Method and system for predicting process information with a parametric model

Abstract

Methods and systems for predicting a complex electric field image with a parametric model are described. For a given input to the parametric model, a latent space representation of a complex electric field image is determined based on dimensional data in a latent space of the parametric model. The given input can be a measured amplitude (e.g., intensity) associated with the complex electric field image. The complex electric field image is predicted based on the latent space representation of the complex electric field image. The predicted complex electric field image includes an amplitude and a phase. The parametric model includes an encoder-decoder architecture. In some embodiments, determining the latent space representation of the electric field image includes minimizing a function that is constrained by a set of electric field images that can be predicted by the parametric model based on the dimensional data in the latent space and the given input.

Description

[0001] Cross-reference to related applications

[0002] This application claims priority to European application 19212419.6, filed on 29 November 2019, and European application 20151169.8, filed on 10 January 2020, the entire contents of which are incorporated herein by reference. Technical Field

[0003] This specification relates to methods and systems for predicting process information using parametric models. Background Technology

[0004] A lithography apparatus is a machine configured to apply a desired pattern onto a substrate. Lithography apparatus can be used, for example, in the manufacture of integrated circuits (ICs). A lithography apparatus can project a pattern (often referred to as a “design layout” or “design”) from a patterning apparatus (e.g., a mask) onto a radiation-sensitive material (resist) layer disposed on a substrate (e.g., a wafer).

[0005] To project a pattern onto a substrate, photolithography equipment can use electromagnetic radiation. The wavelength of this radiation determines the minimum size of the feature that can be formed on the substrate. Typical wavelengths currently used are 365 nm (i-line), 248 nm, 193 nm, and 13.5 nm. Compared to photolithography equipment using radiation with a wavelength of, for example, 193 nm, photolithography equipment using extreme ultraviolet (EUV) radiation with wavelengths in the range of 4 nm to 20 nm (e.g., 6.7 nm or 13.5 nm) can be used to form smaller features on the substrate.

[0006] Low-k1 lithography can be used to process features smaller than the typical resolution limit of lithography equipment. In such a process, the resolution can be expressed as CD = k1 × λ / NA, where λ is the wavelength of the radiation used, NA is the numerical aperture of the projection optics in the lithography equipment, CD is the "critical size" (typically the smallest feature size printed, but in this case, half a pitch), and k1 is an empirical resolution factor. Generally, the smaller k1 is, the more difficult it is to reproduce patterns on the substrate that resemble the shapes and sizes planned by the circuit designer to achieve specific electrical functions and performance.

[0007] To overcome these difficulties, sophisticated fine-tuning steps can be applied to the lithography projection apparatus and / or the design layout. These steps include, for example, but are not limited to, optimization of the NA, customized illumination schemes, use of phase- shifting patterning devices, various kinds of optimization of the design layout such as optical proximity correction (OPC, sometimes also called "optical and process correction") in the design layout, or other methods generally defined as "resolution enhancement techniques" (RET). Alternatively, tight control loops for controlling the stability of the lithography apparatus can be used to improve the reproduction of patterns at low k1. SUMMARY

[0008] Various metrology operations can be used to measure features of a design. These calculations can include, for example, measuring overlay. Overlay can be determined based on computationally intensive determinations of complex electric field images associated with a design. Advantageously, the present methods and systems are configured for (less computationally intensive) prediction of complex electric field images, determination of one or more metrology indicators, and / or other calculations using a parametric model.

[0009] According to an embodiment, a method of predicting an electric field image with a parametric model is provided. The method comprises determining, based on dimensional data in a latent space of the parametric model, a latent space representation of an electric field image for a given input to the parametric model; and predicting the electric field image based on the latent space representation of the electric field image.

[0010] In some embodiments, the electric field image comprises a complex electric field image having an amplitude and a phase.

[0011] In some embodiments, the given input comprises a measured amplitude associated with the complex electric field image.

[0012] In some embodiments, the amplitude comprises an intensity.

[0013] In some embodiments, determining the latent space representation of the electric field image comprises minimizing a function constrained by a set of electric field images that can be predicted by the parametric model based on the dimensional data in the latent space and the given input.

[0014] In some embodiments, the latent space representation of the electric field image comprises a tensor.

[0015] In some embodiments, the parametric model is a machine learning model.

[0016] In some embodiments, the parametric model comprises an encoder-decoder architecture.

[0017] In some embodiments, the encoder-decoder architecture comprises a variable encoder-decoder architecture. The method further comprises training the variable encoder-decoder architecture with a probabilistic latent space, the variable encoder-decoder architecture producing realizations in an output space.

[0018] In some embodiments, the latent space comprises a low-dimensional encoding.

[0019] In some embodiments, the dimensional data in the latent space is encoded by an encoder of the encoder-decoder architecture.

[0020] In some embodiments, the method further comprises training the parameterized model with a training set of complex electric field images.

[0021] In some embodiments, the set of complex electric field images is produced during a through-focus measurement.

[0022] In some embodiments, the training comprises encoding the complex electric field images in the training set into the dimensional data in the latent space, and transforming the dimensional data in the latent space into a deconvolved version of the complex electric field images in the training set, to facilitate verification of the training.

[0023] In some embodiments, the method further comprises iteratively providing an additional complex electric field image as input to the parameterized model. The additional complex electric field image is determined based on how well the deconvolved version of the complex electric field image matches the complex electric field images in the training set.

[0024] In some embodiments, the method further comprises encoding, with the encoder, higher dimensional data associated with the electric field image into the dimensional data in the latent space.

[0025] In some embodiments, predicting the electric field image based on the latent space representation of the electric field image comprises passing the latent space representation of the electric field image through a decoder of the encoder-decoder architecture.

[0026] In some embodiments, the method further comprises determining a metrology metric based on the latent space representation of the electric field image.

[0027] In some embodiments, determining the metrology metric based on the latent space representation of the electric field image comprises providing the latent space representation of the electric field image to a regression network, the regression network being included in the parameterized model or separate from the parameterized model.

[0028] In some embodiments, the metrology metric is overlay.

[0029] In some embodiments, the method further comprises correcting an aberration associated with the metrology apparatus based on the latent space representation of the electric field image and / or the predicted electric field image.

[0030] In some embodiments, the method further comprises determining an adjustment to a semiconductor manufacturing process parameter based on the predicted electric field image, the semiconductor manufacturing process parameter being used to pattern a substrate geometry, the patterning of the substrate geometry being part of a semiconductor manufacturing process.

[0031] According to another embodiment, there is provided a non-transitory computer readable medium having instructions thereon, the instructions when executed by a computer implementing any of the methods described herein.

[0032] According to another embodiment, there is provided a metrology apparatus configured to determine one or more metrology indicators of a semiconductor manufacturing process. The apparatus comprises one or more processors configured to: determine a latent space representation of an electric field image for a given input based on dimensional data in a latent space of a parametric model; predict the electric field image with the parametric model based on the latent space representation of the electric field image; and determine the one or more metrology indicators of the semiconductor manufacturing process based on the predicted electric field image.

[0033] According to another embodiment, there is provided a lithographic cell comprising a metrology apparatus. The metrology apparatus is configured to: determine a latent space representation of an electric field image for a given input based on dimensional data in a latent space of a parametric model; predict the electric field image with the parametric model based on the latent space representation of the electric field image; and determine one or more metrology indicators of the semiconductor manufacturing process based on the predicted electric field image.

[0034] According to another embodiment, there is provided a method for determining one or more metrology indicators for a semiconductor manufacturing process. The method comprises: determining a latent space representation of an electric field image for a given input based on dimensional data in a latent space of a parametric model; predicting the electric field image with the parametric model based on the latent space representation of the electric field image; and determining the one or more metrology indicators for the semiconductor manufacturing process based on the predicted electric field image.

[0035] In some embodiments, the electric field image comprises a complex electric field image having an amplitude and a phase.

[0036] In some embodiments, the one or more determined metrology indicators comprise one or more of: overlay, critical dimension, reconstruction of a three-dimensional profile of a feature of a substrate, or dose or focus of a lithographic apparatus when printing a feature of a substrate with the lithographic apparatus.

[0037] In some embodiments, the given input comprises a measured amplitude associated with the complex electric field image.

[0038] In some embodiments, the amplitude comprises an intensity.

[0039] In some embodiments, the method comprises adjusting one or more semiconductor manufacturing process parameters based on the determined one or more metrology indicators.

[0040] According to another embodiment, there is provided a method for predicting process information with a parametric model, comprising: determining, in a latent space of the parametric model, a latent space representation of a given input to the parametric model; transforming the latent space representation of the given input into a transformed latent space representation of the given input based on a reference latent space representation of the given input; and predicting the process information based on the transformed latent space representation of the given input.

[0041] In some embodiments, the given input is associated with a target and is received from one of a plurality of target characterization apparatuses configured to produce the given input. It is noted that "target" is used broadly and can refer to any feature and / or structure in any substrate, layer or other portion of a device and / or other physical object that is imaged, measured or otherwise characterized in one way or another. This can include, for example, metrology targets and / or other measurement structures. The target can be located, for example, inside or outside a product area on a wafer.

[0042] In some embodiments, the transforming and predicting are configured such that the predicted process information for a target is the same regardless of which of the target characterization apparatuses produced the given input.

[0043] In some embodiments, the transforming comprises one or more mathematical calculations on the latent space representation of the given input.

[0044] In some embodiments, the transforming is performed in the latent space.

[0045] In some embodiments, the reference latent space representation comprises a weighted combination and / or average of latent space representations of previously received inputs to the parametric model or of inputs from the particular target characterization apparatus configured to produce the given input.

[0046] In some embodiments, the process information and the given input are associated with a semiconductor manufacturing process.

[0047] In some embodiments, the predicted process information comprises one or more of a predicted image or a predicted process measurement. It is noted that "image" is used broadly and can refer to any image produced (e.g., measured, predicted, acquired, etc.) during a manufacturing process. This can include, for example, in the context of a semiconductor device, field plane acquisition results, pupil plane acquisition results, and / or other images.

[0048] In some embodiments, the predicted process measurement comprises one or more of a metrology metric, an xyz position, a dimension, an electric field, a wavelength, an illumination and / or detection pupil, a bandwidth, an illumination and / or detection polarization angle, or an illumination and / or detection phase retardation angle.

[0049] In some embodiments, the given input comprises one or more of an input image or an input process measurement.

[0050] According to another embodiment, there is provided a method for predicting process information with a parametric model, comprising: determining, in a latent space of the parametric model, a latent space representation of an optimal set of process parameters associated with a given input to the parametric model based on dimensional data; and predicting the process information based on the latent space representation of the optimal set of process conditions.

[0051] In some embodiments, the predicted process information comprises one or more of a design parameter of a target or a metrology measurement recipe parameter.

[0052] In some embodiments, the design parameter comprises one or more of a critical dimension, a pitch, a segmented pitch, a line geometry, a contact geometry, or a hole geometry associated with the target.

[0053] In some embodiments, the metrology measurement recipe parameter comprises one or more of a wavelength, a bandwidth, a hole, an illumination and / or detection pupil, a bandwidth, an illumination and / or detection polarization angle, an illumination and / or detection phase retardation angle, or a dose for measuring the target.

[0054] In some embodiments, the optimal set of process parameters defines optimal measurement conditions for a metrology metric of the target.

[0055] In some embodiments, the optimal set of process parameters comprises one or more of an intensity, a contrast, an edge response, a diffraction efficiency, or an overlay sensitivity.

[0056] In some embodiments, the given input comprises one or more of a defined design parameter or a defined metrology measurement recipe parameter of the target.

[0057] In some embodiments, the parameterized model is a machine learning model.

[0058] In some embodiments, the parameterized model comprises an encoder-decoder architecture.

[0059] In some embodiments, the latent space comprises a low-dimensional encoding.

[0060] According to another embodiment, there is provided a method for predicting process information with a parameterized model, comprising: determining a latent space representation of a given input to the parameterized model in a latent space of the parameterized model; transforming the latent space representation of the given input into a transformed latent space representation of the given input based on a reference latent space representation of the given input; determining a latent space representation of an optimal set of process parameters associated with the given input based on the transformed latent space representation; and predicting the process information based on the transformed latent space representation of the given input and the latent space representation of the optimal set of process parameters associated with the given input.

[0061] According to another embodiment, there is provided a method for predicting process information with a parameterized model, comprising: determining a latent space representation of a given input to the parameterized model in a latent space of the parameterized model; transforming the latent space representation of the given input into a transformed latent space representation of the given input based on a reference latent space representation of the given input; determining a latent space representation of an electric field image of the given input based on the transformed latent space representation of the given input; determining a latent space representation of an optimal set of process parameters associated with the given input based on the transformed latent space representation; and predicting the process information based on the transformed latent space representation of the given input, the latent space representation of the electric field image, and the latent space representation of the optimal set of process parameters associated with the given input.

[0062] According to another embodiment, there is provided a non-transitory computer readable medium having instructions thereon, which when executed by a computer implement the method of any of the above described embodiments. BRIEF DESCRIPTION OF DRAWINGS

[0063] The accompanying drawings, which are incorporated in and constitute a part of this specification, illustrate one or more embodiments and together with the description, explain these embodiments. Embodiments of the application will now be described, by way of example only, with reference to the accompanying schematic drawings in which corresponding reference symbols indicate corresponding parts or sections, and in which:

[0064] Figure 1 A schematic overview depicting a lithographic apparatus according to an embodiment.

[0065] Figure 2 A schematic overview depicting a lithographic cell according to an embodiment.

[0066] Figure 3 A schematic representation depicting overall lithography according to an embodiment, representing the cooperation between three technologies to optimize semiconductor manufacturing.

[0067] Figure 4 Fig. 1 illustrates an example metrology apparatus, such as a scatterometer, according to an embodiment.

[0068] Figure 5 Fig. 2 illustrates an encoder-decoder architecture according to an embodiment.

[0069] Figure 6 Fig. 3 illustrates an encoder-decoder architecture within a neural network according to an embodiment.

