining an aerial image 64 of the photolithographic mask 14 may include an aerial image acquisition system. Alternatively, the subsystem 118 may include a database or any other memory containing the aerial image 64 of the photolithographic mask 14, and the subsystem 118 may be configured to load the aerial image 64 from the database or memory. The subsystem 118 for obtaining an aerial image 64 of the photolithographic mask 14 may provide the aerial image 64 to the data analysis device 110. The data analysis device 110 includes a processor 112, for example, implemented as a CPU or GPU. Processor 112 may receive aerial image 64 via interface 120. Processor 112 may load program code from memory 114, such as program code for executing a computer-implemented method for detecting defects as described above. Processor 112 may execute the program code.

[0202] 30 illustrates a system 122 for assessing the suitability of defects in a photolithography mask 14 according to an embodiment of the present invention, the system 122 including a subsystem 124 for obtaining a charged particle beam image 126 of the photolithography mask 14 and a data analysis device 110 including at least one memory 114 and at least one processor 112 configured to execute steps of a computer-implemented method 102 for assessing the suitability of defects in a photolithography mask 14 according to an embodiment of the present invention. The subsystem 124 for obtaining the charged particle beam image 126 of the photolithography mask 14 may include a charged particle beam device, such as a helium ion microscope (HIM), a cross-beam device including an FIB and an SEM, or any charged particle imaging device. Alternatively, the subsystem 124 may include a database or any other memory containing the charged particle beam image 126 of the photolithography mask 14, and the subsystem 124 may be configured to load the charged particle beam image 126 from the database or memory. The subsystem 124 for obtaining a charged particle beam image 126 of the photolithography mask 14 may provide the charged particle beam image 126 to the data analysis device 110. The data analysis device 110 includes a processor 112, implemented as, for example, a CPU or a GPU. The processor 112 may receive the charged particle beam image 126 via the interface 120. The processor 112 may load program code from the memory 114, for example, program code for a computer-implemented method for assessing the suitability of defects as described above. The processor 112 may execute the program code.

[0203] Embodiments, examples, and aspects of the present invention can be described by the following clauses.

[0204] 1. A computer-implemented method for simulating an electromagnetic near field of a model of a photolithography mask in a near-field plane, wherein the photolithography mask is illuminated by incident electromagnetic radiation, the photolithography mask includes a mask carrier and a diffraction grating, the diffraction grating includes absorber structures and non-absorber structures that form a pattern on at least a portion of the mask carrier, the photolithography mask includes diffraction grating sections extending between the absorber plane and a mask carrier plane of the photolithography mask, and mask carrier sections extending between the mask carrier plane and a reference plane of the photolithography mask, the method comprising: a) simulating the propagation of electromagnetic waves within a grating section of a photolithographic mask using a wave propagation algorithm that describes the propagation of electromagnetic waves through an inhomogeneous medium; b) simulating the propagation of electromagnetic waves within a mask carrier section of a photolithographic mask using an analytical description of electromagnetic wave propagation within the mask carrier; c) obtaining an electromagnetic near field of a model of the photolithographic mask as simulated propagating electromagnetic waves in the near field next to an absorber plane of the photolithographic mask; A method comprising:

[0205] 2. The wave propagation algorithm

number

number

number

[0206] 3. a) identifying the number M of absorber and non-absorber structure materials that form a pattern of a photolithography mask; b) A subset of the x / y plane at z=z0

number

number

[0207]

number

[0208] 4. The value range D of at least one characteristic function is

number

[0209] 5. The characteristic function is a convex combination of the photolithography mask at each location at z = z0.

number

[0210] 6. A method according to any one of claims 3 to 5, wherein the characteristic function is band-limited.

[0211] 7. The method according to any one of clauses 3 to 6, wherein obtaining the characteristic function comprises decomposing the pattern of the photolithography mask into polygons, representing the polygons by a characteristic function, in particular a binary characteristic function, and applying a low-pass filter to the characteristic function.

[0212] 8. The method of clause 7, wherein applying a low-pass filter comprises applying a spatial analytical Fourier transform to the characteristic function followed by an inverse fast Fourier transform.

[0213] 9. The method of any one of clauses 3 to 8, wherein the analytical Fourier transform used in the wave propagation algorithm is approximated by a fast Fourier transform.

