Method and apparatus for determining overlay - Patents.com

Abstract

Described herein are systems, methods, and media for determining processing parameters associated with a lithography process. In some embodiments, image data of features on a substrate may be acquired, and the image data may be analyzed in Fourier space. Based on this analysis, amplitude and phase may be determined, and overlay of the features may be determined based on the amplitude and phase.

Description

[Technical field]

[0001] CROSS-REFERENCE TO RELATED APPLICATIONS

[0001] This application claims priority to U.S. patent application Ser. No. 63 / 291,759, filed Dec. 20, 2021, the entire contents of which are incorporated by reference into this specification.

[0002] This specification relates to determining parameters (such as overlay) of a process that produces a pattern on a substrate based on an analysis in Fourier space of an image of the substrate to be patterned. [Background technology]

[0003]

[0003] In the manufacturing process of integrated circuits (ICs), unfinished or finished circuit components are inspected to ensure that they are manufactured as designed and free of defects. Inspection systems utilizing optical microscopes or charged particle (e.g., electron) beam microscopes such as scanning electron microscopes (SEMs) can be employed. As the physical size of IC components continues to shrink and their structures continue to become more complex, accuracy and throughput in defect detection and inspection become more important. Overall image quality depends on a combination of high signal detection efficiency, especially of secondary and backscattered electrons. Backscattered electrons have higher emission energy to escape from deeper layers of the sample, so detection of backscattered electrons may be desirable for imaging complex structures such as buried layers, nodes, high aspect ratio trenches or holes in 3D NAND devices. In applications such as overlay metrology, it may be desirable to simultaneously obtain surface information from secondary electrons and buried layer information from backscattered electrons with high quality imaging and efficient collection, highlighting the need to use multiple electron detectors in SEMs. Although multiple electron detectors in various structural arrangements may be used to individually maximize the collection and detection efficiency of secondary and backscattered electrons, the combined detection efficiency remains low and therefore the achieved image quality may be insufficient for high-precision and high-throughput defect inspection and metrology of two- and three-dimensional structures. Summary of the Invention

[0004]

[0004] In one embodiment, a method is provided that includes acquiring image data of a feature on a substrate, analyzing the image data in Fourier space, determining amplitude and phase based on the analysis, and determining overlay based on the amplitude and phase.

[0005] In one aspect, a non-transitory computer program product is provided that includes machine-readable instructions for causing a processor system to perform the methods described herein. In one aspect, a computer program product is provided that includes a computer non-transitory readable medium having instructions recorded thereon, the instructions, when executed by a computer, performing a method or one or more process steps described herein.

[0006]

[0006] In one aspect, there is provided a metrology apparatus for measuring an object in a patterning process, the metrology apparatus configured to perform the methods described herein.

[0007]

[0007] In one aspect, there is provided an inspection apparatus for inspecting an object in a patterning process, the inspection apparatus operable to perform the methods described herein.

[0008] In one aspect, there is provided a system including the metrology apparatus described herein and a computer program product, in an embodiment, the system further includes a lithographic apparatus including a support structure configured to hold a patterning device to modulate a radiation beam and projection optics configured to project the modulated radiation beam onto a radiation sensitive substrate, wherein the object is the substrate.

[0009]

[0009] In one embodiment, a system is provided that includes a hardware processor system and a non-transitory computer-readable storage medium configured to store machine-readable instructions that, when executed, cause the hardware processor system to perform the methods described herein.

[0010]

[0010] The accompanying drawings, which are incorporated in and constitute a part of this specification, illustrate one or more embodiments and, together with the detailed description of the invention, explain these embodiments. The embodiments are described, by way of example only, with reference to the accompanying schematic drawings in which corresponding reference symbols indicate corresponding parts. [Brief description of the drawings]

[0011] [Figure 1] 1 is a schematic diagram of a lithographic projection apparatus according to an embodiment; [Diagram 2]

[0012] 1 shows a schematic diagram of a lithographic cell according to an embodiment. [Diagram 3]

[0013] 1 shows a schematic diagram of holistic lithography, illustrating the collaboration between three techniques to optimize semiconductor manufacturing, according to one embodiment. [Figure 4]

[0014] 1 illustrates a schematic diagram of an embodiment of a charged particle (eg, electron beam) inspection apparatus, according to one embodiment. [Diagram 5]

[0015] 2 illustrates a schematic diagram of another embodiment of an inspection apparatus, according to one embodiment; [Figure 6A]

[0016] 1 depicts an exemplary image of a substrate having features located on a first layer and features located on a second layer formed on the first layer according to one embodiment. [Figure 6B]

[0017] 6B depicts a zoomed-in portion of the image of FIG. 6A according to one embodiment. [Figure 7A]

[0018] 1 is a simplified diagram of a portion of an image of a substrate having stacks of layers, each stack having one or more features, according to one embodiment. [Figure 7B]

[0019] 7B is a simplified illustration of a portion of the image from FIG. 7A, where only the first layer including the first feature is shown, according to one embodiment. [Figure 8]

[0020] 1 corresponds to a method for applying a Fourier transform to a two-dimensional image according to one embodiment. [Figure 9]

[0021] 6B depicts an exemplary amplitude image obtained by applying a Fourier transform to the image of FIG. 6A according to one embodiment. [Figure 10A]

[0022] 1 depicts an exemplary interpolation process for determining the coordinates of a peak in Fourier space with sub-pixel accuracy according to one embodiment. [Figure 10B] 1 depicts an exemplary interpolation process for determining the coordinates of a peak in Fourier space with sub-pixel accuracy according to one embodiment. [Figure 11]

[0023] 6B depicts an exemplary phase image obtained by applying a Fourier transform to the image of FIG. 6A according to one embodiment. [Figure 12]

[0024] 10 is a simplified diagram of a zoomed-in portion of the example amplitude image of FIG. 9 according to one embodiment. [Figure 13]

[0025] 13 corresponds to a method for determining overlay by using a Fourier transform of an image of a patterning device according to one embodiment. [Figure 14]

[0026] 1 is an exemplary graph of a calibration process according to one embodiment. [Figure 15]

[0027] 15 is an image of a phase image of a patterning device including example points used to form the traces of the example graph of FIG. 14 in accordance with one embodiment. [Figure 16]

[0028] 1 is an exemplary graph of a calibration process including ideal exemplary calibration coefficients according to one embodiment. [Figure 17]

[0029] FIG. 1 is a block diagram of an exemplary computer system according to one embodiment. [Figure 18]

[0030] 1 is a schematic diagram of another lithographic projection apparatus according to an embodiment; [Figure 19]

[0031] 1 illustrates a more detailed view of components of a lithographic projection apparatus (eg, the lithographic projection apparatus shown in FIG. 1 or FIG. 18) according to one embodiment. [Figure 20]

[0032] FIG. 2 depicts a detailed view of a source collector module of a lithographic projection apparatus according to an embodiment; DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

[0012]

[0033] Electronic devices are built with circuits formed on pieces of silicon called substrates or wafers. Many circuits may be formed together as repeating patterns of features on the same piece of silicon, called an integrated circuit, or IC. The size of these circuits is decreasing dramatically, thus allowing more circuits to fit on a substrate. For example, an IC chip in a smartphone may be the size of a thumbnail yet contain over 2 billion transistors, each transistor less than 1 / 1000th the size of a human hair.

[0013]

[0034] The manufacture of these tiny ICs is a complex, time-consuming, and costly process, often involving hundreds of individual steps. An error in even one step can result in a defect in the finished IC, rendering it unusable. Thus, one of the goals of a manufacturing process is to avoid such defects and maximize the number of functioning ICs produced by the process, thus increasing the overall yield of the process.

[0014]

[0035] One element of improving yield is monitoring the chip manufacturing process to ensure that a sufficient number of functioning integrated circuits are being produced. One way to monitor the process is to inspect the chip circuit structures at various stages of their formation. Inspection can be done with a scanning electron microscope (SEM). An SEM can be used to image these very small structures, in effect taking a "picture" of the structure. This image can be used to determine if the structure was formed properly, and if it was formed in the proper location. If the structure is defective, the process can be adjusted to make it less likely that the defect will occur again.

[0015]

[0036] To monitor manufacturing process variations, contours of identical pattern features can be determined based on SEM images of substrate patterns. The contours can be stacked and statistically analyzed to determine the variation of a given feature. Some of the contours are usually outliers that can cause degradation of the accuracy measurement / characterization of some manufacturing process features (such as critical dimensions, edge placement errors, or overlay errors, among others), and stacking and averaging the contours "hides" these outliers. This disclosure describes filtering outlier contours before being stacked and statistically analyzed. The contours can be analyzed and in some embodiments a score assesses how close the contour is to the "ideal", and contours with a score below a threshold are filtered and the remaining contours are stacked. Filtering can be done at multiple levels, based on individual points on the contour in the set of inspection contours, or based on the entire geometry of the contour in the set of inspection contours. This may enhance the measurement accuracy / characterization of these manufacturing process features, thereby allowing more optimal adjustments to be made to the manufacturing process to increase device yield, or may have other benefits.