[0070] Figure 7 Fig. 4 illustrates an example camera associated with obtaining complex electric field images according to an embodiment.

[0071] Figure 8 Fig. 5 illustrates an overview of the operation of the present method of predicting electric field images with a parametric model according to an embodiment.

[0072] Figure 9 Fig. 6 illustrates an example of a parametric model according to an embodiment.

[0073] Figure 10 Fig. 7 is a block diagram of an example computer system according to an embodiment.

[0074] Figure 11 Fig. 8 is a block diagram of an example computer system according to an embodiment, Figure 1 Alternative designs of the lithographic apparatus of Fig. 1. DETAILED DESCRIPTION

[0075] Process information (e.g., images, measurements, process parameters, metrology indicators, etc.) can be used to guide various manufacturing operations. Predicting or otherwise determining process information with a parametric model as described herein can be faster, more efficient, require fewer computational resources, and / or have other advantages compared to previous methods of determining process information.

[0076] For example, phase retrieval can be used to determine a complex electric field image. The complex electric field image can be associated with a target. For example, the target can be a portion of a semiconductor device (e.g., a portion of a pattern and / or feature that is patterned in a substrate) and / or other targets. As described above, it is noted that "target" is used broadly and refers to any feature and / or structure in any substrate, layer, or other portion of a device and / or other physical object that is imaged, measured, or otherwise characterized in one way or another. The complex electric field image, in turn, can be used to determine one or more metrology indicators such as overlay and / or other information for the target.

[0077] Phase retrieval includes recovering a complex (valued) electric field from corresponding amplitude (e.g., intensity) measurements (e.g., for a particular target). Phase retrieval is difficult due to a large number of unknowns (e.g., on the order of 10 6 ), nonlinearity, and non-convexity associated with equations satisfied by the complex electric field. Existing methods of phase retrieval include introducing redundancy in the amplitude measurements or utilizing prior knowledge about the complex electric field image (e.g., to reduce the number of unknowns). These and other methods require redundant data, make assumptions, are computationally intensive, and / or have other drawbacks.

[0078] In contrast, the present methods and systems are configured to predict a complex electric field image with a parametric model. Predicting a complex electric field image with a parametric model can be less computationally intensive and / or have other advantages over prior methods. The parametric model is configured to significantly reduce the dimensionality of the phase retrieval problem, which can then be solved using a gradient-based optimization scheme (e.g., in a few dimensions).

[0079] As another example, data from different process sensors and / or tools can be different, even for the same measured or imaged target. Prior attempts to address these differences include physically adjusting one or more components in the sensors or tools, adjusting measurement or imaging parameters on a given sensor or tool, and / or making other adjustments to make data from a particular sensor or tool better aligned with data from other sensors or tools. However, these adjustments are not always applied consistently, rely on human judgment and / or specific placement of physical components, and / or have other drawbacks.

[0080] In contrast, the present methods and systems are configured to determine a latent space representation of a given input to a parametric model in one or more latent spaces or potential spaces of the parametric model (wherein the dimensionality of the data to be analyzed is less than the number of dimensions in the raw data from different sensors and / or tools). The latent space representation is transformed into a transformed latent space representation of the given input based on a reference latent space representation of the given input, and process information is predicted based on the transformed latent space representation of the given input. For example, the given input can be associated with a target and received from one of a plurality of different sensors and / or tools configured to produce the given input. The transformation and prediction are configured such that the predicted and / or otherwise determined process information (e.g., images, measurements, process parameters, metrology indicators, etc.) from the parametric model is the same for the target, independent of which of the sensors and / or tools produced the given input.

[0081] As a third example, operations such as target design, manufacturing recipe determination, and / or other operations often require iterative experimentation, where in each iteration a selected design or recipe variable is purposefully adjusted while other variables remain fixed for the iteration. Each iteration often requires recording and analysis of various measurements. After several iterations, a particular target design or manufacturing recipe can be selected (e.g., by elimination of processes and / or different methods).

[0082] In contrast, the present methods and systems are configured to determine a latent space representation of an optimal set of process parameters associated with a given input to a parametric model based on dimensional data in one or more latent spaces of the parametric model (again, wherein the dimensionality of the data to be analyzed is less than the number of dimensions in the raw data from different experimental iterations). The given input can include, for example, a desired element of a target design or manufacturing recipe or another input (e.g., immutable due to design requirements, machine capabilities, physics, etc.). The present systems and methods are configured to predict and / or otherwise determine process information, such as an optimal target design or manufacturing recipe, based on the latent space representation of the optimal set of process conditions.

[0083] These examples are not intended to be limiting. Note also that one or more parameterized models can perform some or all of the operations described above. For example, one parameterized model can be trained to predict complex electric field images, to transform data from different sensors and / or tools so that data for the same target matches, and to determine optimal target design and / or fabrication recipes; or these different operations can be performed by different parameterized models. The different applications (predicting complex electric field images, transforming data from different sensors and / or tools so that data for the same target from different sensors and / or tools still matches, and determining optimal target design and / or fabrication recipes) can be used together, or they can be used separately.

[0084] Furthermore, although specific reference can be made in this text to the manufacture of ICs, the description herein has many other possible applications. For example, it can be employed in the manufacture of integrated optical systems, guidance and detection patterns for magnetic domain memories, liquid crystal display panels, thin-film magnetic heads, etc. The skilled artisan will appreciate that, in the context of such alternative applications, any use of the terms "reticle," "wafer," or "die" in this text should be considered as interchangeable with the more general terms "mask," "substrate," and "target portion," respectively. In addition, it should be noted that the methods described herein can have many other possible applications in diverse fields such as language processing systems, self-driving cars, medical imaging and diagnostics, semantic segmentation, de-noising, chip design, electronic design automation, etc. The present methods can be applied in any field where quantifying the uncertainty in a machine learning model's prediction is advantageous.

[0085] In the present document, the terms "radiation" and "beam" are used to encompass all types of electromagnetic radiation, including ultraviolet radiation (e.g. with a wavelength of 365, 248, 193, 157 or 126 nm) and EUV (extreme ultra-violet radiation, e.g. having a wavelength in the range of about 5-100 nm).

[0086] The patterning device can comprise or can form one or more design layouts. The design layouts can be generated using a computer aided design (CAD) process. This process is often referred to as electronic design automation (EDA). Most CAD processes follow a predetermine set of design rules, in order to generate functional design layouts / patterning devices. These rules are set based on processing and design limitations. For example, design rules define the space tolerance between devices (such as gates, capacitors, etc.) or interconnect lines, to ensure that the devices or lines do not interact with one another in an undesirable way. One or more of the design rule limitations can be referred to as a “critical dimension” (CD). The critical dimension of a device can be defined as the smallest width of a line or hole, or the smallest space between two lines or two holes. Thus, the CD regulates the overall size and density of the devices designed. One of the goals in device fabrication is to faithfully reproduce the original design intent on the substrate (via the patterning device).

[0087] The term“reticle”,“mask” or“patterning device” as employed herein can be broadly interpreted to refer to any patterning device that can be used to impart a pattern to a beam of radiation, which pattern corresponds to a pattern that is to be created in a target portion of the substrate. The term“light valve” can also be used in such contexts. Examples of other such patterning devices include a programmable mirror array.

[0088] As a brief introduction, Figure 1 A lithographic apparatus LA is schematically depicted. The lithographic apparatus LA includes an illumination system (also referred to as illuminator) IL configured to condition a radiation beam B (e.g., UV, DUV or EUV radiation), a mask support (e.g., a mask table) T constructed to support a patterning device (e.g., a mask) MA and connected to a first positioner PM configured to accurately position the patterning device MA in accordance with certain parameters, a substrate support (e.g., a wafer table) WT configured to hold a substrate (e.g., a resist-coated wafer) W and coupled to a second positioner PW configured to accurately position the substrate support in accordance with certain parameters, and a projection system (e.g., a refractive projection lens system) PS configured to project a pattern imparted to the radiation beam B by patterning device MA onto a target portion C (e.g., including one or more dies) of the substrate W.

[0089] In operation, the illumination system IL receives a radiation beam from a radiation source SO, e.g. via a beam delivery system BD. The illumination system IL can include various types of optical components, such as refractive, reflective, magnetic, electromagnetic, electrostatic, and / or other types of optical components, or any combination thereof, for directing, shaping, and / or controlling the radiation. The illuminator IL can be used to adjust the angular distribution of the radiation beam B, e.g. to have a desired spatial and angular intensity distribution in its cross-section at the plane of the patterning device MA.

[0090] The term "projection system" PS used herein should be broadly interpreted as encompassing various types of projection system, including refractive, reflective, catadioptric, anamorphic, magnetic, electromagnetic and / or electrostatic systems, or any combination thereof, as appropriate for the exposure radiation being used, and / or for other factors such as the use of an immersion liquid or the use of a vacuum. Any use of the term "projection lens" herein can be considered as synonymous with the more general term "projection system" PS.

[0091] The lithographic apparatus LA can be of a type that includes a substrate support WT (also referred to as "wafer stage") arranged to hold a substrate W (also referred to as "wafer") and capable of controlling the position of the substrate W in at least two dimensions. The lithographic apparatus LA can be of a type that includes a substrate support WT arranged to hold a substrate W (also referred to as "wafer") and capable of controlling the position of the substrate W in at least two dimensions. The lithographic apparatus LA can be of a type that includes two or more substrate supports WT (also referred to as "dual stage") as described below.

[0092] The lithographic apparatus LA can also be of a type that includes two or more substrate supports WT (also referred to as "dual stage"). In such "dual stage" machines, substrate supports WT can be used in parallel, and / or steps can be performed simultaneously on two or more substrates W located on the substrate supports WT, while another substrate W on another substrate support WT is being used for exposing a pattern on said another substrate W.

[0093] In addition to the substrate support WT, the lithographic apparatus LA can also include a measurement platform. The measurement platform is arranged to hold a sensor and / or a cleaning device. The sensor can be arranged to measure a property of the projection system PS or a property of the radiation beam B. The measurement platform can hold multiple sensors. The cleaning device can be arranged to clean a part of the lithographic apparatus, for example a part of the projection system PS or a part of a system that provides an immersion liquid. The measurement platform can be movable under the projection system PS when the substrate support WT is away from the projection system PS.

[0094] In operation, the radiation beam B is incident on the patterning device (e.g., mask) MA held on the mask support MT, and is patterned by a patterning device MA that is present on the patterning device MA. Having traversed the mask MA, the radiation beam B passes through the projection system PS which focuses the beam onto a target portion C of the substrate W. With the aid of the second positioner PW and the position measurement system IF, the substrate support WT can be accurately moved, for example, in order to position different target portions C in the path of the radiation beam B. Similarly, the first positioner PM and the possible additional position sensor (which is not explicitly depicted in Figure 1 Figure 1) can be used to accurately position the patterning device MA in relation to the path of the radiation beam B. The patterning device MA and the substrate W can be aligned using mask alignment marks M1, M2 and substrate alignment marks P1, P2. Although the substrate alignment marks P1, P2 (as illustrated) occupy dedicated target portions, the marks can also be located in the spaces between target portions. When the substrate alignment marks P1, P2 are located between target portions C, these marks are referred to as scribe-lane alignment marks.

[0095] Figure 2 A schematic diagram overview of a lithography cell LC is depicted. As shown in Figure 2 Figure 1, the lithographic apparatus LA can form part of a lithocell LC (sometimes also referred to as a litho cell or (litho) cluster), which typically also includes apparatus to carry out pre-exposure processes and post-exposure processes on the substrate W. Conventionally, these apparatus include a spin coater SC configured to deposit a resist layer on a substrate W, a developer DE to develop the resist that has been exposed, a chill plate CH and a bake plate BK, for example, for adjusting the temperature of the substrate W (for example, for adjusting a solvent in the resist layer). A substrate handling apparatus or robot RO picks up substrates W from input / output ports I / O1, I / O2, moves the substrates W between the different process apparatus and delivers the substrates W to a load deck LB of the lithographic apparatus LA. The apparatus in the lithocell, which are often collectively referred to as a track or track coat develop system, are typically under control of a track or track coat develop system control unit TCU, which itself can be controlled by a management control system SCS, which can also control the lithographic apparatus LA, for example, via a lithographic control unit LACU.

[0096] In order to expose a substrate W by the lithographic apparatus LA, Figure 1) correctly and consistently exposed, the inspected substrate is expected to measure properties of the patterned structures, such as overlay errors between subsequent layers, line thicknesses, critical dimensions (CDs), etc. For this purpose, an inspection tool (not shown) can be included in the lithography cell LC. If errors are detected, the exposure of subsequent substrates or other processing steps to be performed on the substrate W can be adjusted, especially if the inspection takes place before other substrates W of the same batch or lot are still to be exposed or processed.

[0097] Inspection apparatuses (which can also be called metrology apparatuses) are used to determine properties of the substrate W( Figure 1 ) and, in particular, how properties of different substrates W vary or how properties associated with different layers of the same substrate W vary from layer to layer. Alternatively, the inspection apparatus is configured to identify defects on the substrate W and can for example be part of the lithocell LC, or can be integrated into the lithographic apparatus LA, or can even be a separate device. The inspection apparatus can measure properties on a latent image (the image in a resist layer after the exposure), or on a semi-latent image (the image in a resist layer after the exposure and a post-exposure bake step PEB), or on a developed resist image (in which the exposed or unexposed parts of the resist have been removed), or even on an etched image (after a pattern transfer step such as etching).

[0098] Figure 3 A schematic representation of the overall lithography is depicted, representing the cooperation between three technologies used to optimize semiconductor manufacturing. Typically, the patterning process in the lithographic apparatus LA is one of the most critical steps in the process, requiring high accuracy in the sizing and placement of structures on the substrate W( Figure 1 ). To ensure this high accuracy, three systems (in this example) can be combined in a so-called "overall" control environment, as schematically depicted in Figure 3 . One of these systems is the lithographic apparatus LA, which is (virtually) connected to a metrology apparatus (e.g. a metrology tool) MT (second system) and to a computer system CL (third system). The "overall" environment can be configured to optimize the cooperation between these three systems to enhance the total process window and provide a tight control loop, ensuring that the patterning by the lithographic apparatus LA stays within the process window. The process window defines a range of process parameters (e.g. dose, focus, overlay) within which a certain manufacturing process yields a defined outcome (e.g. functional semiconductor devices) - typically within which variations in the process parameters in the lithographic or patterning process are allowed.