[0214] 10. The method of clause 9, wherein the wave propagation algorithm takes into account the angle φ of the incident electromagnetic wave relative to the normal to the absorber plane by assuming quasi-periodic boundary conditions in the fast Fourier transform in one or more directions perpendicular to a reference plane of the photolithographic mask.

[0215] 11. Dispersion relations

number

[0216] 12. The method of any one of clauses 9 to 11, wherein the electromagnetic wave within the diffraction grating section has a dispersion relation that depends on the angle φ of the incident electromagnetic wave relative to the normal to the absorber plane in one or more directions perpendicular to the reference plane of the photolithographic mask.

[0217] 13. The dispersion relation within a grating section is such that the phase shift vector α = (α x ,α y) T Using

number

[0218] 14. The method of any one of clauses 1 to 13, wherein the simulated electromagnetic wave is incident on a reference plane, propagates from the reference plane to a mask carrier plane within a mask carrier section of the photolithographic mask, and propagates from the mask carrier plane to an absorber plane within a diffraction grating section of the photolithographic mask.

[0219] 15. The method of any one of clauses 1 to 14, wherein the mask carrier comprises a multilayer in the form of a stack of optical thin films for reflecting an electromagnetic field, and the simulated electromagnetic wave is incident on the absorber plane, propagates from the absorber plane to the mask carrier plane within the grating section of the photolithography mask, is reflected within the multilayer in the mask carrier section of the photolithography mask, and propagates from the mask carrier plane to the absorber plane within the grating section of the photolithography mask.

[0220] 16. The method of clause 15, wherein simulating the reflection of the electromagnetic wave within the multilayer body includes analytical calculation of a reflection coefficient at the mask carrier plane that describes the propagation of the electromagnetic wave within the stack of optical thin films of the multilayer body.

[0221] 17. A method according to clause 16, wherein the reflection coefficients are calculated separately for each medium of the absorbing and non-absorbing structures of the diffraction grating at the mask carrier plane.

[0222] 18. The mask carrier comprises a multilayer in the form of a stack of optical thin films for reflecting electromagnetic waves; a simulated electromagnetic wave incident on the absorber plane, propagating from the absorber plane to the mask carrier plane within a diffraction grating section of the photolithography mask, and reflecting within the multilayer of the mask carrier section of the photolithography mask and propagating from the mask carrier plane to the absorber plane within the diffraction grating of the photolithography mask; To simulate the reflection of an electromagnetic wave in a multilayer body, we use the analytical reflection coefficient r at the mask carrier plane z0. m By the phase term

[0223]

number

number

[0224] 19. A computer-implemented method for simulating an aerial image of a model of a photolithography mask, the method comprising: a) a method for simulating the electromagnetic near field of a model of a photolithography mask according to any one of clauses 1 to 18; b) simulating an aerial image of a model of a photolithography mask by applying to the electromagnetic near field a simulation of an imaging process of a photolithography system in a projection section extending between the near field plane and the wafer plane; A method comprising:

[0225] 20. The method of clause 19, wherein simulating the imaging process includes resampling the simulated electromagnetic near field.

[0226] 21. A computer-implemented method for detecting defects in a photolithography mask, the method comprising: - obtaining an aerial image of a photolithography mask; - simulating an aerial image of a model of a photolithographic mask using the method described in clause 19 or 20; - detecting defects in the photolithography mask by comparing the acquired aerial image with a simulated aerial image; 20. A computer-implemented method comprising:

[0227] 22. A computer-implemented method for assessing the suitability of defects in a photolithography mask, the method comprising: - providing a charged particle beam image of a photolithography mask containing one or more defects; - simulating an aerial image of a model of a photolithography mask using the method of clause 19 or 20, wherein a charged particle beam image is used as the model of the photolithography mask; - assessing the conformance of one or more defects in the photolithography mask using the simulated aerial image; 20. A computer-implemented method comprising:

[0228] 23. A computer readable medium storing a computer program executable by a computing device, the computer program comprising code for performing a method according to any one of clauses 1 to 22.

[0229] 24. A computer program product comprising instructions which, when executed by a computer, cause the computer to carry out the method according to any one of clauses 1 to 22.