[0016]

[0037] In order to enable those skilled in the art to practice the present disclosure, the embodiments of the present disclosure provided as examples to aid in the understanding of the present disclosure will be described in detail with reference to the drawings. In particular, the following figures and examples are not intended to limit the scope of the present disclosure to a single embodiment, and other embodiments are possible by replacing some or all of the elements described or illustrated. Furthermore, when a particular element of the present disclosure can be implemented partially or completely using known components, only the parts of such known components necessary for the understanding of the present disclosure will be described, and detailed descriptions of other parts of such known components will be omitted so as not to obscure the present disclosure. An embodiment described as being implemented in software is not limited thereto, and may include embodiments implemented in hardware or a combination of software and hardware, and vice versa, as would be apparent to a person skilled in the art, unless otherwise specified herein. In this specification, an embodiment showing a single component is not to be considered limiting, and more precisely, the present disclosure is intended to encompass other embodiments including multiple identical components, and vice versa, unless otherwise expressly specified herein. Additionally, applicants do not intend that any term in this specification or claims be given an uncommon or special meaning unless expressly defined as such. Moreover, this disclosure encompasses present and future known equivalents to known components referred to herein for purposes of illustration.

[0017]

[0038] Although specific reference may be made herein to the manufacture of ICs, it should be clearly understood that what is described herein has many other possible applications. For example, it may be used in the manufacture of integrated optical systems, guidance and detection patterns for magnetic domain memories, liquid crystal display (LCD) panels, thin film magnetic heads, etc. Those skilled in the art will understand that in the context of such alternative applications, any use of the terms "reticle," "wafer," or "die" herein should be considered interchangeable with the more general terms "mask," "substrate," and "target portion," respectively.

[0018]

[0039] As used herein, the terms "radiation" and "beam" are used to encompass all types of electromagnetic radiation, including ultraviolet (e.g., having wavelengths of 365, 248, 193, 157, or 126 nm) and EUV (extreme ultraviolet, e.g., having wavelengths in the range of about 5 to 100 nm).

[0019]

[0040] The term "projection optics" as used herein is intended to be broadly interpreted to encompass various types of optical systems including, for example, refractive optics, reflective optics, apertures, and catadioptric optics. The term "projection optics" may also include components that collectively or individually operate according to any of these design types to direct, shape, or control a projection beam of radiation. The term "projection optics" may include any optical component in a lithographic projection apparatus, regardless of where the optical component is located on the optical path of the lithographic projection apparatus. Projection optics may include optical components for shaping, conditioning, or projecting radiation from a source before it passes through a patterning device, or optical components for shaping, conditioning, or projecting radiation after it passes through a patterning device (e.g., a semiconductor). Projection optics generally excludes the source and the patterning device.

[0020]

[0041] A (e.g., semiconductor) patterning device may include or form one or more design layouts. A design layout may be generated utilizing a computer-aided design (CAD) program, a process often referred to as electronic design automation (EDA). Most CAD programs follow a set of predefined design rules to create a functional design layout / patterning device. These rules are set by process and design constraints. For example, design rules define the space tolerance between devices (gates, capacitors, etc.) or interconnect lines to ensure that the devices or lines do not interact with each other in an undesirable way. Design rules may include or specify certain parameters, limits on parameter ranges, or other information. One or more of the design rule constraints or parameters may be referred to as "critical dimensions" (CD). A critical dimension of a device may be defined as the smallest width of a line or hole, or the smallest space between two lines or two holes, or other features. Thus, the CD determines the overall size and density of the designed device. One of the goals in device fabrication is to faithfully reproduce the original design intent on a substrate (via a patterning device).

[0021]

[0042] The terms "mask" or "patterning device" as used herein may be broadly interpreted to refer to any general semiconductor patterning device that can be used to impart an incoming radiation beam with a patterned cross-section that corresponds to the pattern to be produced on a target portion of a substrate, and the term "light valve" is also sometimes used in this context. In addition to traditional masks (transmissive or reflective; binary, phase-shifting, hybrid, etc.), examples of other such patterning devices include programmable mirror arrays and programmable LCD arrays.

[0022]

[0043] An example of a programmable mirror array can be a matrix-addressable surface having a viscoelastic control layer and a reflective surface. The basic principle behind such a device is that (for example) addressed areas of the reflective surface reflect incident radiation as diffracted radiation, while unaddressed areas reflect incident radiation as undiffracted radiation. Using appropriate filters, said undiffracted radiation can be filtered out of the reflected beam, leaving only the diffracted radiation behind, and in this way the beam is patterned according to the addressing pattern of the matrix-addressable surface. The required matrix addressing can be performed using suitable electronic means. An example of a programmable LCD array is given by U.S. Pat. No. 5,229,872, which is incorporated herein by reference.

[0023]

[0044] As used herein, the term "patterning process" generally refers to a process for creating an etched substrate by the application of a specified pattern of light as part of a lithography process. However, since many of the features described herein may provide advantages in forming the printed pattern using an etching (e.g., plasma) process, "patterning process" may also include (e.g., plasma) etching.

[0024]

[0045] As used herein, the term "pattern" refers to an idealized pattern that is etched onto a substrate (eg, a wafer).

[0025]

[0046] As used herein, a "printed pattern" refers to a physical pattern on a substrate that is etched based on a target pattern. The printed pattern may include, for example, valleys, channels, recesses, edges, or other two- and three-dimensional features produced by a lithographic process.

[0026]

[0047] As used herein, the terms "predictive model," "process model," "electronic model," or "simulation model" (which may be used interchangeably) refer to a model that includes one or more models that simulate a patterning process. For example, the models may include an optical model (which may include, e.g., modeling a lens system / projection system used to deliver light in a lithography process and modeling the final optical image of the light onto the photoresist), a resist model (which may include, e.g., modeling the physical effects of the resist, such as chemical effects of light), an optical proximity correction (OPC) model (which may be used, e.g., to create a target pattern and may include sub-resolution resist features (SRAF), etc.), an etch (or etch bias) model (which may include, e.g., simulating the physical effects of an etch process on a printed wafer pattern), or other models.

[0027]

[0048] As used herein, the term "calibrate" means to correct (eg, improve or adjust) or verify something, such as a model.

[0028]

[0049] A patterning system may be a system that includes any or all of the above components and other components configured to perform any or all of the operations associated with these components. A patterning system may include, for example, a lithographic projection apparatus, a scanner, a system configured to apply or remove resist, an etching system, or other system.

[0029]

[0050] By way of introduction, FIG. 1 is a schematic diagram of a lithographic projection apparatus according to an embodiment. The lithographic projection apparatus may include an illumination system IL, a first object table T, a second object table WT, and a projection system PS. The illumination system IL can condition a radiation beam B. In this example, the illumination system also includes a radiation source SO. The first object table (e.g., patterning device table) T may comprise a patterning device holder for holding a patterning device MA (e.g., a reticle) and be connected to a first positioner for accurately positioning the patterning device with respect to the item PS. The second object table (e.g., substrate table) WT may comprise a substrate holder for holding a substrate W (e.g., a resist-coated silicon wafer) and be connected to a second positioner for accurately positioning the substrate with respect to the item PS. The projection system PS (e.g., a refractive, reflective, or catadioptric optical system, e.g., including a lens) may image an irradiated portion of the patterning device MA onto a target portion C of the substrate W (e.g., including one or more dies). Patterning device MA and substrate W may be aligned using, for example, patterning device alignment marks M1, M2 and substrate alignment marks P1, P2.

[0030]

[0051] As depicted, the apparatus may be of a transmissive type (i.e. having a transmissive patterning device). In general, however, it may be of a reflective type, for example (having a reflective patterning device). The apparatus may use different types of patterning devices for conventional masks; examples include a programmable mirror array or an LCD matrix.

[0031]

[0052] A source SO (e.g. a mercury lamp or excimer laser, LPP (Laser Produced Plasma) EUV source) produces a beam of radiation. This beam is fed to an illumination system (illuminator) IL, either directly or after having traversed conditioning means, such as a beam expander, or a beam delivery system (comprising directing mirrors, beam expanders, etc.). The illuminator IL may for example comprise conditioning means AD for setting the outer or inner radial extent (commonly referred to as σ-outer and σ-inner, respectively) of the intensity distribution of the beam. Furthermore it will generally comprise various other components, such as an integrator IN and a condenser CO. In this way, the beam B impinging on the patterning device MA has a desired uniformity and intensity distribution in its cross-section.