[0099] The computer system CL can use the design layout (partially) to be patterned to predict which resolution enhancement techniques to use, and perform computational lithography simulations and calculations to determine which mask layouts and lithography equipment settings achieve the maximum total process window (within) of the patterning process. Figure 3 (Depicted by double arrows in the first scale SC1). Typically, resolution enhancement techniques are arranged to match the patterning possibilities of the lithography equipment LA. The computer system CL can also be used to detect where the lithography equipment LA is currently operating within the process window (e.g., using input from the metrology tool MT) to predict whether defects might be attributable to, for example, suboptimal processing (in...). Figure 3 (This is depicted by the arrow pointing to "0" in the second ruler SC2).

[0100] The measurement device (tool) MT can provide input to the computer system CL for accurate simulation and prediction, and can provide feedback to the lithography equipment LA to identify possible drifts in, for example, the calibration status of the lithography equipment LA. Figure 3 (The middle part is depicted by multiple arrows in the third ruler SC3).

[0101] During photolithography, it is desirable to frequently measure the resulting structure, for example, for process control and verification. Tools used to perform such measurements include measurement tools (devices) MT. Different types of measurement tools MT for performing such measurements are well known, including scanning electron microscopes or various forms of scatterometer measurement tools MT. A scatterometer is a multi-functional instrument that allows the measurement of parameters of the photolithography process by means of a sensor in the pupil or a plane conjugate to the pupil of the scatterometer's objective lens (measurements are generally referred to as pupil-based measurements), or by means of a sensor in the image plane or a plane conjugate to the image plane, in which case measurements are generally referred to as image- or field-based measurements. Such scatterometers and associated measurement techniques are further described in patent applications US20100328655, US2011102753A1, US20120044470A, US20110249244, US20110026032, or EP1628164A, which are incorporated herein by reference in their entirety. For example, the aforementioned scatterer can use light from soft X-rays and the visible to near-IR wavelength range to measure the characteristics of a substrate, such as a grating.

[0102] In some examples, scatterometer MT is an angle-resolved scatterometer. In these embodiments, scatterometer reconstruction methods can be applied to the measured signals to reconstruct or calculate properties of gratings and / or other features in the substrate. This reconstruction can result, for example, from simulating the interaction of the scattered radiation with a mathematical model of the target structure and comparing the simulated results with the measured results. The parameters of the mathematical model are adjusted until the simulated interaction produces a diffraction pattern similar to that observed from the real target.

[0103] In some embodiments, scatterometer MT is a spectroscopic scatterometer MT. In these embodiments, the spectroscopic scatterometer MT can be configured so that radiation emitted by the radiation source is directed onto a target feature of the substrate and reflected or scattered radiation from the target is directed to a spectrometer detector that measures the spectrum of the reflected radiation (i.e. measures the intensity as a function of wavelength). From this data, the structure or profile of the target that produced the detected spectrum can be reconstructed, for example by rigorous coupled wave analysis and non-linear regression or by comparison with a library of simulated spectra.

[0104] In some embodiments, scatterometer MT is an ellipsometric scatterometer. Ellipsometric scatterometers allow determination of parameters of a lithography process by measuring the scattered radiation for each polarization state. Such metrology devices (MT) emit polarized light (such as linear, circular or elliptical) by using, for example, appropriate polarization filters in the illumination section of the metrology device. Sources suitable for use in metrology devices can also provide polarized radiation. Various embodiments of existing ellipsometric scatterometers are described in U.S. Patent Applications 11 / 451,599, 11 / 708,678, 12 / 256,780, 12 / 486,449, 12 / 920,968, 12 / 922,587, 13 / 000,229, 13 / 033,135, 13 / 533,110, and 13 / 891,410, which are incorporated by reference in their entirety.

[0105] In some embodiments, scatterometer MT is adapted to measure the overlay of two misaligned gratings or periodic structures (and / or other target features of the substrate) by measuring the reflection spectrum and / or asymmetry in the detection configuration, which is related to the degree of overlay. The two (typically superimposed) grating structures can be applied in two different layers (not necessarily consecutive layers) and the grating structures can be formed substantially at the same location on the wafer. The scatterometer can have a symmetric detection configuration as described, for example, in patent application EP 1,628,164 A, so that any asymmetry can be clearly distinguished. This provides a way to measure misalignment in the gratings. Further examples of measuring overlay can be found in PCT patent application publication number WO 2011 / 012624 or U.S. patent application US 20160161863, which are incorporated by reference in their entirety.

[0106] Other parameters of interest can be focus and dose. Focus and dose can be determined simultaneously by scatterometry (or alternatively by scanning electron microscopy) as described in U.S. Patent Application US2011-0249244, which is incorporated herein by reference in its entirety. A single structure (e.g., a feature in a substrate) can be used that has a unique combination of critical dimension and sidewall angle measurements for each point in an energy matrix for focus (FEM, also known as focus exposure matrix). If these unique combinations of critical dimension and sidewall angle are obtainable, then the focus and dose values can be uniquely determined from these measurements.

[0107] The metrology target can be a composite grating and / or other set of features in a substrate that is formed by a lithographic process (typically in resist, but also after e.g., an etching process). Typically, the pitch and line width of the structures in the grating are dependent on the measurement optics (especially the NA of the optics) to be able to capture the diffraction orders from the metrology target. The diffraction signal can be used to determine a shift between two layers (also known as "overlay") or can be used to reconstruct at least part of the original grating as produced by the lithographic process. Such reconstruction can be used to provide a guide to the quality of the lithographic process and can be used to control at least part of the lithographic process. The target can have smaller sub-segments that are configured to mimic the dimensions of a functional part of the design layout in the target. Due to such sub-segments, the target will behave more like the functional part of the design layout so that the overall process parameter measurement is similar to the functional part of the design layout. The target can be measured in an underfill mode or in an overfill mode. In underfill mode, the measurement beam produces a spot that is smaller than the total target. In overfill mode, the measurement beam produces a spot that is larger than the total target. In such overfill mode, it is also possible to measure different targets simultaneously, thereby determining different process parameters simultaneously.

[0108] The overall measurement quality of a lithographic parameter using a particular target is determined at least partly by the measurement recipe used to measure such lithographic parameter. The term "substrate measurement recipe" can include one or more parameters of the measurement itself, one or more parameters of the pattern being measured, or both. For example, if the measurement used in the substrate measurement recipe is a diffraction-based optical measurement, one or more of the parameters of the measurement can include the wavelength of the radiation, the polarization of the radiation, the angle of incidence of the radiation relative to the substrate, the direction of the radiation relative to the pattern on the substrate, etc. One of the criteria to select a measurement recipe can for example be the sensitivity of one of the measurement parameters to a process bias. More examples are described in U.S. Patent Application US2016-0161863 and published U.S. Patent Application US 2016 / 0370717 Al, which are incorporated herein by reference in their entirety.

[0109] Figure 4 An example metrology apparatus (tool) MT, such as a scatterometer, is illustrated. The MT comprises a broadband (white light) radiation projector 40 which projects radiation onto a substrate 42. Reflected or scattered radiation is passed to a spectrometer detector 44 which measures the spectrum 46 of the mirror reflected radiation (i.e. measures the intensity as a function of wavelength). From this data the structure or profile which gave rise to the measured spectrum can be reconstructed 48, e.g. by rigorous coupled wave analysis and non-linear regression or by comparison with a library of simulated spectra, as shown at the bottom of Figure 3 In general, for reconstruction, the general form of the structure is known and some parameters are assumed from knowledge of the process by which the structure was manufactured, leaving only a few parameters of the structure to be determined from the scatterometry data. Such a scatterometer can be configured as a normal-incidence scatterometer or an oblique-incidence scatterometer, for example.

[0110] It is often desirable to be able to computationally determine how a patterning process will produce a desired pattern on a substrate. Computational determination can include modeling, for example. Models and / or simulations can be provided for one or more portions of a manufacturing process. For example, it is desirable to be able to simulate a lithography process that transfers a patterning device pattern onto a resist layer of a substrate and the pattern produced in the resist layer after development of the resist, simulate metrology operations such as determination of an overlay, and / or perform other simulations. The purpose of the simulation can be to accurately predict, for example, metrology targets (e.g. overlay, critical dimension, reconstruction of a three-dimensional profile of features of a substrate, dose or focus of a lithographic apparatus when printing features of a substrate with the lithographic apparatus, etc.), manufacturing process parameters (e.g. edge placement, aerial image intensity slope, sub-resolution assist features (SRAFs), etc.), and / or other information that can then be used to determine whether a contemplated or target design has been implemented. The contemplated design is typically defined as a pre-optical proximity correction design layout, which can be provided as a standardized digital file format such as GDSII, OASIS, or another file format.

[0111] Modeling can be used to determine one or more metrology targets (e.g., to perform overlay and / or other metrology measurements), to configure one or more features of a patterning device pattern (e.g., to perform optical proximity correction), to configure one or more features of illumination (e.g., to change one or more characteristics of a spatial / angular intensity distribution of illumination, such as to change a shape), to configure one or more features of a projection optics (e.g., numerical aperture, etc.), and / or for other purposes. Such determination and / or configuration can generally be referred to as, for example, mask optimization, source optimization, and / or projection optimization. Such optimization can be performed independently or in different combinations, etc. One such example is source- mask optimization (SMO), which involves configuring one or more features of a patterning device pattern and one or more features of illumination. Optimization can use, for example, the parametric models described herein to predict values of various parameters, including images, etc.

[0112] In some embodiments, the optimization process of a system can be represented as a cost function. The optimization process can include finding a set of parameters (design variables, process variables, etc.) of the system that minimizes the cost function. The cost function can have any suitable form depending on the goal of the optimization. For example, the cost function can be a weighted root mean square (RMS) of deviations of certain characteristics (evaluation points) of the system from expected values (e.g., ideal values) of these characteristics. The cost function can also be a maximum of these deviations (i.e., worst deviation). The term “evaluation points” should be interpreted broadly to include any characteristics of the system or manufacturing method. Due to the applicability of the implementation of the system and / or method, the design and / or process variables of the system can be limited to a finite range and / or can be interdependent. In the case of a lithographic projection apparatus, the constraints are often associated with physical properties and characteristics of the hardware, such as tunable ranges and / or patterning device manufacturability design rules. The evaluation points can include physical points on a resist image on a substrate, as well as non-physical characteristics, such as, for example, dose and focus.

[0113] In some embodiments, the present systems and methods can include performing one or more empirical models of the operations described herein. The empirical models can predict an output based on a correlation between various inputs (e.g., one or more characteristics of a complex electric field image, one or more characteristics of a design layout, one or more characteristics of a patterning device, one or more characteristics of illumination used in a lithographic process, such as wavelength, etc.).

[0114] As an example, the empirical model can be a parametric model and / or other model. The parametric model can be a machine learning model and / or any other parametric model. In some embodiments, the machine learning model may, for example, be and / or include mathematical equations, algorithms, graphs, charts, networks (e.g., neural networks), and / or other tools and machine learning model components. For example, the machine learning model can be and / or include one or more neural networks having an input layer, an output layer, and one or more intermediate or hidden layers. In some embodiments, the one or more neural networks can be and / or include a deep neural network (e.g., a neural network having one or more intermediate or hidden layers between the input layer and the output layer).

[0115] As an example, the one or more neural networks can be based on a collection of large neural units (or artificial neurons). The one or more neural networks can not strictly mimic the way a biological brain works (e.g., through clusters of large biological neurons connected by axons). Each neural unit of a neural network can be connected to many other neural units of the neural network. Such connections can either reinforce or inhibit their effect on the activation state of the connected neural units. In some embodiments, each individual neural unit can have a summation function that combines the values of all its inputs together. In some embodiments, each connection (or the neural unit itself) can have a threshold function such that a signal must exceed a threshold value before it is allowed to propagate to other neural units. These neural network systems can be self-learning and trained after, rather than explicitly programmed, and can do significantly better in certain problem-solving domains compared to traditional computer programs. In some embodiments, the one or more neural networks can include multiple layers (e.g., where signal paths traverse from front-end layers to back-end layers). In some embodiments, backpropagation techniques can be utilized by the neural network, where forward stimulation is used to reset weights for “front-end” neural units. In some embodiments, stimulation and inhibition of the one or more neural networks can be a more free-flowing, where connections interact in a more chaotic and complex manner. In some embodiments, intermediate layers of the one or more neural networks include one or more convolutional layers, one or more recurrent layers, and / or other layers.

[0116] One or more neural networks can be trained using a training dataset (e.g., ground-based data) (i.e., their parameters are determined). Training data can include a set of training samples. Each sample can be a pair consisting of an input object (typically an image, measurement, tensor, or vector (which may be called a feature tensor or vector)) and a desired output value (also called a management signal). The training algorithm analyzes the training data and adjusts the behavior of the neural network by tweaking its parameters (e.g., the weights of one or more layers) based on the training data. For example, given a set of training data (x1, y1), (x2, y2), ..., (x...), the training data can be trained using a training dataset (e.g., ground-based data). N y N A set of N training samples of the form )} makes x i Let y be the feature tensor / vector of the i-th example and y i In the case of managing signals, the training algorithm seeks a neural network g: X→Y, where X is the input space and Y is the output space. Feature tensors / vectors are n-dimensional tensors / vectors representing the numerical features of some object (e.g., a complex electric field image). The tensor / vector space associated with these vectors is often called the feature or latent space. After training, the neural network can be used to make predictions using new samples.