[0230] 25. A system for simulating the electromagnetic near field of a model of a photolithography mask, the system comprising: a data analysis device including at least one memory and at least one processor configured to perform the steps of the computer-implemented method of any one of clauses 1 to 18.

[0231] 26. A system for simulating an aerial image of a model of a photolithography mask, the system comprising a data analysis device including at least one memory and at least one processor configured to perform the steps of the computer-implemented method of clause 19 or 20.

[0232] 27. A system for detecting defects in a photolithography mask, the system comprising: - a subsystem for acquiring an aerial image of a photolithography mask; - a data analysis device comprising at least one memory and at least one processor configured to perform the steps of the computer-implemented method according to clause 21; A system comprising:

[0233] 28. A system for assessing the suitability of defects in a photolithography mask, the system comprising: - a subsystem for acquiring a charged particle beam image of a photolithography mask; - a data analysis device comprising at least one memory and at least one processor configured to perform the steps of the computer-implemented method according to clause 22; A system comprising:

[0234] In summary, the present invention relates to a computer-implemented method 54, 54', 54'', 54''' for generating an aerial image 64 of a model of a photolithography mask 14 under illumination with an incident electromagnetic wave 22, the method including: a) approximately simulating propagation of the incident electromagnetic wave 22 within a first section 25 of the photolithography mask 14 including a plurality of structures; b) analytically or numerically simulating propagation of the simulated electromagnetic wave 22 from step a) within a second section 27 of the photolithography mask 14; c) simulating a representation of an electromagnetic near field 20 of the model of the photolithography 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 photolithography mask 14. [Explanation of symbols]

[0235] 10, 10' Photolithography System 12 Radiation source 14 Photolithography mask 16 Illumination optical system 17 Projection optical system 18 Wafer Plane 19 Projection Section 20 Near Field 22 Electromagnetic Waves 24 Diffraction Grating 25 First Section 26 Structure 27 Second Section 28 Nonstructural 30 Structural Plane 32 Boundary plane 34 Reference plane 38 Multilayer body 40 Optical thin film 42 Capping Layer 44 Effective mirror surface 46 substrate layers 48 Mask Carrier 50 Main propagation direction 52 Near-field plane 54, 54', 54'', 54''' Computer-Implemented Method 56 First compartment simulation step 58 Second compartment simulation step 60 Near-field generation steps 61 Characteristic Function Step 62 Binary Characteristic Functions 63 Aerial image generation step 64 Spatial image 66 Approximation error 68 Band-limited characteristic function 70 Root Mean Square Error 72 Maximum error 74 Root Mean Square Error 76 Maximum error 78 Upper horizontal axis 79 iterations 80 Lower horizontal axis 81 Optimization 82 Vertical axis 83 Acquired aerial image 84 Normal 85 Metrology Systems 86 Wave Vector 87 Uncalibrated aerial image 88 Reference space image 89 Calibration Steps 90 Comparison results 91 Defect Map 92 model 93 Defects 94 elements 95 rigorous simulation steps 96 Computer-Implemented Methods 97 Registration Results 98 Aerial Image Step 99 Registration 100 Defect Detection Steps 101 Further Information Steps 102 Computer-Implemented Methods 103 Computer-Implemented Methods 104 Imaging Steps 105 Analysis Steps 106 Assessment Steps 107 Application Steps 108 System 110 Data Analysis Device 112 processors 114 memory 116 System 118 Subsystems 120 Interface 122 System 124 Subsystems 126 Charged particle beam image 128 machine learning models 130 Training Models 132 Reference space image 134 Acquired spatial image 136 Machine Learning Models 138 Defect Map 140 model pairs 142 Flawless Model 144 Defect Model 146 Aerial Image Pair 148 Defect-free aerial image 150 Defect Aerial Image 152 Noise 154 Noisy Aerial Image Pairs 156 Noisy, defect-free aerial images 158 Noisy Defect Aerial Image

Claims

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

2. 2. The method of claim 1, wherein in step a) the propagation of the incident electromagnetic wave within the first section (25) of the photolithographic mask (14) is approximately simulated using the Helmholtz equation.