[0032]

[0053] In some embodiments, the source SO may be located within the housing of the lithographic projection apparatus (as is often the case when the source SO is, for example, a mercury lamp), but it may also be located remotely from the lithographic projection apparatus. For example, the radiation beam it produces may be directed into the apparatus (for example by means of suitable directing mirrors). This latter scenario may be the case when the source SO is, for example, an excimer laser (for example based on KrF, ArF or F2 lasing).

[0033]

[0054] The beam B may then intersect the patterning device MA, which is held on a patterning device table T. After traversing the patterning device MA, the beam B may pass through a lens PL, which focuses the beam B onto a target portion C of the substrate W. Using the second positioning means (and interferometric measurement means IF), the substrate table WT can be precisely moved, e.g. to position a different target portion C in the path of the beam B. Similarly, the first positioning means can be used to precisely position the patterning device MA with respect to the path of the beam B, e.g. after mechanical retrieval of the patterning device MA from a patterning device library or during a scan. In general, the movement of the tables T, WT can be realised using a long-stroke module (coarse positioning) and a short-stroke module (fine positioning). However, in the case of a stepper (as opposed to a step-and-scan tool), the patterning device table T may be connected to a short-stroke actuator or may be fixed.

[0034]

[0055] The depicted tool can be used in two different modes: step mode and scan mode. In step mode, the patterning device table T remains essentially stationary and the entire patterning device image is projected onto the target portion C in one movement (i.e. in a single "flash"). The substrate table WT can be shifted in the x or y direction so that a different target portion C can be irradiated by the beam B. In scan mode, essentially the same scenario applies, except that a given target portion C is not exposed in a single "flash". Instead, the patterning device table T is movable in a given direction (e.g. the "scan direction", i.e. the "y" direction) with a speed v, such that the projection beam B is caused to scan over the patterning device image. In parallel, the substrate table WT is simultaneously moved in the same or opposite direction with a speed V=Mv, where M is the magnification factor of the lens (typically M=1 / 4 or 1 / 5). In this way, a relatively large target portion C can be exposed, without having to compromise on resolution.

[0035]

[0056] FIG. 2 shows a schematic diagram of a lithographic cell LC. As shown in FIG. 2, the lithographic apparatus LA may form part of a lithographic cell LC, sometimes also called a litho-cell or (litho) cluster, which often also includes apparatus for performing pre-exposure and post-exposure processes on a substrate W. Conventionally, these include a spin coater SC configured to deposit a resist layer, a developer DE for developing the exposed resist, a cooling plate CH and a bake plate BK for adjusting the temperature of the substrate W, e.g., to adjust the solvent in the resist layer. A substrate handler or robot RO picks up substrates W from input / output ports I / O1, I / O2, moves them between the different process apparatus, and transports the substrates W to a loading bay LB of the lithographic apparatus LA. The devices in the litho-cell (often also collectively called the track) are typically under the control of a track control unit TCU, which itself may be controlled by a supervisory control system SCS, which may also control the lithographic apparatus LA, e.g. via a lithography control unit LACU.

[0036]

[0057] In order for a substrate W (FIG. 1) to be exposed by lithographic apparatus LA to be accurately and consistently exposed, it is desirable to inspect the substrate to measure properties of the patterned structures such as the placement of feature edges, overlay errors between subsequent layers, line thickness, critical dimensions (CD), etc. For this purpose, an inspection tool (not shown) may be included in the lithocell LC. If an error is detected, particularly if inspection is performed before other substrates W of the same batch or lot have yet to be exposed or processed, adjustments may be made, for example, to the exposure of subsequent substrates or other processing steps performed on substrate W.

[0037]

[0058] In order to determine the properties of the substrate W (FIG. 1), in particular how the properties of different substrates W vary or how the properties related to different layers of the same substrate W vary from layer to layer, an inspection apparatus, sometimes also called a metrology apparatus, is used. The inspection apparatus may alternatively be constructed to identify defects on the substrate W and may for example be part of a lithocell LC or integrated in the lithography apparatus LA or may be a stand-alone device. The inspection apparatus may measure the properties using a real substrate (e.g. a charged particle -SEM-image of the wafer pattern) or an image of the real substrate, on a latent image (image of the resist layer after exposure), on a semi-latent image (image of the resist layer after a post-exposure bake step PEB), on a developed resist image (where the exposed or unexposed parts of the resist have been removed), on an etched image (after a pattern transfer step such as etching) or in other ways.

[0038]

[0059] FIG. 3 shows a schematic diagram of holistic lithography, which represents the cooperation between three techniques for optimizing semiconductor manufacturing. Typically, the patterning process in the lithographic apparatus LA is one of the most critical steps in the process, which demands high accuracy in the dimensioning and placement of the structures on the substrate W (FIG. 1). To guarantee this high accuracy, three systems (in this example) may be combined in a so-called "holistic" control environment, as illustrated diagrammatically in FIG. 3. One of these systems is the lithographic apparatus LA, which is (virtually) connected to a metrology apparatus (e.g., metrology tool) MT (second system) and a computer system CL (third system). The "holistic" environment may be configured to optimize the cooperation between these three systems to improve the overall process window, and to provide a tight control loop to ensure that the patterning performed by the lithographic apparatus LA stays within the process window. A process window defines the range of process parameters (e.g., dose, focus, overlay) within which a particular manufacturing process produces a defined result (e.g., a functioning semiconductor device) and typically within this range the process parameters of a lithography or patterning process are allowed to vary.

[0039]

[0060] The computer system CL can use (part of) the design layout to be patterned to predict which resolution enhancement technique should be used and perform computational lithography simulations and calculations to determine which mask layout and lithography apparatus settings will achieve the largest overall process window of the patterning process (depicted in FIG. 3 by the double arrow at the first scale SC1). Typically, the resolution enhancement techniques are positioned to match the patterning capabilities of the lithography apparatus LA. The computer system CL may also be used to detect (e.g., using input from the metrology tool MT) where the lithography apparatus LA is currently operating in the process window and predict whether there may be defects due to, for example, sub-optimal processing (depicted in FIG. 3 by the arrow pointing to "0" at the second scale SC2).

[0040]

[0061] The metrology tool MT can provide input to the computer system CL to enable accurate simulation and prediction, and can, for example, provide feedback to the lithography apparatus LA to identify possible drift in the calibration status of the lithography apparatus LA (depicted in Figure 3 by multiple arrows at a third scale SC3).

[0041]

[0062] In lithographic processes, it is frequently desirable to perform measurements of fabricated structures, e.g. for process control and verification. Tools for performing such measurements include metrology tools (apparatus) MT. Different types of metrology tools MT for performing such measurements are known, including scanning electron microscopes (SEMs) or various forms of scatterometer metrology tools MT. In some embodiments, the metrology tool MT is or includes a SEM.

[0042]

[0063] In some embodiments, the metrology tool MT is or includes a spectroscopic scatterometer, an ellipsometric scatterometer, or other light-based tool. A spectroscopic scatterometer may be configured such that radiation emitted by a radiation source is directed to a target feature of the substrate, and reflected or scattered radiation from the target is directed to a spectrometer detector, which measures the spectrum of the specularly reflected radiation (i.e. a measurement of the intensity as a function of wavelength). From this data, the structure or profile of the target giving rise to the detected spectrum can be reconstructed, for example by rigorous coupled wave analysis and nonlinear regression, or by comparison with a library of simulated spectra. An ellipsometric scatterometer allows the determination of parameters of the lithography process by measuring the scattered radiation of each polarization state. Such metrology tools (MT) emit polarized light (such as linear, circular, or elliptical), for example by using an appropriate polarizing filter in the illumination section of the metrology device. A source suitable for the metrology device may also provide polarized radiation.

[0043]

[0064] As noted above, fabricated devices (e.g., patterned substrates) may be inspected at various points during manufacture. FIG. 4 shows a schematic of a generalized embodiment of a charged particle (e.g., electron beam) inspection apparatus 50. In some embodiments, the inspection apparatus 50 may be an electron beam or other charged particle inspection apparatus (e.g., the same as or similar to a scanning electron microscope (SEM)) that produces an image of a structure exposed or transferred onto a substrate (e.g., part or all of a structure of a device such as an integrated circuit). A primary electron beam 52 emitted from an electron source 54 is focused by a condenser lens 56 and then passes through a beam deflector 58, an E×B deflector 60, and an objective lens 62 to illuminate a substrate 70 on a substrate table ST at a focal point.