[0117] As described herein, the method and system comprise a parameterized model (e.g., a machine learning model, such as a neural network) employing a variable encoder-decoder architecture. In the middle (e.g., intermediate layers) of the model (e.g., a neural network), the model formulates a low-dimensional encoding (e.g., a latent space), which encapsulates information within the model's inputs (e.g., complex electric field images and / or other inputs associated with patterns or other features of a semiconductor manufacturing process). The system and method leverage the low dimensionality and compactness of the latent space for direct determination within it.

[0118] With the help of non-restrictive examples, Figure 5 The diagram illustrates a (variable) encoder-decoder architecture 50. The encoder-decoder architecture 50 has an encoding section 52 (encoder) and a decoding section 54 (decoder). Figure 5 In the example shown, the encoder-decoder architecture 50 can output the predicted complex electric field image 56.

[0119] Using another non-restrictive example, Figure 6 The diagram illustrates the encoder-decoder architecture 50 within the neural network 62. The encoder-decoder architecture 50 includes an encoding section 52 and a decoding section 54. Figure 6In some embodiments, x represents an encoder input (e.g., an input complex electric field image and / or extracted features of the input complex electric field image) and x' represents a decoder output (e.g., a predicted output image and / or predicted features of the output image). In some embodiments, x' can represent, for example, an output from an intermediate layer of a neural network (as opposed to a final output of the overall model), and / or other outputs. In some embodiments, x' can represent, for example, an output from an intermediate layer of a neural network (as opposed to a final output of the overall model), and / or other outputs. In some embodiments, x' can represent, for example, an output from an intermediate layer of a neural network (as opposed to a final output of the overall model), and / or other outputs. Figure 6 In some embodiments, z represents a latent space 64 and / or a low-dimensional encoding (tensor / vector). In some embodiments, z is or is related to a latent variable.

[0120] In some embodiments, the low-dimensional encoding z represents one or more features of the input (e.g., a complex electric field image). The one or more features of the input can be considered key or determinative features of the input. A feature can be considered a key or determinative feature of the input due to, for example, its relative predictiveness and / or other characteristics as compared to other features of the desired output. The one or more features (dimensions) represented in the low-dimensional encoding can be predetermined (e.g., by a process designer when creating the machine learning model), determined by a previous layer of the neural network, adjusted by a user via a user interface associated with the systems described herein, and / or can be determined by other methods. In some embodiments, the number of features (dimensions) represented by the low-dimensional encoding can be predetermined (e.g., by a process designer when creating the machine learning model), determined based on outputs from a previous layer of the neural network, adjusted by a user via a user interface associated with the systems described herein, and / or determined by other methods.

[0121] It should be noted that while reference is made throughout this specification to machine learning models, neural networks, and / or encoder-decoder architectures, the machine learning models, neural networks, and encoder-decoder architectures are merely examples, and the operations described herein can be applied to different parameterized models.

[0122] As described above, process information (e.g., images, measurements, process parameters, metrology metrics, etc.) can be used to guide various manufacturing operations. Predicting and / or otherwise determining process information with a relatively low dimensionality of a latent space can be faster, more efficient, require fewer computational resources, and / or have other advantages as compared to previous methods of determining process information.

[0123] The low-dimensional latent space can be advantageously used to predict, for example, complex electric field images. Predicting a complex electric field image requires determining and / or predicting a phase associated with the complex electric field image based on a corresponding amplitude measurement. Typical phase retrieval includes recovering a complex (valued) electric field x e C from a corresponding amplitude (e.g., intensity) measurement y e R of (e.g., of a particular target) according to the following equation n :

[0124] y = |Ax| + e (1)

[0125] where y e R m is a measurement vector, A e C m×n is a measurement matrix, e e R m is a measurement noise (where e represents noise), x is a complex-valued electric field that is perfectly focused at the camera level, A is a defocus operator that propagates the focused electric field to the out-of-focus electric field (as a result of introducing measurement diversity by moving the camera along z), y is a set of amplitude measurements (e.g., the modulus of the electric field at the camera for various defocus positions (potentially contaminated by random noise e)), m is the size of the measured signal, and n is the size of the recovered complex-valued electric field. As an example: n can be the number of pixels of the camera (assuming that one wants to recover the electric field of all pixels and not a subset of the electric field), and m = K*n, where K is a positive integer corresponding to the number of (through-focus) measurements (i.e., the number of different acquisitions taken with the camera).

[0126] Phase recovery is difficult due to the large number of unknowns associated with the complex electric field (e.g., on the order of 10 6 ), nonlinearity, and non-convexity (which combine to make phase recovery a high-dimensional problem). Existing methods for phase recovery include introducing redundancy in the amplitude measurements (e.g., taking more amplitude measurements of x than the dimension of the true signal, such that m > n; typically in the form of an oversampled Fourier transform, a short-time Fourier transform, random Gaussian measurements, an encoded diffraction pattern using a random mask or structured illumination, a wavelet transform, and / or a Gabor frame), or exploiting prior knowledge about the complex electric field image (e.g., exploiting knowledge about the true signal x (prior information) such as sparsity, gradient sparsity (total variation), smoothness, compactness, non-negativity, etc. to reduce the number of unknowns).

[0127] As an example, phase recovery is used to recover an aberrated complex-valued electric field at a camera (e.g., a sensor used to acquire the electric field image). Figure 7 An example camera 70 associated with obtaining a complex electric field image is illustrated (e.g., the above-described variables x, y, and A are related to the camera 70). Figure 7 A pupil 72 (and associated aberrations 74), a target 76, and an illumination 78 are also illustrated. Having access to the full field (both amplitude and phase) allows the aberrations 74 (obtained from a separate calibration step) to be divided out via an inverse transform of the pupil 72, followed by a forward transform of the camera 70, to computationally remove the aberrations 74 associated with the corresponding sensor (e.g., the camera 70). The cleaned-up image at the camera 70 allows for more robust overlay and / or other metrology parameter determination.

[0128] These and other previous approaches require redundant data, make assumptions, are computationally intensive, and / or have other drawbacks. Redundant measurements result in reduced throughput. Incorporating assumptions and / or other previous information into the phase retrieval problem introduces bias since it is not determined directly from the data. Furthermore, the computational cost of resources is high due to the large dimensionality of x, making iterative phase retrieval operations very time consuming.

[0129] Advantageously, the present systems and methods solve the phase retrieval problem using the parametric model described herein. The parametric model is configured to predict electric field images, predict metrology metrics (e.g., instead of or in addition to the metrology equipment described above), and / or predict other information. The parametric model is configured to significantly reduce the dimensionality of the phase retrieval problem, which can then be solved in a few dimensions using simple gradient-based optimization schemes and / or using other methods.

[0130] Figure 8 An overview method 80 illustrates the operations of the present method for predicting electric field images with a parametric model. At operation 82, a parametric model is trained. At operation 84, a latent space representation of an electric field image is determined for a given input. At operation 86, an electric field image is predicted with the parametric model. At operation 88, one or more metrology metrics are determined based on the latent space representation of the electric field image, the predicted electric field image, and / or other information. The operations of method 80 presented below are intended to be illustrative. In some embodiments, method 80 can be accomplished with one or more additional operations not described, and / or without one or more of the Figure 8 operations discussed. For example, method 80 can not require determining one or more metrology metrics. Additionally, the order in which the operations of method 80 are illustrated in FIG. 8 and described below is not intended to be limiting. In some embodiments, one or more portions of method 80 can be implemented (e.g., by simulation, modeling, etc.) in one or more processing devices (e.g., one or more processors). The one or more processing devices can include one or more devices that perform some or all of the operations of method 80 in response to instructions stored electronically on an electronic storage medium. The one or more processing devices can include one or more devices configured via hardware, firmware, and / or software designed to perform, for example, one or more of the operations of method 80.

[0131] At operation 82, a parameterized model is trained. The parameterized model can be a machine learning model that includes a neural network, an encoder-decoder architecture, and / or other components. The encoder-decoder architecture can be a variable encoder-decoder architecture and / or other architectures. In some embodiments, the parameterized model can be a variable deep neural network (DNN) that includes an encoder, a decoder, and / or other components. In some embodiments, such or similar structure of the parameterized model facilitates including uncertainty in predictions according to the parameterized model and / or has other advantages. For example, because the parameterized model is variable (as described herein), the parameterized model is able to output multiple predictions for a single input. This in turn means that an uncertainty metric can be determined for those implementations, e.g., by computing a standard deviation of those implementations.

[0132] Operation 82 includes training the variable encoder-decoder architecture with the probabilistic latent space, which produces implementations in the output space. In some embodiments, the latent space includes low-dimensional encodings (e.g., as described above). The parameterized model is trained with existing data (e.g., measured and / or simulated complex field images with amplitudes and phases, corresponding metrology metrics, etc.) and / or other information. In some embodiments, the parameterized model can model (or functionally model) one or more of the pupil, the aberration, the target, or the illumination as shown in FIG. 1. Figure 7 In some embodiments, the parameterized model is trained with existing data (e.g., measured and / or simulated complex field images with amplitudes and phases, corresponding metrology metrics, etc.) and / or other information. In some embodiments, the parameterized model can model (or functionally model) one or more of the pupil, the aberration, the target, or the illumination as shown in FIG. 1.

[0133] When the parameterized model is trained, the latent space forms a compressed continuous representation of, e.g., encoded images, which facilitates performance of various operations in the latent space. Advantageously, the latent space is low-dimensional (e.g., compared to the image space). Various operations can include, e.g., determining a latent space representation of a complex field image (as described below) and / or other operations. For example, at least due to the low-dimensional value of the latent space (relative to the image space), this can be computationally cheaper in the latent space compared to the image space.

[0134] In some embodiments, the dimensional data in the latent space is encoded by an encoder of the encoder-decoder architecture. In some embodiments, the predictions and / or other outputs from the parametric model are produced by a decoder of the encoder-decoder architecture. As described herein, the encoder includes a portion of the parametric model configured to transform the model input into the dimensional data in the latent space, and the decoder includes a different portion of the parametric model configured to transform the dimensional data in the latent space into the output realization. The transformations can include, for example, encoding, decoding, projecting, mapping, etc. By way of non-limiting practical example, in some embodiments, the model input can be complex electric field images and / or other information associated with a semiconductor device manufacturing process. The dimensional data in the latent space can include multi-dimensional tensors and / or vectors associated with the model input. The output realization can include predicted complex electric field images, metrology indicators, and / or other information. In some embodiments, the predictions include decoding the multi-dimensional tensors and / or vectors of the dimensional data into the output realization with one or more layers and / or one or more nodes of a neural network.

[0135] The parametric model is trained with (known, e.g., measured and / or modeled) electric field images, corresponding amplitude measurements, metrology indicators, and / or other information. The electric field images and corresponding amplitude measurements and / or metrology indicators can form, for example, input object / output value training pairs (as described herein). The electric field images can be complex electric field images and / or other electric field images. The complex electric field images can include amplitudes, phases, and / or other information. The amplitudes can be, for example, absolute values of complex-valued electric fields. The amplitudes can include, for example, image intensities, and / or other parameters that are functions of the amplitudes. The phases can include, for example, angles of complex-valued electric field vectors in a complex plane. Thus, the parametric model is trained with corresponding amplitude and / or phase information and / or other information of the electric field images.

[0136] In some embodiments, operation 82 includes training the parametric model with a set of complex electric field images. In some embodiments, the set of complex electric field images is produced during a through-focus measurement and / or in other ways. For example, in some embodiments, the set of complex electric field images is produced by adjusting a position of a camera (e.g., camera 70 shown in Figure 7 ) to obtain images that are up and / or down in the “z” direction (e.g., as shown in Figure 7 ).

[0137] In some embodiments, operation 82 includes encoding higher dimensional data associated with the electric field images into dimensional data in a latent space with an encoder. In some embodiments, operation 82 includes encoding complex electric field images, measured amplitudes, metrology indicators, and / or other information in a training set into dimensional data in a latent space, and transforming the dimensional data in the latent space into de-noised versions of the complex electric field images in the training set, predicted metrology indicators, and / or performing other operations to facilitate verification of the training.

[0138] In some embodiments, operation 82 includes iteratively providing additional complex electric field images as input to the parameterized model. The additional complex electric field images are determined based on how well the de-noised versions of the complex electric field images match the complex electric field images in the training set. For example, during training, if the de-noised (or predicted) complex electric field images match the corresponding input images, then fewer or no additional complex electric field images can need to be provided for training. Conversely, if the de-noised (or predicted) complex electric field images do not match or do not match well the corresponding input images, then several additional complex electric field images can need to be provided for training. This process can be repeated any number of times until the parameterized model is sufficiently trained.

[0139] Figure 9 An example of a parameterized model 90 is illustrated. The parameterized model 90 can be, for example, a variational autoencoder. The parameterized model 90 uses an encoder or encoder network 91 to encode an input (x) 92 (e.g., a complex electric field image with amplitude and phase, measured amplitude, metrology indicator, and / or other input) into a continuous representation (also referred to as a latent space (z) 93), and uses a decoder or decoder network 94 to produce a corresponding output (x’) 95 (a predicted image with predicted amplitude and phase, and / or other output such as a predicted metrology indicator). For example, the encoder or encoder network 91 (E(x)) can be configured to transform an input image x e C n into a lower dimensional, real-valued continuous latent vector z e R k (k « n). C can represent a space of complex-valued numbers, R can represent a space of real-valued numbers, k can represent a dimensionality of the latent space, and n can represent a dimensionality of the input space. The encoder or encoder network 91 can also model uncertainty in such latent vectors. The decoder or decoder network 94 D(z) can be configured to transform the continuous latent vector back into a copy of the input image x’.

[0140] As Figure 9As shown in FIG. 1, in some embodiments, the parametric model 90 includes a portion 96 configured to determine one or more metrology indicators (v) 97 (which is further described below). Training of both the networks 91, 94 and the portion 96 uses input object / output value pairs (e.g., electric field images and corresponding amplitude measurements and / or metrology indicators as described above).