3. 2. The method of claim 1, wherein in step a) the propagation of the incident electromagnetic wave within the first section (25) of the photolithography mask (14) is approximately simulated using a machine learning model.

4. The method of claim 2 , wherein the Helmholtz equation is approximated using a forward Helmholtz equation.

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

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

7. 7. The method of claim 6, wherein the first section (25) of the photolithography mask (14) is decomposed into different materials by defining characteristic functions for each material that are indicative of the presence of said material in different locations of the first section (25) of the photolithography mask (14), and at least one characteristic function is non-binary.

8. 38. The method of claim 37, wherein the characteristic function forms an affine connection at each location of the first section (25) of the photolithographic mask.

9. The method according to claim 7 or 8, wherein the characteristic function is band-limited.

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

11. The method of claim 10 , wherein applying the low-pass filter comprises applying a spatial analytical Fourier transform to the characteristic function, followed by applying an inverse Fourier transform.

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

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

14. 14. The method of claim 13, wherein the electromagnetic wave (22) in the first section (25) has a dispersion relation depending on the angle of the incident electromagnetic wave (22).

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

16. The method of any one of claims 1 to 15, wherein the photolithographic mask (14) is a transmissive photolithographic mask.

17. 16. The method according to any one of claims 1 to 15, wherein the photolithography mask (14) is a reflective photolithography mask and the second section (27) comprises a multilayer (38) in the form of a stack of optical thin films (40) for reflecting the electromagnetic waves (22).

18. 18. The method of claim 17, wherein simulating the reflection of the electromagnetic wave (22) within the multilayer body (38) comprises analytical or numerical calculation of a reflection coefficient at a boundary (32) between the second section (27) and the first section (25) of the photolithography mask (14), the reflection coefficient describing the propagation of the electromagnetic wave (22) within the stack of optical thin films (40) of the multilayer body (38).

19. 20. The method of claim 18, wherein the reflection coefficient at the boundary (32) is calculated separately within and outside the structure (26) of the first section (25) of the photolithography mask (14).

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

21. 21. The method of any one of claims 1 to 20, further comprising adjusting at least one parameter of the method to minimize differences between one or more reference aerial images of one or more photolithographic masks and corresponding generated aerial images of corresponding models of the one or more photolithographic masks, wherein the at least one parameter is from a group comprising mask parameters and optical parameters.

22. 22. The method of claim 1, further comprising: registering one or more reference aerial images of the photolithographic mask to corresponding generated aerial images of the model of the photolithographic mask; and reporting at least one registration parameter.

23. 23. The method of claim 21 or 22, wherein the one or more reference aerial images comprise a focal stack of a photolithographic mask.

24. 1. A computer-implemented method (96) for improving the design of a photolithography mask (14), repairing a photolithography mask (14), determining the quality of a photolithography mask (14), performing measurements on a photolithography mask (14), detecting or assessing defects in a photolithography mask (14), or selecting illumination settings for a photolithography system, the method comprising: - generating an aerial image (64) of a model of said photolithographic mask (14) using a method according to any one of claims 1 to 23; - analyzing the generated aerial image (64) accordingly; - using analysis results to improve the design of the photolithography mask (14), to repair the photolithography mask (14), to determine the quality of the photolithography mask (14), to detect or assess defects in the photolithography mask (14), or to select illumination settings for a photolithography system; A computer-implemented method (96) comprising:

25. 24. A computer-implemented method for training a machine learning model that maps a model of a photolithographic mask to an aerial image of the photolithographic mask, the method comprising: generating aerial images of a plurality of models of the photolithographic mask using the method of any one of claims 1 to 23; and training the machine learning model using training data that includes the generated aerial images.

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

27. A computer readable medium storing a computer program executable by a computing device, the computer program comprising code for performing the method of any one of claims 1 to 26.

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

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

30. 1. A system (116) for improving a model of a photolithography mask (14), repairing a photolithography mask (14), determining the quality of a photolithography mask (14), performing measurements on a photolithography mask (14), detecting or assessing defects in a photolithography mask (14), or selecting illumination settings for a photolithography system, the system comprising: A system (116) comprising a data analysis device (110) including at least one memory (114) and at least one processor (112) configured to perform the steps of the computer-implemented method (96) of claim 24.

Images