[0044]

[0065] When the substrate 70 is irradiated with the electron beam 52, secondary electrons are generated from the substrate 70. The secondary electrons are deflected by the E×B deflector 60 and detected by the secondary electron detector 72. A two-dimensional electron beam image can be obtained by detecting electrons generated from the sample, for example, in synchronization with a two-dimensional scan of the electron beam by the beam deflector 58, or a repeated scan of the electron beam 52 in the X-direction or the Y-direction by the beam deflector 58 (with a continuous movement of the substrate 70 in the other of the X-direction or the Y-direction by the substrate table ST). Thus, in some embodiments, the electron beam inspection apparatus has an electron beam field of view defined by the angular range over which the electron beam can be provided by the electron beam inspection apparatus (e.g., the angular range over which the deflector 60 can provide the electron beam 52). The spatial range of the field of view is therefore the spatial range over which the angular range of the electron beam can impinge on a surface (where the surface may be stationary or may move relative to the field of view).

[0045]

[0066] As shown in FIG. 4, the signal detected by the secondary electron detector 72 may be converted to a digital signal by an analog-to-digital (A / D) converter 74, and the digital signal may be sent to an image processing system 76. In some embodiments, the image processing system 76 may have a memory 78 for storing all or a portion of the digital image for processing by a processing unit 80. The processing unit 80 (e.g., specially designed hardware, or a combination of hardware and software, or a computer-readable medium including software) is configured to convert or process the digital image into a data set representative of the digital image. In some embodiments, the processing unit 80 is configured or programmed to cause the performance of the operations described herein (e.g., SEM inspection). Additionally, the image processing system 76 may have a storage medium 82 configured to store the digital image and the corresponding data set in a reference database. A display device 84 may be connected to the image processing system 76 so that an operator can use a graphical user interface to perform the required operations of the equipment.

[0046]

[0067] FIG. 5 shows a schematic diagram of another embodiment of a charged particle inspection apparatus. The apparatus is used to inspect a sample 90 (such as a patterned substrate) on a sample stage 89, and includes a charged particle beam generator 81, a condenser lens module 99, a probe-forming objective lens module 83, a charged particle beam deflection module 88, a secondary charged particle detector module 85, an image forming module 86, or other components. The charged particle beam generator 81 generates a primary charged particle beam 91. The condenser lens module 99 focuses the generated primary charged particle beam 91. The probe-forming objective lens module 83 focuses the focused primary charged particle beam onto a charged particle beam probe 92. The charged particle beam deflection module 88 scans the formed charged particle beam probe 92 over the surface of an area of ​​interest on the sample 90 fixed on the sample stage 89. In some embodiments, the charged particle beam generator 81, the condenser lens module 83, and the probe-forming objective lens module 83, or equivalent designs, alternatives, or any combination thereof, together form a charged particle beam probe generator that generates a scanning charged particle beam probe 92.

[0047]

[0068] The secondary charged particle detector module 85 detects secondary charged particles 93 (possibly along with other reflected or scattered charged particles from the sample surface) emitted from the sample surface when impacted by the charged particle beam probe 92 and generates a secondary charged particle detection signal 94. The imaging module 86 (e.g., a computing device) is coupled to the secondary charged particle detector module 85 to receive the secondary charged particle detection signal 94 from the secondary charged particle detector module 85 and form at least one scanned image in response thereto. In some embodiments, the secondary charged particle detector module 85 and the imaging module 86, or equivalent designs, alternatives, or any combination thereof, together form an imaging device that forms a scanned image from the detected secondary charged particles emitted from the sample 90 impacted by the charged particle beam probe 92.

[0048]

[0069] In some embodiments, the monitoring module 87 is coupled to the imaging module 86 of the imaging device to monitor, control, etc., the patterning process or to derive parameters for designing, controlling, monitoring, etc., of the patterning process using the scanned images of the sample 90 received from the imaging module 86. In some embodiments, the monitoring module 87 is configured or programmed to cause the performance of the operations described herein. In some embodiments, the monitoring module 87 includes a computing device. In some embodiments, the monitoring module 87 includes a computer program configured to provide the functionality described herein. In some embodiments, the probe spot size of the electron beam in the system of FIG. 3 is significantly larger, e.g., compared to the CD, such that the probe spot is large enough to allow for high inspection speeds. However, the large probe spot may result in reduced resolution.

[0049]

[0070] As described above, it may be desirable to use one or more tools to produce results that can be used to, for example, design, control, monitor, etc., a patterning process. One or more tools may be provided for use in computer-controlled, design, etc., one or more aspects of a patterning process, such as pattern design for a patterning device (e.g., including the addition of sub-resolution assist features or optical proximity correction), illumination for a patterning device, etc. Thus, in a system for computer-controlled, design, etc., of a manufacturing process involving patterning, the components or processes of the manufacturing system may be described by various functional modules or models. In some embodiments, one or more electronic (e.g., mathematical, parameterized, etc.) models may be provided that describe one or more steps or devices of the patterning process. In some embodiments, a simulation of the patterning process may be performed using one or more electronic models to simulate how the patterning process forms a patterned substrate using a design pattern provided by a patterning device.

[0050]

[0071] For example, images from the systems of Figures 4 or 5 may be processed to extract dimensions, shape, contour, or other information describing edges of objects representing semiconductor device structures in the images. The shape, contour, or other information may be quantified by metrics such as edge placement error (EPE), CD, etc., at user-defined cutlines or other locations. These shape, contour, or other information may be used, for example, to optimize the patterning process.

[0051]

[0072] In some embodiments, the optimization of the patterning process can be expressed as a cost function. The optimization process may include finding a set of parameters (design variables, process variables, etc.) of the patterning process that minimizes the cost function. The cost function may have any suitable form depending on the goal of the optimization. For example, the cost function may be a weighted root mean square (RMS) of deviations of certain features (value points) of the system from intended values ​​(e.g., ideal values) of these features. The cost function may also be the maximum of these deviations (i.e., the worst deviation). The term "value point" shall be interpreted broadly to include any feature of the system or fabrication method. The design or process variables of the patterning process may be limited to a finite range or may be interdependent for the practicality of the implementation of the system or method. In the case of a lithographic projection apparatus, the constraints are often related to the physical properties and characteristics of the hardware, such as adjustable ranges or manufacturability design rules of the patterning device. The value points may include physical points in an image of the substrate, as well as non-physical features, such as one or more etch parameters, dose, and focus.

[0052]

[0073] In an etch system, as an example, the cost function (CF) can be expressed as:

number

[0053]

[0074] The cost function may represent any one or more suitable characteristics of the patterning system, the patterning process, the lithographic apparatus, the lithographic process, or the substrate, such as focus, CD, image shift, image distortion, image rotation, stochastic variation, throughput, local CD variation, process window, inter-layer features, or a combination thereof. In some embodiments, the cost function may include a function that represents one or more characteristics of the resist image. For example, p (z 1 , z 2 , …, z N) is simply the distance between a point in the resist image and the intended location of that point (i.e., the edge placement error EPE, e.g., after etching or other processes). p (z 1 , z 2 , …, z N )) The parameters (e.g., design variables) may include any adjustable parameters of the etch system, source, patterning device, projection optics, adjustable parameters such as dose, focus, etc.

[0054]

[0075] There are constraints on the parameters (e.g., design variables), which means that (z 1 , z 2 , …, z N )∈Z, where Z is a set of possible values ​​of the design variables. One possible constraint on the design variables may be imposed by a desired throughput of the lithographic projection apparatus. Without such a constraint imposed by the desired throughput, the optimization may result in a set of unrealistic design variable values. The constraints are not to be construed as essential.

[0055]

[0076] Contours of pattern features are often determined based on images of substrate patterns. These contours are used to determine various key performance indicators (KPIs), which in turn are used to monitor semiconductor manufacturing process variations. As an example, edge placement error (EPE) has become a common KPI used to monitor process variations. EPE includes contributions from imaging (e.g., critical dimension (CD) and pattern placement variations) as well as contributions from overlay.

[0056]

[0077] Contours are stacked (e.g., overlaid) and analyzed to determine the variability of a given feature. However, some of the contours may be outliers, and stacking and averaging of contours "hides" these outliers. For example, contours may have varying quality (e.g., the image may not provide a clean, clear view of the edge of the feature used to determine the contour, and may instead be pixelated and obscured, etc.) or may have varying confidence levels (e.g., the image may be obscured from a lower quality image exactly where the contour is in the image, or a particular manufacturing process may produce anomalous results). Outliers may be caused by, for example, process variations or tool state drift, due to inherent differences in the pattern design (some patterns are more challenging for imaging, analysis, or other extraction tasks), or for other reasons. If these outlier contours make it into the stack and analysis is performed based on them, the outlier contours may affect any decisions drawn from the stack. Using EPE as an example, an EPE calculation based on a stack that includes outlier profiles can result in erroneous (or at least less accurate) EPE determinations and unnecessary manufacturing process adjustments to some manufacturing parameters when the "budget" for such adjustments might be best used for other manufacturing parameters.