[0141] The parametric model 90 learns to encode complex electric field images x in a low-dimensional latent space. Given a low-dimensional input vector z, in the latent space, a decoder or decoder network 94 can produce a new sample D(z). The variable nature of the parametric model 90 enables prediction of uncertainty, which in turn can be used for uncertainty-guided training (active learning). This can include providing more training examples of complex electric field images to the parametric model 90 that produces greater uncertainty. In some embodiments, a loss function used to train the parametric model 90 can be a typical (L1, L2, etc.) norm or similar measure of similarity learned directly from data, e.g., using a discriminator network.

[0142] In some embodiments, the parametric model 90 can be sufficiently trained. In these embodiments, the operation 82 can be configured to improve the parametric model 90 by training the parametric model 90 with new and / or additional input object / output value pairs. In some embodiments, the parametric model 90 can be partially trained or untrained. In these embodiments, the operation 82 can facilitate at least partial training of the parametric model 90, including formation of the latent space 93.

[0143] Returning to Figure 8 At operation 84, a latent space representation of an (e.g., complex) electric field image is determined for a given input to the parametric model. The given input includes measured amplitudes associated with the electric field image, and / or other information. The latent space representation of the electric field image includes continuous latent tensors, vectors, and / or other latent space representations. The latent space representation is determined based on dimensional data and / or other information in the latent space of the parametric model. In some embodiments, determining the latent space representation of the electric field image includes minimizing a function constrained by a set of electric field images that can be predicted by the parametric model based on the dimensional data in the latent space and the given input.

[0144] For example, to ultimately predict a complex electric field image (which includes both amplitude and phase) from only amplitude measurements y, determining the latent space representation of the electric field image includes minimizing the following objective function:

[0145]

[0146] where Range(D) is the set of latent images that can be produced by the pre-trained decoder D. It is noted that x' (described above) represents any predicted image, and x * The particular image that minimizes the norm in equation (2) is the image that contains the amplitudes and phases that are being attempted to recover. The above minimization problem can be equivalently formulated in the lower-dimensional latent space representation as follows:

[0147]

[0148] The variable z * is or represents a latent space representation of the electric field image. The continuous representation of the latent space facilitates efficient guiding of the search for the optimal z * based on gradients, for example.

[0149] At operation 86, the electric field image is predicted with the parametric model. The electric field image is predicted based on the latent space representation of the electric field image and / or other information. In some embodiments, predicting the electric field image based on the latent space representation of the electric field image (e.g., operation 86) includes passing the latent space representation of the electric field image through a decoder of an encoder-decoder architecture.

[0150] For example, as shown in Figure 9 , the decoder or decoder network 94 (or a portion of the decoder network 94) maps, projects, decodes, or otherwise transforms the low-dimensional latent space representation of the electric field image in the latent space 93 to the output 95. In some embodiments, the output 95 is predicted based on the dimensional data in the latent space 93, the low-dimensional latent space representation of the complex electric field image in the latent space 93, and / or other information. The output 95 in this example is the predicted complex electric field image. In other words, once z * is found, the complex electric field image is predicted by the solution (to z * ) by forward pass through the decoder, such that

[0151] x * = D(z * ). (4)

[0152] As the optimization is performed on the lower-dimensional latent space variable z, the computational cost of resources is substantially reduced compared to previous approaches. Other advantages are expected.

[0153] Returning to Figure 8At operation 88, one or more metrology metrics are determined based on the predicted electric field images, the low-dimensional latent space representation of the (complex) electric field images, the dimensional data in the latent space, and / or other information. The determination can include prediction and / or other determination. For example, in some embodiments, a metrology metric such as overlay can be predicted and / or otherwise determined based on an average of the intensity range of one or more predicted complex electric field images over a region of interest. As another example, in some embodiments, a metrology metric can be predicted and / or otherwise determined based on the latent space representation of the electric field images. In some embodiments, determining one or more metrology metrics based on the latent space representation of the electric field images includes providing the latent space representation of the electric field images to a regression network and / or other predictor included in or separate from the parametric model. In some embodiments, other low-dimensional data in the latent space can be provided to such a regression network for prediction and / or other determination of one or more metrology metrics.

[0154] For example, as shown in Figure 9 , a portion 96 of the parametric model 90 (including a regression network R(z) in this example) is configured to determine one or more metrology metrics 97 based on the low-dimensional data in the latent space 93. In the example shown in Figure 9 , the one or more metrology metrics include overlay (v). The overlay can be determined based on the eigenvectors in the latent space (e.g., the latent space representation of the complex electric field images) and / or other information. For example, the overlay can be determined by means of z * The metrology metrics are predicted and / or otherwise determined by forward pass of the regression network (e.g., portion 96) such that:

[0155] v * = R(z * ). (5)

[0156] It is noted that v (described herein) can be any predicted metrology metric (such as overlay), and v * is a particular metrology metric corresponding to the image x * .

[0157] As Figure 9As shown in FIG. 1, in some embodiments, the parametric model 90 can be configured such that additional information inherently embedded in the initial signal x (e.g., input 92) can be extracted via an additional step, such as, for example, feeding it to a regression network (e.g., portion 96). In this way, metrology information can be extracted simultaneously with phase retrieval. Here, an example metrology indicator is overlay, but the method can be extended to any additional scalar parameter embedded in x (e.g., one or more metrology indicators and / or other indicators). In some embodiments, the one or more metrology indicators include one or more of: overlay, critical dimension, reconstruction of a three-dimensional profile of a feature of a substrate, dose or focus of a lithography apparatus in imprinting a feature of a substrate with the lithography apparatus, alignment, and / or other metrology indicators.

[0158] By way of non-limiting practical example, the present systems, methods, apparatus, etc. described herein can be used to determine one or more metrology indicators of a semiconductor manufacturing process (e.g., overlay as described above) using a parametric model (e.g., 90 as shown in FIG. 1) that includes one or more machine learning algorithms (e.g., encoder network 91, decoder network 94, portion 96, etc. as shown in FIG. 1). Figure 9 Figure 9 By way of non-limiting practical example, the present systems, methods, apparatus, etc. described herein can be used to determine one or more metrology indicators of a semiconductor manufacturing process (e.g., overlay as described above) using a parametric model (e.g., 90 as shown in FIG. 1) that includes one or more machine learning algorithms (e.g., encoder network 91, decoder network 94, portion 96, etc. as shown in FIG. 1). Figure 10 Figure 9 Figure 9 Figure 10 ​​​​The recommendation can be displayed on a display 112 (shown in FIG. 1 and described above) and / or another display (shown in FIG. 1 and described above). Such an operation can be performed by, for example, part 96 of parameterized model 90 and / or other components described herein.

[0159] Returning to Figure 8 In some embodiments, a metrology apparatus can perform operation 88 and / or some or all of the other operations described herein. For example, in some embodiments, a metrology apparatus can be configured to determine one or more metrology indicators of a semiconductor manufacturing process. The apparatus can include one or more processors configured to: determine, based on dimensional data in a latent space of a parameterized model, a latent space representation of an electric field image for a given input; predict, with the parameterized model, the electric field image based on the latent space representation of the electric field image; and determine, based on the predicted electric field image, one or more metrology indicators of the semiconductor manufacturing process. In some embodiments, the metrology apparatus can be included in a lithography cell (e.g., as described herein) or the metrology apparatus can be standalone.

[0160] In some embodiments, operation 88 includes determining, based on the one or more predicted electric field images, the one or more metrology indicators, and / or other information, an adjustment to a semiconductor manufacturing process parameter used to pattern a substrate geometry, the patterning of the substrate geometry being part of the semiconductor manufacturing process. In some embodiments, the one or more semiconductor manufacturing process parameters include one or more of a mask design, a pupil shape, a dose, a focus, and / or other parameters. In some embodiments, the one or more semiconductor manufacturing process parameters include a mask design, such that the mask design is changed from a first mask design to a second mask design. Other examples are contemplated related to several different aspects of an integrated circuit manufacturing process and / or other processes.

[0161] The principles described herein (e.g., utilizing the relatively lower dimensionality of a latent space in a trained parameterized model to predict and / or otherwise determine process information) can have a number of additional applications (e.g., in addition to and / or instead of the complex electric field prediction application shown in FIG. 1 and described above). For example, the present systems and methods can be used to reconcile data from different process sensors and / or tools, which can be different even for the same measurement or imaging target. Figure 8

[0162] ​The present methods and systems are configured to determine a latent space representation of a given input to a parametric model in a latent space of the parametric model in which there are fewer dimensions of data to be analyzed than the number of dimensions in raw data from different sensors and / or tools. The latent space representation is transformed based on a reference latent space representation of the given input into a transformed latent space representation of the given input; and process information is predicted based on the transformed latent space representation of the given input. For example, the given input can be associated with a target and received from one of a plurality of different sensors and / or tools configured to produce the given input. The transformation and prediction are configured such that the predicted and / or otherwise determined process information (e.g., images, measurements, process parameters, metrology indicators, etc.) from the parametric model of the target is the same regardless of which of the sensors and / or tools produced the given input.

[0163] In some embodiments, the given input is associated with a target and received from one of a plurality of target characterization devices configured to produce the given input. The target characterization devices can include various sensors and / or tools configured to produce data about the target. The data can include images, values of various indicators, and / or other information. In some embodiments, the given input includes one or more of an input image, an input process measurement and / or a series of process measurements and / or other information. In some embodiments, the latent space representation can be a tensor, a vector, and / or other latent space representation (e.g., something having fewer dimensions than the number of dimensions associated with the given input).

[0164] In some embodiments, the transformation includes one or more mathematical calculations performed on the latent space representation of the given input. In some embodiments, the transformation is performed in the latent space. The mathematical calculations can include: a transformation (e.g., in x, y, z, or other equivalent dimensions in the latent space); a (high order) polynomial modeling that covers all dimensions (or a subset of the dimensions), such as scaling, rotation, etc.; and / or other operations.

[0165] In some embodiments, the reference latent space representation includes a weighted combination of previously received inputs to the parametric model, an average of latent space representations of previously received inputs to the parametric model, a latent space representation of an input from a particular target characterization device configured to produce the given input, and / or other latent space representations. For example, a user can have a particular sensor and / or other tool (e.g., a “golden tool”) that the user believes consistently provides the most accurate and / or otherwise optimal data for a target. The reference latent space representation can be an encoding of data (e.g., images and / or other data) from that sensor and / or other tool.

[0166] By way of non-limiting example, a given input can include an image (e.g., any image associated with or produced during semiconductor manufacturing). The image can be encoded by an encoder architecture of the parametric model into a low-dimensional vector representing the image in a latent space. Various dimensions of the vector can be transformed in the latent space such that the vector representing the input image matches a reference latent space representation (as one example, other transformations are also possible). The transformed vector can then be decoded into predicted process information.

[0167] In some embodiments, the predicted process information includes one or more of a predicted image, a predicted process measurement (e.g., a metric value), and / or other information. In some embodiments, the predicted process measurement includes one or more of a metrology metric, an xyz position, a dimension, an electric field, a wavelength, an illumination and / or detection pupil, a bandwidth, an illumination and / or detection polarization angle, an illumination and / or detection phase delay angle, and / or other process measurement.

[0168] In this example, the parametric model can have a variable encoder-decoder architecture with a probabilistic latent space that produces a realization in an output space. In some embodiments, the latent space includes low-dimensional encoding (e.g., as described above). The parametric model is trained with existing data (e.g., measurements and / or simulation data (e.g., images, values of various metrics, etc.) from different target characterization devices (e.g., sensors, tools, etc.) for the same target, etc.) and / or other information. The training data can include, for example, different data from a wide range of different target characterization devices but for the same target. As described above, the dimensional data in the latent space is encoded by an encoder of the encoder-decoder architecture. In some embodiments, predictions and / or other outputs from the parametric model are produced by a decoder of the encoder-decoder architecture.

[0169] Further, the training of the parametric model can be extended for (matching) characterization devices of any type of measurement (field, pupil, position, height, etc.). The training of the parametric model can also be extended by adding target characterization device constants (e.g., machine constants from calibration and / or hardware specifications) to the training data. In such embodiments, the latent space transformation can potentially be used to determine root causes of biases between characterization devices. For example, systematic shifts in positioning (such as xyz positions of a stage and / or optical components) can be transformed into overlay errors.

[0170] As another example, the present methods and systems can be configured to determine, based on dimensional data in a latent space of a parametric model (again, in which the latent space has fewer dimensions of data to be analyzed than the number of dimensions in the raw data from different experimental iterations), a latent space representation of an optimal set of process parameters associated with a given input to the parametric model, and predict process information based on the latent space representation of the optimal set of process conditions.

[0171] In some embodiments, the given input includes one or more of defined design parameters of the target, defined metrology measurement recipe parameters, and / or other defined information. The defined design parameters and / or defined metrology measurement recipe parameters can be unchangeable due to, for example, design requirements (e.g., specific dimensions, shapes, locations, and / or other requirements for one or more features of the target design), machine capabilities, materials used during the manufacturing process, scientific limitations (e.g., laws of physics, chemistry, etc.), required relationships between elements of the target design, required steps in the manufacturing recipe, or other inputs.

[0172] In some embodiments, the predicted process information includes one or more of design parameters of the target, metrology measurement recipe parameters, and / or other parameters. In some embodiments, the design parameters include one or more of critical dimensions, pitches, segmented pitches, line geometries, contact geometries, hole geometries, and / or other information associated with the target. In some embodiments, the metrology measurement recipe parameters include one or more of wavelengths, bandwidths, apertures, illumination and / or detection pupils, bandwidth, illumination and / or detection polarization angles, illumination and / or detection phase delay angles, dose, and / or other recipe parameters for measuring the target.

[0173] In some embodiments, the optimal set of process parameters defines optimal measurement conditions for measuring a metrology target of a metrology target. The optimal measurement conditions can be, for example, the ideal measurement conditions for measuring a particular metrology target, and / or other measurement conditions. In some embodiments, the optimal set of process parameters includes one or more of intensity, contrast, edge response, diffraction efficiency, overlay sensitivity, and / or other process parameters. By way of non-limiting example, each process parameter can have an ideal working range: intensity: in the middle of the measurement camera range; contrast: high contrast to surrounding features; edge response: as low overshoot at the edge as possible; diffraction efficiency: as high as possible (but related to intensity); overlay sensitivity: above a minimum threshold, e.g. absolute sensitivity > 0.2 (on a scale of 0 to 2), and also related to intensity. Given these constraints, it can be determined how to project these parameters back onto the measurement conditions. This will give a probability distribution on the measurement conditions, where the probability distribution optimally matches the optimal process parameters.