[0057]

[0078] FIG. 6A depicts an example image of a substrate having features located on a first layer and features located on a second layer formed on the first layer, according to one embodiment. As seen in FIG. 6A, image 600 depicts a patterned substrate. For example, the substrate may be patterned by using a lithography apparatus as described with respect to FIG. 1, 18, or 19. In some embodiments, the substrate depicted by image 600 may be a stacked image including two or more layers stacked on top of each other. Each layer may include features formed through a patterning process. In some embodiments, the features may be the same or similar in size, shape, orientation, or other aspects, but may also differ slightly or significantly in one or more of the aforementioned aspects.

[0058]

[0079] In some embodiments, features formed on a first layer of a substrate may be buried significantly below one or more additional layers. For example, a patterned substrate may have a first layer including a first feature and a second layer including a second feature. The height of the second layer, which may be above the first layer, which may be above the substrate, may be 100 nm or more, 200 nm or more, 300 nm or more, 400 nm or more, 500 nm or more, or other heights.

[0059]

[0080] In some embodiments, patterning parameters such as overlay between features of a stack (e.g., a stack formed on a substrate via a lithographic patterning process) can be determined by analyzing a scanning electron microscope (SEM) image of a patterning device including the stack. Some techniques for determining patterning parameters such as overlay include direct contouring and cross-correlation against reference features, or other techniques employed by using the metrology systems and metrology techniques described herein. However, current techniques can produce inaccurate results when features included in the stack overlap or when the intensity of one of the features is much greater than the intensity of another feature. The latter case, for example, can be particularly prevalent when features of a "lower" layer are embedded under features of an "upper" layer. In such cases, the intensity of the features of the upper layer will be much stronger than the intensity of the features of the lower layer.

[0060]

[0081] As an example, referring to FIG. 6A, image 600 may depict an SEM image of a device patterned using a patterning process via a lithography apparatus as described herein. Image 600 depicts an image taken from "above" (e.g., looking down on the top surface of the device). From this view, a first feature 602 may be visible. The first feature 602 may be a horizontal grid. For example, the pattern of first features 602 and 604 between successive instances of first feature 602 may be represented using a two-dimensional sine function.

[0061]

[0082] In image 600, a first feature 602 may be formed on an upper layer of a patterning device. During the patterning process, a lithographic apparatus may form several layers on a substrate, where some or all of these layers may include features formed by the patterning process. If the height of one layer (or layers) formed on another layer is sufficiently high (e.g., greater than 300 nm, greater than 400 nm, greater than 500 nm, or greater than other heights), the intensity in image 600 of the feature formed on the upper layer may be greater than the intensity of the feature formed on the lower layer. In some cases, due to the intensity of the feature formed on the upper layer, a portion of the image signal (e.g., forming image 600) corresponding to the feature on the upper layer may cast a shadow on a portion of the image signal corresponding to the feature on the lower layer. As a result, the feature on the upper layer may be easily visible while the feature on the lower layer may be fully or partially hidden.

[0062]

[0083] Portion 610 may represent an example region of image 600. For example, in some cases, features on an upper layer and a lower layer may be periodic within a unit cell (each set of features having the same or different periods). Thus, portion 610 may depict a unit cell in which features on an upper layer and / or features on a lower layer may be symmetrically arranged. As seen in FIG. 6B, a zoomed-in version of portion 610 is depicted that includes feature 602 having width w1 along the Y axis. Between some pairs of features 602, there may be a feature 604 having width w2. In some cases, w1=w2, but otherwise w1≠w2. Additionally, in some cases, w1 and / or w2 may vary along the X and / or Y axes.

[0063]

[0084] In FIG. 6B, an instance of feature 620 formed on a lower layer (e.g., a layer formed closer to the upper surface of the substrate than the layer in which feature 602 is formed) may be partially visible. As can be seen from the dashed boundary of feature 620, feature 620 may be rectangular having a first side length w3 and a second side length w4, where w3>w4. However, the foregoing is merely exemplary, as one skilled in the art will recognize other shapes and sizes of shapes may be used. In addition to feature 602 being of different shapes and sizes, feature 620 may also be oriented in different ways. For example, relative to the orientation of feature 602 (e.g., uniform along the X-axis), feature 620 may be rotated by an angle θ.

[0064]

[0085] In some embodiments, there may be a relative shift between feature 602 and feature 620. The relative shift may be from their expected positions due to separate transfer of the features onto the substrate with overlay error. In this example, feature 602 is located in a higher (e.g., "upper") layer on the substrate than feature 620 (which is located on the lower layer). Thus, in some embodiments, feature 620 may be generated in a first, lower layer during a first run of a patterning process, and may be generated in a second layer that is higher than the first, lower layer during a second run of the patterning process. However, in some cases, features 602 and 620 may be formed in the same layer.

[0065]

[0086] Importantly, the unit cell has or can have geometric symmetry about an axis or point. For example, FIG. 6B can have mirror symmetry about an axis and / or point and rotational symmetry about the point. In some embodiments, the unit cell has geometric symmetry of a certain feature (such as overlay). In some cases, the unit cell can have zero overlay when it is geometrically symmetric. However, instead, the unit cell can have zero overlay for some geometric asymmetry. Appropriate offsets and calculations will then be used to compensate for the unit cell having zero overlay when it has some geometric asymmetry. The unit cell should be able to change in symmetry (e.g., become asymmetric), or should become even asymmetric, or should go from an asymmetric situation to a symmetric situation depending on some feature value.

[0066]

[0087] As an example, an overlay error for a unit cell when a feature on an upper layer is shifted in the X direction relative to a feature on a lower layer can result in an overlay error in the X direction (i.e., an X-direction overlay error). Of course, a feature on a lower layer could be shifted relative to a feature on an upper layer, or both could be shifted relative to each other. In any case, the result is an X-direction overlay error. However, as should be appreciated from this unit cell arrangement, a pure relative shift in the Y direction between a feature on an upper layer and a feature on a lower layer would not change the geometric symmetry of this unit cell. However, with appropriate geometric arrangements, overlay in the two directions of the unit cell or overlay between various combinations of portions can change the symmetry, and can also be determined.

[0067]

[0088] FIG. 7A is a schematic diagram of a portion of an image of a substrate having a stack of layers, each having one or more features, according to one embodiment. FIG. 7B is a schematic diagram of a portion of an image from FIG. 7A, according to one embodiment, where only the first layer including the first feature is shown. As seen in FIG. 7A, an image 700 of a portion of a substrate having a stack of layers is depicted. The substrate may be substantially similar to the substrate described above with reference to FIG. 6A. The image 700 includes first features 602, which may form a series of parallel lines along the Y-axis. The first features 602 may be separated by features 604. In some embodiments, the features 604 may be open spaces, but alternatively the features 604 may be formed in an upper layer.

[0068]

[0089] Also seen in image 700 can be feature 620 located on a lower layer of the stack. Similar to Figures 6A and 6B, feature 620 can be rectangular in shape and can be rotated by an angle θ relative to feature 602 on the upper layer of the stack.

[0069]

[0090] For simplicity, image 750 of FIG. 7B depicts an example of image 700 of FIG. 7A, where there are no features on an upper layer (e.g., feature 602). For example, this may refer to an image of a substrate prior to a patterning process that forms an upper layer on a lower layer. In image 750, feature 620 is shown, where a space 702 is formed between each instance of feature 620. In a scenario where feature 620 is symmetric about an axis (e.g., an axis rotated by an angle θ), feature 620 may have a distance d2 between the centers of consecutive instances of the feature. Additionally, a vector extending from the center of a feature to the center of an adjacent feature (e.g., directly above that feature along the Y axis) may have a distance d1 therebetween.

[0070]

[0091] Depending on the height of the lower layer, the upper layer, any intermediate layers, or a combination thereof, the clarity of features located on the lower layer in the captured image (e.g., image 600) may be too weak to precisely measure one or more process parameters. For example, particles of the incident beam may be scattered in the upper layer or layers, thereby achieving the resolution of the incident beam when it reaches the lower layer. Thus, the signal-to-noise ratio (SNR) may be very low for the lower layer. It may also be that the incident particles of the metrology beam may not be able to penetrate the lower layer, and therefore may not be reflected back to the image detector to reconstruct the image. As a result, defects and / or other aberrations may go undetected, affecting product performance. For example, feature 620 may be difficult to detect in image 600 because it may be on a layer located below the layer containing features 602 and 604. The difficulty in detection may manifest itself when using image processing techniques to identify physical characteristics of the patterned features (e.g., to detect edges representing the edges of the features, to detect the centroids of the features, etc.). Due to low signal-to-noise ratio (SNR) in the image signal in areas where embedded features are expected to be present, various feature detectors (e.g., edge detectors, Sobel filters, etc.) may not be able to determine that those features are present in the captured image.

[0071]

[0092] FIG. 8 corresponds to a method of applying a Fourier transform to a two-dimensional image according to one embodiment. In FIG. 8, at 800, image data may be acquired, including an image of a patterned substrate. For example, the image data may be captured by using a metrology system such as those described herein with respect to FIGS. 3-5. In some cases, the image data may be SEM image data, including an SEM image of a patterning device. The example image 600 of FIG. 6A may be an SEM image. In some embodiments, the image data may be captured and stored in a memory and retrieved for post-processing.

[0072]

[0093] At 802, a two-dimensional (2D) Fourier transform, which for simplicity may also be referred to herein interchangeably as a 2D Fast Fourier Transform (FTT) or FTT, may be applied to the image data. A 2D FFT may transform a function in a 2D image plane (e.g., the xy plane) into a function in a 2D frequency plane. For example, consider a 2D function f(x,y). The 2D FFT (F(u,v)) of this function is represented by Equation (1):

number

[0073]

[0094] The Fourier transform converts a function into a composition of basis functions. In particular, with respect to the image analysis (e.g., metrology process) examples described herein, the Fourier transform converts an image captured in image space into a composition of basis functions. For example, equations (2) and (3) represent the Fourier transforms of two different 2D sine functions:

number

number

[0074]

[0095] In terms of a 2D image (e.g., image 600), the process of applying a Fourier transform to an image involves processing the pixel values ​​of each pixel that forms the image. For example, image 600 may be an n×m pixel image, and thus each pixel (n,m) may have a value I that represents the intensity of light incident on that pixel via a measurement process. nm The 2D FFT operates in two steps: (1) a 1D FFT is performed along each column of pixels in the image; and (2) a 1D FFT is performed along each row of pixels in the image.

[0075]

[0096] The resulting image may be a 2D image of Fourier coefficients, where there are n x m Fourier coefficients. The lower frequency coefficients are located towards the corners of the image, while the higher frequency coefficients are located closer towards the center of the image. In some cases, the frequencies may be shifted so that the lower frequency coefficients are located towards the center and the higher frequency coefficients are located towards the corners.

[0076]

[0097] By applying a 2D FFT, an amplitude spectrum and a phase spectrum in Fourier space that together represent the image in image space can be generated. An amplitude image representing the amplitude spectrum can be calculated by calculating the amplitude for each frequency of the function in image space.

[0077]

[0098] As an example (in one dimension for simplicity), applying a Fourier transform to a function f(x)=A×cos(nx+φ) may generate a frequency spectrum of the function f(u)=A / 2exp(-iφ) for u=±n / 2π. Thus, an amplitude image may be calculated by plotting the amplitude (e.g., magnitude) of each frequency. In this way, the original signal in space may be represented by a set of frequencies combined with a sine function. A phase image may be generated by calculating the phase shift between signals with various frequencies.

[0078]

[0099] Therefore, for a 2D function f(x,y) in image space, a 2D FFT function F(u,v), where u and v are spatial frequencies, is calculated by using Equation (1). In other words, the 2D function f(x,y) can be expressed as a linear combination of harmonic functions.

[0079]

[0100] The 2D FTT function F(u,v) is a complex function containing real and imaginary parts. The amplitude spectrum is

number

number

[0080]

[0101] At 804, a first plot of the amplitude spectrum and a second plot of the phase spectrum may be generated. In some embodiments, the first plot and the second plot, or data representing the first plot, the second plot, or the first plot and the second plot, may be stored in a memory (e.g., memory 1706). In some embodiments, the first plot and / or the second plot may be rendered in a user interface that is displayed on a display device. The first plot of the amplitude spectrum may be interchangeably referred to herein as an "amplitude image," and the second plot of the phase spectrum may be interchangeably referred to herein as a "phase image." In some embodiments, the phase image and the amplitude image are generated in parallel, but may alternatively be generated sequentially.

[0081]

[0102] FIG. 9 depicts an exemplary amplitude image obtained by applying a Fourier transform to the image of FIG. 6A according to one embodiment. Amplitude image 900 includes a first set of peaks 902 and a second set of peaks 904. Each peak in the first set of peaks 902 may represent a Fourier transform of an image signal of an image related to a feature on the "upper" layer. For example, peak 910 is a peak from the first set of peaks 902 that is related to a feature on the "upper" layer (e.g., a horizontal grating). On the other hand, each peak in the second set of peaks 904 may represent (a portion of) a Fourier transform of (a portion of) an image signal of an image related to a feature on the "lower" layer. For example, peaks 912 and 914 are peaks from the second set of peaks 904 that is related to a feature on the "lower" layer (e.g., a rectangular feature). Each peak displayed in amplitude image 900 may be associated with a particular location in Fourier space (e.g., defined by u and v coordinates). Furthermore, each point in Fourier space (eg, {u,v}) may correspond to a sinusoidal signal in image space.

[0082]

[0103] The technical improvement of performing a Fourier transform on the image of the patterned substrate in image space is that "even if embedded features produce very weak image signals that contribute to the overall final output image, these weak signals still have a defined representation in Fourier space that can be separated and analyzed."

[0083]

[0104] Based on the amplitude image, a determination can be made as to which peaks are associated with which features. In some embodiments, one or more optimization algorithms can be used to fit the two sets of peaks to the peaks in the amplitude image. In some embodiments, the peak locations and amplitudes are fitted using a non-linear least squares optimization routine, where the peak shapes are fitted using a Gaussian function on a constant background representing the noise level, but the peak shapes can also be described by a Fourier transform of a window function applied to the image in image space. The two sets of peaks can correspond to peaks associated with features on the "upper" layer (e.g., feature 602) and peaks associated with features on the "lower" layer (e.g., feature 620). Based on the fitted peaks, the peak locations of each peak in Fourier space can be obtained with sub-pixel accuracy.

[0084]

[0105] Fitting the peaks may allow sub-pixel accuracy to be obtained for each peak, allowing sub-pixel phase resolution in determining the phase associated with a given peak. The greater the accuracy in determining the phase, the greater the accuracy in determining patterning process metrics such as overlay. As an example, referring to FIG. 10A, an amplitude image 1000 shows a 3×3 pixel array. Each pixel in the pixel array may have a height and width D, which may vary and are set equal to one another for simplicity.

[0085]

[0106] In the magnitude image 1000, each circle represents the center of a given pixel in Fourier space. For example, the nine pixels can be labeled via their center coordinates (e.g., pixel center: {u i ,v j}, where i=0,1,2 and j=0,1,2). If a peak in the magnitude image is at one of the center positions of one of the pixels, it is possible to determine the phase of the corresponding pixel.

number

[0086]

[0107] As an example, referring to FIG. 10B, an amplitude image 1050 represents a subsection of the pixel array seen in FIG. 10A. The amplitude image 1050 depicts a diamond representing a peak (e.g., one of peaks 910, 912, 914 in FIG. 9) generated, for example, via a Fourier transform of the image signal. To determine the coordinates of the diamond, an interpolation process is performed to calculate the relative position of the diamond to the centers of the surrounding pixels. For example, vectors r11, r12, r21, and r22 may be calculated. Vector r11 represents the distance from center {u1,v1} to the diamond, vector r12 represents the distance from center {u1,v2} to the diamond, vector r21 represents the distance from center {u2,v1} to the diamond, and vector r22 represents the distance from center {u2,v2} to the diamond. These vectors (e.g., vectors r11-r22) and the centers of each pixel are related to the coordinate position of the diamond, {u * ,v *} can then be used to determine the coordinate position {u * ,v *} is the phase of the peak (e.g.

number

[0087]

[0108] In some embodiments, each peak and its corresponding peak location in Fourier space may be sorted into one of two sets of peaks: a first set of peaks associated with features formed on an upper layer of the imaged device (e.g., a substrate including a stack of layers) (e.g., first set of peaks 902), or a second set of peaks associated with features formed on a lower layer of the imaged device (e.g., second set of peaks 904). The peak locations may then be used to calculate phase from the phase image.