[0174] In other words, a user can input a defined target design or metrology measurement recipe parameter into a parameterized model. The parameterized model can be configured such that there is an optimal set of process parameters that define the input optimal measurement conditions. The parameterized model can then predict (determine) one or more (e.g. recommended) design parameters and / or metrology recipe parameters of the input based on the input, the optimal set of process parameters, and / or other information. In other words, the model is configured to output the optimal possible target design (e.g. parameters defining such optimal possible target design) and / or optimal possible measurement settings (e.g. parameters) of a metrology apparatus given the constraints input by the user into the model (based on the model having been trained to know what the optimal set of process parameters are in relation to the input).

[0175] In this example, the parameterized model can have a variable encoder-decoder architecture with a probabilistic latent space that produces realizations in an output space. In some embodiments, the latent space includes low-dimensional encodings (e.g., as described above). The parameterized model is trained with existing data (e.g., measured and / or simulated target designs, corresponding known measurement conditions and / or metrology metrics, etc.) and / or other information. As described above, the dimensional data in the latent space is encoded by an encoder of the encoder-decoder architecture. In some embodiments, predictions and / or other outputs from the parameterized model are produced by a decoder of the encoder-decoder architecture. It should be noted that the parameterized model can be trained for predicting and / or otherwise determining any target design and / or metrology measurement recipe parameters. For example, the parameterized model can be used for overlay target designs and / or metrology measurement recipe parameters, post-etch target images and / or metrology measurement recipe parameters, image-based overlay target designs and / or metrology measurement recipe parameters, focus target designs and / or metrology measurement recipe parameters, alignment target designs and / or metrology measurement recipe parameters, and / or other target and / or metrology measurement recipe parameters.

[0176] As described above, one parameterized model can be trained to predict complex electric field images, transform data from different sensors and / or tools so that data for the same target matches, and determine optimal target designs and / or manufacturing recipes; or these different operations can be performed by different parameterized models. The different applications (predicting complex electric field images, transforming data from different sensors and / or tools so that data for the same target from different sensors and / or tools still matches, and determining optimal target designs and / or manufacturing recipes) can be used together, or they can be used separately.

[0177] As an example, the present systems and methods can be configured to predict process information such as matching data for the same target from different sensors and / or tools and optimal target designs and / or manufacturing recipes (e.g., a combination of two of the three applications described herein) with a single parameterized model. In other words, the parameterized model can be configured to jointly optimize target design and metrology recipe parameters. This can include determining a latent space representation of a given input to the parameterized model in a latent space of the parameterized model (e.g., as described above). The latent space representation of the given input can be transformed into a transformed latent space representation of the given input based on a reference latent space representation of the given input (e.g., as described above).

[0178] A latent space representation of an optimal set of process parameters associated with the given input can be determined based on the transformed latent space representation of the given input (e.g., as described above), and process information can be predicted based on the transformed latent space representation of the given input and the latent space representation of the optimal set of process parameters associated with the given input.

[0179] As another example, the present systems and methods can be configured to predict process information such as complex electric field images, matching data from different sensors and / or tools and optimal target design and / or manufacturing recipes for the same target (e.g., a combination of all three applications described herein) to a single parametric model. This can include determining a latent space representation of a given input to the parametric model in a latent space of the parametric model (e.g., as described herein); transforming the latent space representation of the given input to a transformed latent space representation of the given input based on the reference latent space representation of the given input; determining a latent space representation of an electric field image of the given input based on the transformed latent space representation of the given input; determining a latent space representation of an optimal set of process parameters associated with the given input based on the transformed latent space representation; predicting process information based on the transformed latent space representation of the given input, the latent space representation of the electric field image, and the latent space representation of the optimal set of process parameters associated with the given input; and / or other operations.

[0180] It is noted that while the description herein often refers to a (single) latent space, such should not be considered limiting. The principles described herein can be applied with and / or to any number of latent spaces. For example, the systems, methods, (metrology) apparatuses, non-transitory computer-readable media, etc. described herein can be configured such that determining a latent space representation of an electric field image is performed for a given input to a parametric model based on dimensional data in one or more latent spaces of the parametric model (or one or more parametric models). The electric field image is determined based on the latent space representation of the electric field image and / or other information. As described above, in some embodiments, the electric field image includes a complex electric field image having an amplitude and a phase, and the given input includes a measured amplitude associated with the complex electric field image. Determining the latent space representation of the electric field image includes minimizing a function constrained by a set of electric field images that can be predicted by the parametric model based on the dimensional data in the one or more latent spaces and the given input.

[0181] The one or more latent spaces can be used serially (e.g., to analyze data and / or make a first prediction, followed by a second prediction, etc.), in parallel (e.g., to analyze data and / or make predictions simultaneously), and / or in other ways. Advantageously, separate latent spaces of a parametric model can be more robust than a single latent space. For example, separate latent spaces can focus on particular properties of a data set, such as one latent space for recovering phase, another latent space for classification based on measurement parameters, etc. A combined latent space can be configured to capture possibilities, while in the case of separate latent spaces, each individual latent space can be configured (e.g., trained to) focus on particular topics and / or aspects of a data set. Individual latent spaces can potentially be simpler, but better at capturing information (e.g., when set up accordingly).

[0182] In some embodiments, the one or more latent spaces can include at least two latent spaces, a plurality of latent spaces, and / or other numbers of latent spaces, with individual latent spaces corresponding to different mechanisms of the parametric model. The different mechanisms of the parametric model can include an encoding mechanism (e.g., 91 shown in FIG. 4), a decoding mechanism (e.g., 94 shown in FIG. 4), a complex electric field parameter determination mechanism (e.g., a mechanism that determines input 92 and / or other features of the input electric field image shown in FIG. 4), a phase recovery mechanism (e.g., similar to and / or the same as 94 shown in FIG. 4), and / or other mechanisms. In some embodiments, the different mechanisms can correspond to different operations made by the parametric model (or one or more parametric models). By way of non-limiting example, in some embodiments, a plurality of latent spaces can be used in parallel, such as one latent space for image encoding and / or decoding, another latent space for predicting aberrations, another latent space for fitting scheme settings (e.g., predicting or recommending process set points), etc. Individual latent spaces corresponding to different mechanisms can be more robust than a single latent space associated with multiple mechanisms. Figure 9 Figure 9 Figure 9 Figure 9 Figure 9

[0183] ​​​​​In some embodiments, separate latent spaces can be associated with different independent parameters and corresponding dimensional data to a given input to the parametric model. Separate latent spaces corresponding to different independent parameters can also be more robust than a single latent space associated with multiple parameters. For example, in some embodiments, the present systems and methods can include a first latent space that phase recovery (as described herein), and a separate second latent space that processes tool-related (i.e., due to optical differences) interference. The first latent space can be configured (e.g., trained) to perform phase recovery, and (unrelated to such first latent space) the second latent space can be configured (e.g., trained) to process image differences caused by tool-specific properties. It is noted that this is merely one possible example, and is not intended to be limiting. Numerous other possible examples are contemplated.

[0184] Figure 10 FIG. 1 is a block diagram that illustrates a computer system 100 that can perform and / or assist in implementing the methods, processes, systems, or devices disclosed herein. Computer system 100 includes a bus 102 or other communication mechanism for communicating information, and a processor 104 (or multiple processors, not shown) coupled with bus 102 for processing information. Computer system 100 also includes a main memory, such as a random access memory (RAM) or other dynamic storage device 106, coupled to bus 102 for storing information and instructions to be executed by processor 104. Main memory 106 also can be used for storing temporary variables or other intermediate information during execution of instructions to be executed by processor 104. Computer system 100 further includes a read only memory (ROM) 108 or other static storage device coupled to bus 102 for storing static information and instructions for processor 104. A storage device 110, such as a magnetic disk or optical disk, is provided and coupled to bus 102 for storing information and instructions.

[0185] Computer system 100 can be coupled via bus 102 to a display 112, such as a cathode ray tube (CRT) or flat panel or touch panel display for displaying information to a computer user. An input device 114, including alphanumeric and other keys, is coupled to bus 102 for communicating information and command selections to processor 104. Another type of user input device is cursor control 116, such as a mouse, a trackball, or cursor direction keys for communicating direction information and command selections to processor 104 and for

[0186] According to one embodiment, portions of the methods described herein, or variations thereof, can be performed by computer system 100 in response to processor 104 executing one or more sequences of one or more instructions contained in main memory 106. Such instructions can be read into main memory 106 from another computer-readable medium, such as storage device 110. Execution of the sequences of instructions contained in main memory 106 causes processor 104 to perform the process steps described herein. One or more processors in a multi-processing arrangement can also be employed to execute the sequences of instructions contained in main memory 106. In alternative embodiments, hard-wired circuitry can be used in place of or in combination with software instructions. Thus, the description herein is not limited to any specific combination of hardware circuitry and software.

[0187] The term "computer readable medium" as used herein refers to any medium that participates in providing instructions to processor 104 for execution. Such a medium can take many forms, including but not limited to, non-volatile media, volatile media, and transmission media. Non-volatile media includes, for example, optical or magnetic disks, such as storage device 110. Volatile media includes

[0188] Various forms of computer readable media can be involved in carrying one or more sequences of one or more instructions to processor 104 for execution. For example, the instructions can initially be carried on a magnetic disk of a remote computer. The remote computer can load the instructions into its dynamic memory and send the instructions over a telephone line using a modem. A modem local to computer system 100 can receive the data on the telephone line and use an infra-red transmitter to convert the data to an infra-red signal. An infra-red detector coupled to bus 102 can receive the data carried in the infra-red signal and place the data on bus 102. Bus 102 carries the data to main memory 106, from which processor 104 retrieves and executes the instructions. The instructions received by main memory 106 can optionally be stored on storage device 110 either before or after execution by processor 104.

[0189] Computer system 100 can also include a communication interface 118 coupled to bus 102. Communication interface 118 provides a two-way data communication coupling to a network link 120 that is connected to a local network 122. For example, communication interface 118 can be an integrated services digital network (ISDN) card or a modem to provide a data communication connection to a corresponding type of telephone line. As another example, communication interface 118 can be a local area network (LAN) card to provide a data communication connection to a compatible LAN. Wireless links can also be implemented. In any such implementation, communication interface 118 sends and receives electrical, electromagnetic or optical signals that carry digital data streams representing various types of information.

[0190] Network link 120 typically provides data communication through one or more networks to other data devices. For example, network link 120 can provide a connection to a

[0191] Computer system 100 can send messages and receive data, including program code, through the network(s), network link(s) 120, and communication interface(s) 118. In the Internet example, a server 130 might transmit a requested code for an application program through the Internet 128, the ISP 126, the local network 122 and the communication interface 118. One such downloaded application provides all or part of the functionality described herein. The received code can be executed by processor 104 as it is received, and / or stored in storage device 110, or other non-volatile storage for later execution. In this manner, computer system 100 can obtain application code in the form of a carrier wave.

[0192] Figure 11 For Figure 1 Detailed view of an alternative design of the lithographic projection apparatus LA shown in Fig. 1. Figure 1 Involving DUV radiation, due to the use of lenses and the use of transparent mask plates, Figure 11 Involving a lithographic apparatus using EUV radiation, due to the use of mirrors and reflective mask plates. As Figure 11As shown in the figure, the lithographic projection apparatus can include a source SO, an illumination system IL, and a projection system PS. The source SO is configured to make available (e.g., generate) a vacuum environment in an enclosure 220 of the source SO. An EUV (e.g., radiation emitting) plasma 210 can be formed by a discharge, for example. EUV radiation can be produced by a gas or vapor, for example Xe gas, Li vapor or Sn vapor, in which the plasma 210 is created to emit radiation in the EUV range of the electromagnetic spectrum. The plasma 210 is created by, for example, a discharge causing at least partially ionized plasma. For high efficiency of radiation generation, a fractional pressure of, for example, 10 Pa of Xe, Li, Sn vapor or any other suitable gas or vapor can be required. In some embodiments, a plasma of excited tin (Sn) is provided to generate EUV radiation.

[0193] Radiation emitted by the plasma 210 is passed from a source chamber 211 into a collector chamber 212 via an optionally present gas barrier or contaminant trap 230 (also referred to as contaminant barrier or foil trap in some instances) located in or behind an opening in the source chamber 211. The contaminant trap 230 can include a channel structure. The chamber 211 can include a radiation collector CO, which can be for example a grazing incidence collector. The radiation collector CO has an upstream radiation collector side 251 and a downstream radiation collector side 252. Radiation that traverses the collector CO can be reflected off a grating filter 240 to be focused in a virtual source point IF along an optical axis indicated by the line "O". The virtual source point IF is commonly referred to as the intermediate focus, and the source SO is arranged so that the intermediate focus IF is located at or near the opening 221 in the enclosure 220. The virtual source point IF is an image of the radiation emitting plasma 210.

[0194] Subsequently, the radiation traverses an illumination system IL which can include a facetted field mirror device 22 and a facetted pupil mirror device 24 arranged to provide a desired angular distribution of the radiation beam 21, at the patterning device MA, and to provide a desired uniformity of radiation intensity at the patterning device MA. Upon reflection of the beam of radiation 21 at the patterning device MA, held by the support structure (bench) T, a patterned beam 26 is formed and the patterned beam 26 is imaged by the projection system PS via reflective elements 28, 30 onto a substrate W held by the substrate table WT. More elements than shown can generally be present in illumination optics unit IL and projection system PS. Depending on the type of lithography apparatus, the grating filter 240 can or can not be present. Also, more mirrors can be present than the ones shown in each of the figures. For example, in a projection system PS, additional reflective elements 25 can be present, for example, to correct for aberrations. Figure 11 The reflective elements shown in the figure can be more than 1 to 6 additional reflective elements.

[0195] AsFigure 11 The collector optic CO illustrated in the middle is depicted as a nested collector with grazing incidence reflectors 253, 254 and 255, merely as an example of a collector (or collector mirror). The grazing incidence reflectors 253, 254 and 255 are disposed axially symmetric around the optical axis O, and this type of collector optic CO can be used in combination with a discharge produced plasma source, often referred to as a DPP source.

[0196] Further embodiments are disclosed in the following numbered list of aspects:

[0197] 1. A method for determining one or more metrology indicators for a semiconductor manufacturing process, the method comprising:

[0198] determining a latent space representation of an electric field image for a given input based on dimensional data in a latent space of a parametric model;

[0199] predicting the electric field image based on the latent space representation of the electric field image with the parametric model; and

[0200] determining the one or more metrology indicators for the semiconductor manufacturing process based on the predicted electric field image.

[0201] 2. The method according to aspect 1, wherein the electric field image comprises a complex electric field image having an amplitude and a phase.

[0202] 3. The method according to any one of aspects 1 to 2, wherein the determined one or more metrology indicators comprise one or more of: an overlay, a critical dimension, a reconstruction of a three-dimensional profile of a feature of a substrate, or a dose or focus of a lithographic apparatus in imprinting a feature of a substrate with the lithographic apparatus.

[0203] 4. The method according to any one of aspects 1 to 3, wherein the electric field image comprises a complex electric field image, and wherein the given input comprises a measured amplitude associated with the complex electric field image.

[0204] 5. The method according to aspect 4, wherein the amplitude comprises an intensity.

[0205] 6. The method according to any one of aspects 1 to 5, further comprising adjusting one or more semiconductor manufacturing process parameters based on the determined one or more metrology indicators.

[0206] 7. A method for predicting an electric field image with a parametric model, the method comprising:

[0207] determining, based on dimensional data in a latent space of the parametric model, a latent space representation of an electric field image for a given input to the parametric model; and

[0208] predicting the electric field image based on the latent space representation of the electric field image.

[0209] 8. The method of aspect 7, wherein the electric field image comprises a complex electric field image having an amplitude and a phase.

[0210] 9. The method of aspect 8, wherein the given input comprises a measured amplitude associated with the complex electric field image.

[0211] 10. The method of aspect 9, wherein the amplitude comprises an intensity.

[0212] 11. The method of any one of aspects 7-10, wherein determining the latent space representation of the electric field image comprises minimizing a function constrained by a set of electric field images that can be predicted by the parametric model based on the dimensional data in the latent space and the given input.

[0213] 12. The method of aspect 11, wherein the latent space representation of the electric field image comprises a tensor.

[0214] 13. The method of any one of aspects 7-12, wherein the parametric model is a machine learning model.

[0215] 14. The method of any one of aspects 7-13, wherein the parametric model comprises an encoder-decoder architecture.

[0216] 15. The method of aspect 14, wherein the encoder-decoder architecture comprises a variational encoder-decoder architecture, the method further comprising training the variational encoder-decoder architecture with a probabilistic latent space, the variational encoder-decoder architecture producing realizations in an output space.

[0217] 16. The method of aspect 15, wherein the latent space comprises a low-dimensional encoding.

[0218] 17. The method of any one of aspects 14-16, wherein the dimensional data in the latent space is encoded by an encoder of the encoder-decoder architecture.

[0219] 18. The method of any one of aspects 14-17, further comprising training the parametric model with a training set of complex electric field images.

[0220] 19. The method of aspect 18, wherein the set of complex electric field images is generated based on a defocus measurement and, optionally, a complex electric field image reconstruction algorithm.

[0221] 20. The method of aspect 18 or 19, wherein the training comprises:

[0222] encoding the complex electric field images in the training set into the dimensional data in the latent space, and

[0223] transforming the dimensional data in the latent space into a de-noised version of the complex electric field images in the training set to facilitate verification of the training.

[0224] 21. The method of aspect 20, further comprising iteratively providing an additional complex electric field image as input to the parametric model, the additional complex electric field image being determined based on how well the de-noised version of the complex electric field image matches the complex electric field images in the training set.

[0225] 22. The method of any of aspects 16-21, further comprising encoding higher dimensional data associated with the electric field image into the dimensional data in the latent space with the encoder.

[0226] 23. The method of any of aspects 14-22, wherein predicting the electric field image based on the latent space representation of the electric field image comprises passing the latent space representation of the electric field image through a decoder of the encoder-decoder architecture.

[0227] 24. The method of any of aspects 7-23, further comprising determining a metrology metric based on a latent space representation of an electric field image.

[0228] 25. The method of aspect 24, wherein determining the metrology metric based on the latent space representation of the electric field image comprises providing the latent space representation of the electric field image to a regression network, the regression network being included in the parametric model or separate from the parametric model.

[0229] 26. The method of aspect 24 or 25, wherein the metrology metric is overlay.

[0230] 27. The method of any of aspects 7-26, further comprising correcting an aberration associated with a metrology device based on the latent space representation of the electric field image and / or a predicted electric field image.

[0231] 28. The method of any one of aspects 7 to 27, further comprising determining, based on the predicted electric field image, an adjustment to a semiconductor manufacturing process parameter used to pattern a substrate geometry as part of the semiconductor manufacturing process.

[0232] 29. A non-transitory computer readable medium having instructions thereon, the instructions when executed by a computer implementing the method of any one of aspects 1 to 28.

[0233] 30. A metrology apparatus configured to determine one or more metrology indicators of a semiconductor manufacturing process, the metrology apparatus comprising one or more processors configured to:

[0234] determine, based on dimensional data in a latent space of a parametric model, a latent space representation of an electric field image for a given input;

[0235] predict, with the parametric model, the electric field image based on the latent space representation of the electric field image; and

[0236] determine, based on the predicted electric field image, the one or more metrology indicators of the semiconductor manufacturing process.

[0237] 31. A lithographic cell comprising a metrology apparatus, the metrology apparatus configured to:

[0238] determine, based on dimensional data in a latent space of a parametric model, a latent space representation of an electric field image for a given input;

[0239] predict, with the parametric model, the electric field image based on the latent space representation of the electric field image; and

[0240] determine, based on the predicted electric field image, the one or more metrology indicators of the semiconductor manufacturing process.

[0241] 32. A method for predicting process information with a parametric model, comprising:

[0242] determining, in a latent space of a parametric model, a latent space representation to a given input to the parametric model;

[0243] transforming the latent space representation of the given input into a transformed latent space representation of the given input based on a reference latent space representation of the given input; and

[0244] predicting process information based on the transformed latent space representation of the given input.

[0245] 33. The method of clause 32, wherein the given input is associated with a target and is received from one of a plurality of target characterisation devices configured to produce the given input; and

[0246] wherein the transformation and prediction are configured such that the predicted process information for a target is the same regardless of which of the target characterisation devices produced the given input.

[0247] 34. The method of clause 32 or 33, wherein the transformation comprises one or more mathematical calculations on the latent space representation of the given input.

[0248] 35. The method of any one of clauses 32 to 34, wherein the transformation is performed in the latent space.

[0249] 36. The method of any one of clauses 32 to 35, wherein the reference latent space representation comprises a weighted combination and / or average of latent space representations of previously received inputs to the parametric model, or a latent space representation of an input from a particular target characterisation device configured to produce the given input.

[0250] 37. The method of any one of clauses 32 to 36, wherein the process information and the given input are associated with a semiconductor manufacturing process.

[0251] 38. The method of any one of clauses 32 to 37, wherein the predicted process information comprises one or more of a predicted image or a predicted process measurement.

[0252] 39. The method of clause 37, wherein the predicted process measurement comprises one or more of a metrology metric, an xyz position, a dimension, an electric field, a wavelength, an illumination and / or detection pupil, a bandwidth, an illumination and / or detection polarisation angle, or an illumination and / or detection phase retardation angle.

[0253] 40. The method of any one of clauses 32 to 39, wherein the given input comprises one or more of an input image or an input process measurement.

[0254] 41. A method for predicting process information with a parametric model, comprising:

[0255] determining, in a latent space of the parametric model, a latent space representation of an optimal set of process parameters associated with a given input to the parametric model based on dimensional data; and

[0256] predicting the process information based on the latent space representation of the optimal set of process conditions.

[0257] 42. The method of clause 41, wherein the predicted process information comprises one or more of a design parameter or a metrology measurement recipe parameter for the target.

[0258] 43. The method of clause 42, wherein the design parameter comprises one or more of a critical dimension, pitch, segmented pitch, line geometry, contact geometry, or hole geometry associated with the target.

[0259] 44. The method of clause 42, wherein a metrology measurement recipe parameter comprises one or more of a wavelength, bandwidth, aperture, illumination and / or detection pupil, bandwidth, illumination and / or detection polarization angle, illumination and / or detection phase delay angle, or dose used to measure the target.

[0260] 45. The method of any one of clauses 41 to 44, wherein the optimal set of process parameters defines optimal measurement conditions for a metrology metric measuring the target.

[0261] 46. The method of clause 45, wherein the optimal set of process parameters comprises one or more of an intensity, contrast, edge response, diffraction efficiency, or overlay sensitivity.

[0262] 47. The method of any one of clauses 41 to 46, wherein the given input comprises one or more of a defined design parameter or a defined metrology measurement recipe parameter for the target.

[0263] 48. The method of any one of clauses 32 to 47, wherein the parameterized model is a machine learning model.

[0264] 49. The method of any one of clauses 42 to 48, wherein the parameterized model comprises an encoder-decoder architecture.

[0265] 50. The method of any one of clauses 32 to 49, wherein the latent space comprises a low-dimensional encoding.

[0266] 51. A method for predicting process information with a parameterized model, comprising:

[0267] determining, in a latent space of the parameterized model, a latent space representation of a given input to the parameterized model;

[0268] transforming the latent space representation of the given input into a transformed latent space representation of the given input based on a reference latent space representation of the given input;

[0269] determining, based on the transformed latent space representation of the given input, a latent space representation of an optimal set of process parameters associated with the given input; and

[0270] predicting the process information based on the transformed latent space representation of the given input and the latent space representation of the optimal set of process parameters associated with the given input.

[0271] 52. A method for predicting process information with a parametric model, comprising:

[0272] determining, in a latent space of a parametric model, a latent space representation of a given input to the parametric model;

[0273] transforming the latent space representation of the given input into a transformed latent space representation of the given input based on a reference latent space representation of the given input;

[0274] determining, based on the transformed latent space representation of the given input, a latent space representation of an electric field image of the given input;

[0275] determining, based on the transformed latent space representation, a latent space representation of an optimal set of process parameters associated with the given input; and

[0276] predicting the process information based on the transformed latent space representation of the given input, the latent space representation of the electric field image, and the latent space representation of the optimal set of process parameters associated with the given input.

[0277] 53. A non-transitory computer readable medium having instructions thereon, the instructions, when executed by a computer, implement the method of any one of aspects 32 to 52.

[0278] 54. A non-transitory computer readable medium having instructions thereon, the instructions, when executed by a computer, cause the computer to:

[0279] determining, based on dimensional data in one or more latent spaces of a parametric model, a latent space representation of an electric field image of a given input to the parametric model; and

[0280] predicting the electric field image based on the latent space representation of the electric field image.

[0281] 55. The non-transitory computer readable medium of aspect 54, wherein the one or more latent spaces comprise at least two latent spaces.

[0282] 56. The non-transitory computer-readable medium of any of aspects 54 or 55,

[0283] wherein the one or more latent spaces comprise a plurality of latent spaces, wherein individual latent spaces of the plurality of latent spaces correspond to different mechanisms of the parameterized model.

[0284] 57. The non-transitory computer-readable medium of aspect 56, wherein the different mechanisms of the parameterized model comprise an encoding mechanism, a decoding mechanism, a complex electric field parameter determination mechanism, and / or a phase recovery mechanism.

[0285] 58. The non-transitory computer-readable medium of any of aspects 54 to 57, wherein the one or more latent spaces comprise at least two latent spaces associated with different independent parameters of the given input to the parameterized model and corresponding dimensionality data.

[0286] 59. The non-transitory computer-readable medium of any of aspects 54 to 58, wherein the electric field image comprises a complex electric field image having an amplitude and a phase, and the given input comprises a measured amplitude associated with the complex electric field image.

[0287] 60. The non-transitory computer-readable medium of any of aspects 54 to 59, wherein determining the latent space representation of the electric field image comprises minimizing a function constrained by a set of electric field images that are predictable by the parameterized model based on the dimensionality data in the one or more latent spaces and the given input.

[0288] 61. A method for predicting an electric field image with a parameterized model, the method comprising:

[0289] determining, based on dimensionality data in one or more latent spaces of the parameterized model, a latent space representation of an electric field image for a given input to the parameterized model; and

[0290] predicting the electric field image based on the latent space representation of the electric field image.

[0291] 62. The method of aspect 61, wherein the one or more latent spaces comprise at least two latent spaces.

[0292] 63. The method of any of aspects 61 or 62, wherein the one or more latent spaces comprise a plurality of latent spaces, wherein individual latent spaces of the plurality of latent spaces correspond to different mechanisms of the parameterized model.

[0293] 64. The method of aspect 63, wherein the different mechanisms of the parameterized model comprise an encoding mechanism, a decoding mechanism, a complex electric field parameter determination mechanism, and / or a phase recovery mechanism.

[0294] 65. The method of any one of aspects 61 to 64, wherein the one or more latent spaces comprise at least two latent spaces associated with different independent parameters and corresponding dimensional data of the given input to the parameterized model.

[0295] 66. The method of any one of aspects 61 to 65, wherein the electric field image comprises a complex electric field image having an amplitude and a phase, and the given input comprises a measured amplitude associated with the complex electric field image.

[0296] 67. The method of any one of aspects 61 to 66, wherein determining the latent space representation of the electric field image comprises minimizing a function constrained by a set of electric field images that can be predicted by the parameterized model based on the dimensional data in the one or more latent spaces and the given input.