[0088]

[0109] 11 is a schematic diagram of a phase image 1100. The phase image 1100 depicts the phase associated with each peak from the magnitude image 900. For example, point 1110 corresponds to peak 910, point 1112 corresponds to peak 912, and point 1114 corresponds to peak 914. The phase represents the shift in image space of a sinusoidal signal having a propagation vector k that extends from the center of the feature (e.g., the origin in Fourier space) to the point of interest. For example, referring to FIG. 12, the magnitude image 1200 depicts a vector k that extends from the origin in Fourier space to the point of interest in Fourier space. The magnitude of the vector k is related to the unit cell dimension of the feature as seen by equation (4):

number

[0089]

[0110] The phase is the shift of the sinusoidal signal in image space relative to the vector k. The phase φ is

number

[0090]

[0111] In some cases, due to symmetries in the original image (e.g., periodicity in the pattern of features 602, 604, 620), the phase angles may cancel each other out, and therefore areas of the phase image 1100 that contain data points may represent areas where there is an asymmetry in the pattern. This asymmetry may be the result of overlay or other asymmetries resulting from the lithographic printing process.

[0091]

[0112] In terms of processing a 2D image signal, calculating the phase may involve performing an FFT on the original signal (e.g., the (x,y) pixel data) and extracting the real component of the transformed signal and the imaginary part of the transformed signal. Then, the amplitude of vector k is

number

number

[0092]

[0113] In some embodiments, one or more image processing steps may be performed prior to analyzing the phase image 1100. For example, a thresholding filter may be applied to reduce noise in the image signal.

[0093]

[0114] In some embodiments, the overlay error, represented by the vector ΔOV, may be projected onto a point of interest in the magnitude image, for example as seen in Figure 12. The overlay error may be projected with respect to a reference overlay condition (such that overlay(OVL)=0). The overlay error is related to the extracted phase φ according to equation (5):

number

number

[0094]

[0115] With reference to FIG. 13, the method may begin at 1300, where image data representative of an image of a patterning device (e.g., a substrate having a stack in which one or more layers of the stack have been patterned using a lithographic patterning process) may be acquired. In some embodiments, the image data may be SEM image data representative of a SEM image. For example, the image data (e.g., SEM image data) may be captured by using a metrology system such as described above in connection with FIGS. 3-5. In some embodiments, one or more pre-processing steps (noise reduction, grayscale, cropping, scaling, windowing, etc.) may be performed on the image. FIG. 6A depicts one example of an image of a patterning device.

[0095]

[0116] At 1302, amplitude and phase images may be generated based on applying an FFT to image data of a patterning device (e.g., a substrate with a stack). The Fourier transform decomposes the aerial image of the patterning device into a set of sine functions. Thus, the amplitude image may depict an image of the intensity of each sine function at a particular frequency. Since the intensity at each point corresponds to the amplitude of the sine function at a given 2D frequency, the amplitude of each point in Fourier space may be determined.

[0096]

[0117] At 1304, the locations of the peaks in the magnitude image may be fitted. For example, a 2D sine function may be fitted to the peaks. For example, a Gaussian function with noise may be used. Based on this fitting function, the location in Fourier space of each peak may be extracted. In some embodiments, the peaks may be stored with an associated location vector in Fourier space.

[0097]

[0118] The overlay vector is calculated based on a k-vector that extends from an origin in Fourier space to a point of interest in Fourier space. For example, as seen in FIG. 12, vector k originates at point p1 and extends to point p2. The vector may be represented by 2D frequency values ​​of a location in Fourier space. Thus, vector k may originate at point p1 with frequency coordinates u1,v1 (e.g., u1=0, v1=0) and extend to point p2 with frequency coordinates u,2,v2 (e.g., u2=u, v2=v). The corresponding coordinates in Fourier space of each peak may then be stored (e.g., {p1:u1,v1}, {p2:u2,v2}). In some embodiments, multiple sets of peaks may be fitted to the amplitude of the amplitude image 900. Each set of peaks may correspond to a feature of a layer of the patterning device. For example, a first set of peaks can be fitted to features on a first layer of a patterning device and a second set of peaks can be fitted to features on a second layer of the patterning device. Whether a peak is fitted to the first set of peaks or the second set of peaks can be used to determine which peak refers to a feature in an "upper" layer or a feature in a "lower" layer.

[0098]

[0119] At 1306, a phase may be extracted from a phase image (e.g., phase image 1100) of a given peak. In some embodiments, the phase may be extracted from a peak representing a feature on a "lower" layer of the imaged substrate (e.g., a patterning device including a stack). In some cases, the phase may be extracted from a peak representing a feature on an "upper" layer of the imaged substrate. In some embodiments, the phase associated with each peak (or some of the peaks) may be stored in association with a frequency location in Fourier space (e.g., {p1:u1,v1,φ1},{p2:u2,v2,φ2}). The phase may be

number

[0099]

[0120] To achieve sufficient accuracy, phase extraction may involve interpolating neighboring pixels in the phase image. In some embodiments, u,v are fractional coordinates. The interpolated pixel may be close to a phase wrapping boundary (e.g., where the phase jumps from a value near +π to a value near -π). A phase unwrapping algorithm may be applied to accurately interpolate the phase.

[0100]

[0121] At 1308, the extracted phase may be translated into an overlay value. The overlay may be related to the extracted phase via equation (5). Simulations further show that a calibration (or sensitivity constant) may be required to improve the accuracy of the overlay calculation. A calibrated version of equation (6) is shown below:

number

[0101]

[0122] To perform the calibration, multiple images (e.g., SEM images) of a patterning device may be acquired. The images may be based on simulations of a particular patterning device, or the images may be images of a physical patterning device. As an example, graph 1400 shows three (3) traces: trace 1402, trace 1404, and trace 1406. Each trace is associated with a k-vector orientation. For example, trace 1402 is associated with a k-vector orientation of -21 degrees, trace 1404 is associated with a k-vector orientation of 48 degrees, and trace 1406 is associated with a k-vector orientation of 138 degrees. As an example, referring to image 1500 of FIG. 15, a k-vector orientation of 21 degrees points to point 1502 in image 1500, a k-vector orientation of 48 degrees points to point 1504, and a k-vector orientation of 138 points to point 1506. Referring again to FIG. 14, the x and y components of overlay can be calculated by using equation (5) for each peak (e.g., k-vector orientations −21 degrees, 48 ​​degrees, 138 degrees) of each image of the multiple images of the patterning device. These points are plotted in graph 1400. For each set of points corresponding to a given k-vector orientation, a linear function can be fitted to the data points to obtain calibration coefficients for the overlay calculation. For example, the calibration coefficient for a phase angle of −21 degrees should be 1.6, while the calibration coefficients for phase angles of 48 degrees and 138 degrees should be 0.54 and 0.86, respectively. In some embodiments, a higher level model can be used to relate phase to overlay.

[0102]

[0123] In some embodiments, the process of extracting phase from the phase image may involve interpolation techniques to interpolate phase values ​​on the phase image grid to (sub-pixel accuracy) peak positions obtained by fitting the peak positions(s) on the amplitude image. Some embodiments may use linear nearest neighbor or bicubic interpolation. Other embodiments may also include 1D or 2D phase unwrapping techniques to produce a more accurate interpolation of the phase.

[0103]

[0124] The effect that calibration has can be seen, for example, with respect to Figure 16. In Figure 16, graph 1600 depicts an example of data points calculated for a phase of 138 degrees. The slope of these data points is 0.86. However, the dashed line depicts how the fitted data points would behave if the slope were ideal (e.g., slope = 1.0).