[0297] 68. A metrology apparatus comprising one or more processors configured to:

[0298] determine, based on dimensional data in one or more latent spaces of a parameterized model, a latent space representation of an electric field image for a given input to the parameterized model; and

[0299] predict the electric field image based on the latent space representation of the electric field image.

[0300] 69. The metrology apparatus of aspect 68, wherein the one or more latent spaces comprise at least two latent spaces.

[0301] 70. The metrology apparatus of any one of aspects 68 or 69, wherein the one or more latent spaces comprise a plurality of latent spaces, wherein individual latent spaces of the plurality of latent spaces correspond to different mechanisms of the parameterized model.

[0302] 71. The metrology apparatus of aspect 70, wherein the different mechanisms of the parameterized model comprise an encoding mechanism, a decoding mechanism, a complex electric field parameter determination mechanism, and / or a phase recovery mechanism.

[0303] 72. The metrology apparatus of any one of aspects 68 to 71, wherein the one or more latent spaces comprise at least two latent spaces associated with different independent parameters and corresponding dimensional data of the given input to the parameterized model.

[0304] 73. The metrology apparatus of any of aspects 68 to 72, wherein the electric field image comprises a complex electric field image having an amplitude and a phase, and the given input comprises a measured amplitude associated with the complex electric field image.

[0305] 74. The metrology apparatus of any of aspects 68 to 73, wherein determining the latent space representation of the electric field image comprises minimizing a function constrained by a set of electric field images that are predictable by the parametric model based on the dimensional data in the one or more latent spaces and the given input.

[0306] 75. A method for determining one or more metrology indicators of a semiconductor manufacturing process using a machine learning algorithm, comprising:

[0307] receiving, using a control circuit, a feature vector associated with an unknown electric field image, wherein the feature vector represents values corresponding to a latent space representation of an electric field image;

[0308] inputting, using the control circuit, the feature vector into a machine learning model, wherein the machine learning model comprises a generative classifier for identifying a known electric field image based on a labeled feature vector corresponding to a latent space representation of an electric field image, wherein the known electric field image is a higher dimensional representation of the latent space representation of the electric field image;

[0309] receiving, using the control circuit, a first prediction from the machine learning model, wherein the first prediction indicates whether the first feature vector corresponds to the known electric field image; and

[0310] in response to the first prediction indicating that the first feature vector corresponds to the known electric field image, generating a recommendation for a metrology indicator of a semiconductor manufacturing process corresponding to the known electric field image for display on a user interface.

[0311] The concepts disclosed herein can be modeled mathematically or simulated for any general imaging system used to image sub-wavelength features, and can be used especially for emerging imaging technologies capable of producing shorter and shorter wavelengths. Emerging technologies already in use include extreme ultraviolet (EUV), DUV lithography capable of producing 193 nm wavelengths by using ArF lasers and even 157 nm wavelengths by using fluorine lasers. In addition, EUV lithography can produce wavelengths in the range of 20 nm to 5 nm by using synchrotrons or by shooting high energy electrons at a material (solid or plasma) to produce photons in this range.

[0312] While the concepts disclosed herein can be used for imaging on substrates such as silicon wafers, it should be appreciated that the disclosed concepts can be used with any type of lithographic imaging system, e.g., a lithographic imaging system for imaging on substrates other than silicon wafers, and / or a metrology system. Moreover, combinations and sub-combinations of the disclosed elements can include separate embodiments. For example, predicting a complex electric field image and determining a metrology indicator such as overlay can be performed by the same parametric model and / or different parametric models. These features can include separate embodiments, and / or these features can be used together in the same embodiments.

[0313] While specific reference can be made in this text to embodiments of the application in the context of a metrology apparatus, embodiments of the application can be used in other apparatuses. Embodiments of the application can form part of a mask inspection apparatus, a lithographic apparatus, or any apparatus that measures or processes an object such as a wafer (or other substrate) or a mask (or other patterning device). These apparatuses can be generally referred to as lithographic tools. Such lithographic tools can use vacuum conditions or ambient (non-vacuum) conditions.

[0314] While the above can have specifically referred to use of embodiments of the application in the context of optical lithography, it will be appreciated that the application is not limited to optical lithography where the context allows, and can be used in other applications such as imprint lithography. While particular embodiments of the application have been described, it will be apparent to those skilled in the art that the application can be practiced with different and alternative ways. The above descriptions are intended to be illustrative, and not restrictive. Accordingly, one skilled in the art will appreciate that modifications to the described embodiments can be practiced without departing from the scope of the claims set forth below.

Claims

1. A method for determining one or more metrology indicators for a semiconductor manufacturing process, the method comprising: determining, based on dimensional data in a latent space of a parametric model, a latent space representation of an electric field image for a given input; predicting, with the parametric model, the electric field image based on the latent space representation of the electric field image; and determining, based on the predicted electric field image, the one or more metrology indicators for the semiconductor manufacturing process. The electric field image comprises a complex electric field image having an amplitude and a phase.

2. The method of claim 1, wherein, The determined one or more metrology indicators comprise one or more of: an overlay, a critical dimension, a reconstruction of a three-dimensional profile of a feature of a substrate, or a dose or focus of a lithographic apparatus in imprinting a feature of a substrate with the lithographic apparatus.

3. The method of claim 1 or 2, wherein, The electric field image comprises a complex electric field image, and wherein the given input comprises a measured amplitude associated with the complex electric field image.

4. The method of claim 1 or 2, wherein, The amplitude comprises an intensity.

5. The method of claim 4, wherein, Adjusting one or more semiconductor manufacturing process parameters based on the determined one or more metrology indicators.

6. The method of any one of claims 1, 2, and 5, further comprising:

7. A method for predicting an electric field image with a parametric model, the method comprising: determining, based on dimensional data in a latent space of the parametric model, a latent space representation of an electric field image for a given input to the parametric model; and predicting, based on the latent space representation of the electric field image, the electric field image. The electric field image comprises a complex electric field image having an amplitude and a phase. The given input comprises a measured amplitude associated with a complex electric field image.

8. The method of claim 7, wherein, The amplitude comprises an intensity.

9. The method of claim 8, wherein, Determining the latent space representation of the electric field image comprises minimizing a function constrained by a set of electric field images, the set of electric field images being predictable by the parametric model based on the dimensional data in the latent space and the given input.

10. The method of claim 9, wherein, The latent space representation of the electric field image comprises a tensor.

11. The method of any one of claims 7-10, wherein, The parametric model is a machine learning model.

12. The method of claim 11, wherein, The parametric model comprises an encoder-decoder architecture.

13. The method of any one of claims 7-10 and 12, wherein, The encoder-decoder architecture comprises a variational encoder-decoder architecture, the method further comprising training the variational encoder-decoder architecture with a probabilistic latent space, the variational encoder-decoder architecture producing a realization in an output space.

14. The method of any one of claims 7-10 and 12, wherein, The latent space comprises a low-dimensional encoding.

15. The method of claim 14, wherein, The dimensional data in the latent space is encoded by an encoder of the encoder-decoder architecture.

16. The method of claim 15, wherein, The parametric model is trained with a training set of complex electric field images.

17. The method of claim 14, wherein, The set of complex electric field images is produced based on an off-focus measurement, and a complex electric field image reconstruction algorithm.

18. The method of claim 14, further comprising: The training comprises:

19. The method of claim 18, wherein, encoding the complex electric field images in the training set into the dimensional data in the latent space, and 20. The method of claim 19, wherein, transforming the dimensional data in the latent space into a de-noised version of the complex electric field images in the training set to facilitate a verification of the training. ​ ​ 21. The method of claim 20, wherein, The method further comprises iteratively providing an additional complex electric field image as input to the parametric model, the additional complex electric field image being determined based on how well the de-noised version of the complex electric field image matches the complex electric field images in the training set.

22. The method of any of claims 16, 19-21, further comprising: encoding, associated with the electric field image, into the dimensional data in the latent space using the encoder.

23. The method of claim 14, wherein, predicting the electric field image based on the latent space representation of the electric field image comprises passing the latent space representation of the electric field image through a decoder of the encoder-decoder architecture.

24. The method of any one of claims 7-10, 12, 15, 16, 19-21, and 23, further comprising: determining a metrology indicator based on a latent space representation of an electric field image.

25. The method of claim 24, wherein, determining the metrology indicator based on the latent space representation of the electric field image comprises providing the latent space representation of the electric field image to a regression network, the regression network being comprised in the parametric model or separate from the parametric model.

26. The method of claim 24, wherein, the metrology indicator is an overlay.

27. The method of any one of claims 7-10, 12, 15, 16, 19-21, 23, 25-26, further comprising: correcting an aberration associated with a metrology apparatus based on the latent space representation of the electric field image and / or the predicted electric field image.

28. The method of any one of claims 7-10, 12, 15, 16, 19-21, 23, 25-26, further comprising: determining an adjustment to a semiconductor manufacturing process parameter for patterning a substrate geometry as part of the semiconductor manufacturing process based on the predicted electric field image.

29. A non-transitory computer readable medium having instructions thereon, the instructions, when executed by a computer, implementing a method according to any one of claims 1 to 28.

30. A metrology apparatus configured to determine one or more metrology indicators of a semiconductor manufacturing process, the metrology apparatus comprising one or more processors configured to: determine a latent space representation of an electric field image for a given input based on dimensional data in a latent space of a parametric model; predict the electric field image based on the latent space representation of the electric field image with the parametric model; and determine the one or more metrology indicators of the semiconductor manufacturing process based on the predicted electric field image.

31. A lithographic cell comprising a metrology apparatus configured to: determine a latent space representation of an electric field image for a given input based on dimensional data in a latent space of a parametric model; predict the electric field image based on the latent space representation of the electric field image with the parametric model; and determine one or more metrology indicators of a semiconductor manufacturing process based on the predicted electric field image.

32. A non-transitory computer readable medium having instructions thereon, the instructions, when executed by a computer, causing the computer to: determine a latent space representation of an electric field image for a given input to a parametric model based on dimensional data in one or more latent spaces of the parametric model; and predict the electric field image based on the latent space representation of the electric field image.

33. The non-transitory computer-readable medium of claim 32, wherein, the one or more latent spaces comprise at least two latent spaces.

34. The non-transitory computer readable medium of any one of claims 32 or 33, wherein, The one or more latent spaces comprise a plurality of latent spaces, wherein individual ones of the plurality of latent spaces correspond to different mechanisms of the parametric model.

35. The non-transitory computer-readable medium of claim 34, wherein, The different mechanisms of the parametric model comprise an encoding mechanism, a decoding mechanism, a complex electric field parameter determination mechanism, and / or a phase recovery mechanism.

36. The non-transitory computer-readable medium of any one of claims 32, 33, and 35, wherein, The one or more latent spaces comprise at least two latent spaces associated with different independent parameters and corresponding dimensional data of the given input to the parametric model.

37. The non-transitory computer-readable medium of any one of claims 32, 33, and 35, wherein, The electric field image comprises a complex electric field image having an amplitude and a phase, and the given input comprises a measured amplitude associated with the complex electric field image.

38. The non-transitory computer-readable medium of any one of claims 32, 33, and 35, wherein, Determining the latent space representation of the electric field image comprises minimizing a function constrained by a set of electric field images that can be predicted by the parametric model based on the dimensional data in the one or more latent spaces and the given input.

39. A method for predicting an electric field image with a parametric model, the method comprising: determining, based on dimensional data in one or more latent spaces of the parametric model, a latent space representation of an electric field image of a given input to the parametric model; and predicting the electric field image based on the latent space representation of the electric field image.

40. The method of claim 39, wherein the one or more latent spaces comprise at least two latent spaces.

41. The method of claim 39 or 40, wherein, The one or more latent spaces comprise a plurality of latent spaces, wherein individual ones of the plurality of latent spaces correspond to different mechanisms of the parametric model.

42. The method of claim 41, wherein, The different mechanisms of the parametric model comprise an encoding mechanism, a decoding mechanism, a complex electric field parameter determination mechanism, and / or a phase recovery mechanism.

43. The method of any one of claims 39, 40, and 42, wherein, The one or more latent spaces comprise at least two latent spaces associated with different independent parameters and corresponding dimensional data of the given input to the parametric model.

44. The method of any one of claims 39, 40, and 42, wherein, The electric field image comprises a complex electric field image having an amplitude and a phase, and the given input comprises a measured amplitude associated with the complex electric field image.

45. The method of any one of claims 39, 40, and 42, wherein, Determining the latent space representation of the electric field image comprises minimizing a function constrained by a set of electric field images that can be predicted by the parametric model based on the dimensional data in the one or more latent spaces and the given input.

46. A metrology apparatus comprising one or more processors configured to: determine, based on dimensional data in one or more latent spaces of a parametric model, a latent space representation of an electric field image of a given input to the parametric model; and predict the electric field image based on the latent space representation of the electric field image.

47. The apparatus of claim 46, wherein, The one or more latent spaces comprise at least two latent spaces.

48. The apparatus of claim 46 or 47, wherein, The one or more latent spaces comprise a plurality of latent spaces, wherein individual ones of the plurality of latent spaces correspond to different mechanisms of the parametric model.

49. The apparatus of claim 48, wherein the different mechanisms of the parametric model comprise an encoding mechanism, a decoding mechanism, a complex electric field parameter determination mechanism, and / or a phase recovery mechanism.

50. The apparatus of any of claims 46, 47, and 49, wherein, The one or more latent spaces comprise at least two latent spaces associated with different independent parameters and corresponding dimensional data of the given input to the parametric model.

51. The apparatus of any of claims 46, 47, and 49, wherein, The electric field image comprises a complex electric field image having an amplitude and a phase, and the given input comprises a measured amplitude associated with the complex electric field image.

52. The apparatus of any one of claims 46, 47, and 49, wherein, Determining the latent space representation of the electric field image comprises minimizing a function constrained by a set of electric field images that can be predicted by the parametric model based on the dimensional data in the one or more latent spaces and the given input.

Images