[0104]

[0125] In some embodiments, the relationship between phase and overlay may be estimated by using one or more machine learning models. For example, a machine learning model may be trained to predict overlay between features (e.g., features on various layers of a stack) based on phase. Machine learning techniques that may be used within the systems described herein may include, but are not limited to (and no other list is implied to be limiting): Ordinary Least Squares Regression (OLSR), Linear Regression, Logistic Regression, Stepwise Regression, Multivariate Adaptive Regression Splines (MARS), Locally Estimated Scatterplot Smoothing (LOESS), Instance-Based Algorithms, k-Nearest Neighbor (KNN), Learning Vector Quantization (LVQ), Self-Organizing Maps (SOM), Locally Weighted Learning (LWL), and others. ), Regularization algorithms, Ridge Regression, Least Absolute Shrinkage and Selection Operator (LASSO), Elastic Net, Least-Angle Regression (LARS), Decision Tree Algorithms, Classification and Regression Trees (CART), Iterative Dichotomizer 3 (ID3), C4.5 and C5.0 (different versions of powerful approaches), Chi-squared Automatic Interaction Detection (CHAID), Decision Stump, M5, Conditional Decision Trees, Naive Bayes, Gaussian Naive Bayes, Causality Networks (CN), Multinomial Naive Bayes, Averaged One-Dependence Estimators (AODE), Bayesian Belief Networks (BBN), Bayesian Networks (BN), k-Means, k-Median, K-Cluster, Expectation Maximization (EM), Hierarchical Clustering, Association Rules Learning algorithms, A-priori algorithms, Eclat algorithms, artificial neural network algorithms, Perceptron, Back-Propagation, Hopfield networks, Radial Basis Function networks (RBFN), deep learning algorithms, Deep Boltzmann machines (DBM), Deep Belief networks (DBN), Convolutional Neural Networks (CNN), deep metric learning, Stacked Auto-Encoders, Dimensionality Reduction algorithms, Principal Component Analysis (PCA), Principal Component Regression (PCR), Partial LeastSquares Regression (PLSR), Collaborative Filtering (CF), Latent Affinity Matching (LAM), Cerebri Value Computation (CVC), Multidimensional Scaling (MDS), Projection Pursuit, Linear Discriminant Analysis (LDA), Mixture Discriminant Analysis (MDA), Quadratic Discriminant Analysis (QDA), Flexible Discriminant Analysis (FDA), Ensemble Algorithms, Boosting, Bootstrapped Aggregation (Bagging), AdaBoost, Stacked Generalization (blending), Gradient Boosting Machines (GBM), Gradient Boosted Regression Trees (GBRT), Random Forest, Computational intelligence (evolutionary algorithms, etc.), Computer Vision (CV), Natural Language Processing (NLP), Recommender Systems, Reinforcement Learning, Graphical Models, or separable convolutions (e.g. depth-separable convolutions) convolutions, spatially separable convolutions).

[0105]

[0126] In some embodiments, the machine learning model may be trained based on training data formed of multiple SEM images. Each of the SEM images may depict a patterned device having features on a first lower layer and a second upper layer. The various images may depict similar patterns or different patterns, and the extent of overlay may vary across the various images. In some embodiments, the overlay may be a known value (either preprogrammed or pre-set during the fabrication process) or a measured value (e.g., by using one or more metrology processes).

[0106]

[0127] In some embodiments, an FFT may be applied to the SEM image. For each SEM image captured in image space, the FFT may generate an amplitude image and a phase image that represent the SEM image in Fourier space. Using the method described above in connection with FIG. 12, the acquired amplitude image may be fitted to determine the peak location in Fourier space for each peak that represents a feature in image space. Peaks corresponding to features on the "upper" layer and peaks corresponding to features on the "lower" layer may be identified and used to determine the phase angle that represents a given point in the amplitude image that corresponds to the peak of the feature (e.g., a feature from the lower layer). The phase may then be used to calculate overlay or overlay error.

[0107]

[0128] The machine learning model may be trained during a training phase and validated to ensure accuracy. Once training and validation are complete, the trained model may be deployed and used to calculate overlay between features of a stack based on an input topology.

[0108]

[0129] Referring to Figure 17, a computer system 1700 is shown. The computer system 1700 includes a bus 1702 or other communication mechanism for communicating information and a processor 1704 (or multiple processors 1704 and 1705) coupled to the bus 1702 for processing information. The computer system 1700 also includes a main memory 1706 (such as a random access memory (RAM) or other dynamic storage device) coupled to the bus 1702 for storing information and instructions executed by the processor 1704. The main memory 1706 may also be used to store temporary variables or other intermediate information during execution of instructions executed by the processor 1704. The computer system 1700 further includes a read only memory (ROM) 1708 or other static storage device coupled to the bus 1702 for storing static information and instructions for the processor 1704. A storage device 1710 , such as a magnetic disk or optical disk, is provided and coupled to bus 1702 for storing information and instructions.

[0109]

[0130] The computer system 1700 may be coupled via bus 1702 to a display 1712, such as a cathode ray tube (CRT), flat panel, or touch panel display, for displaying information to a computer user. An input device 1714, including alphanumeric and other keys, is coupled to the bus 1702 for communicating information and command selections to the processor 1704. Another type of user input device is a cursor control 1716, such as a mouse, trackball, or cursor direction keys, for communicating directional information and command selections to the processor 1704 and for controlling cursor movement on the display 1712. This input device generally has two degrees of freedom in two axes, a first axis (e.g., x) and a second axis (e.g., y), allowing the device to be positioned in a plane. A touch panel (screen) display may be used as an input device.

[0110]

[0131] The computer system 1700 may be suitable to function as a processing unit herein in response to the processor 1704 executing one or more sequences of instructions contained in the main memory 1706. Such instructions may be read into the main memory 1706 from another computer-readable medium, such as the storage device 1710. Execution of a sequence of instructions contained in the main memory 1706 causes the processor 1704 to perform the processes described herein. One or more processors in a multi-processing arrangement may be employed to execute a sequence of instructions contained in the main memory 1706. In some alternative embodiments, hardwired circuitry may be used in place of or in combination with software instructions. Thus, some embodiments are not limited to any specific combination of hardware circuitry and software.

[0111]

[0132] The term "computer-readable medium" as used herein refers to any medium that participates in providing instructions to the processor 1704 for execution. Such media may take many forms, including but not limited to non-volatile media, volatile media, and transmission media. Non-volatile media include optical or magnetic disks, such as the storage device 1710. Volatile media include dynamic memory, such as the main memory 1706. Transmission media include coaxial cables, copper wire and fiber optics, including the wires that comprise the bus 1702. Transmission media may also take the form of acoustic or light waves, such as those generated during radio frequency (RF) and infrared (IR) data communications. Common forms of computer-readable media include, for example, a floppy disk, a flexible disk, a hard disk, a magnetic tape, any other magnetic medium, a CD-ROM, a DVD, any other optical medium, punch cards, paper tape, any other physical storage medium with a pattern of holes, a RAM, a PROM, and an EPROM, a FLASH-EPROM, any other memory chip or cartridge, a carrier wave as described herein, or any other medium from which a computer may read.

[0112]

[0133] Various forms of computer readable media may be involved in carrying one or more sequences of one or more instructions to the processor 1704 for execution. For example, the instructions may initially reside 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 the computer system 1700 can receive the data on the telephone line and use an infrared transmitter to convert the data to an infrared signal. An infrared detector coupled to the bus 1702 can receive the data carried in the infrared signal and place the data on the bus 1702. The bus 1702 carries the data to the main memory 1706, from which the processor 1704 reads and executes the instructions. The instructions received by the main memory 1706 may optionally be stored on a storage device 1710 either before or after execution by the processor 1704.

[0113]

[0134] The computer system 1700 may also include a communications interface 1718 coupled to the bus 1702. The communications interface 1718 also provides a two-way data communication coupling to a network link 1720 that is connected to a local network 1722. For example, the communications interface 1718 may 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, the communications interface 1718 may be a Local Area Network (LAN) card to provide a data communication connection to a compatible LAN. A wireless link may also be implemented. In such an implementation, the communications interface 1718 sends and receives electrical, electromagnetic or optical signals that carry digital data streams representing various types of information.

[0114]

[0135] The network link 1720 typically provides data communication through one or more networks to other data devices. For example, the network link 1720 may provide a connection through a local network 1722 to a host computer 1724 or to data equipment operated by an Internet Service Provider (ISP) 1726. The ISP 1726 in turn provides data communication services through the worldwide packet data communication network now commonly referred to as the "Internet" 1728. Both the local network 1722 and the Internet 1728 use electrical, electromagnetic or optical signals that carry digital data streams. The signals through the various networks and the signals on the network data link 1720 and through the communication interface 1718, which carry the digital data to and from the computer system 1700, are exemplary forms of carrier waves transporting the information.

[0115]

[0136] The computer system 1700 may send messages and receive data, including program code, through the network(s), network data link 1720, and communication interface 1718. In the Internet example, a server 1730 may send a requested code for an application program through the Internet 1728, the ISP 1726, the local network 1722 and the communication interface 1718. According to one or more embodiments, one such downloaded application may provide, for example, the methods disclosed herein. The received code may be executed by the processor 1704 once received, and / or stored in the storage device 1710, or other non-volatile storage for later execution. In this manner, the computer system 1700 may obtain the application code in the form of a carrier wave.

[0116]

[0137] FIG. 18 is a schematic diagram of another lithographic projection apparatus (LPA) that may be used for or facilitate one or more of the operations described herein. The LPA shown in FIG. 18 is similar or the same as the apparatus shown in FIG. 1. The LPA may include a source collector module SO, an illumination system (illuminator) IL configured to condition a radiation beam B (e.g., EUV radiation), a patterning device table T, a substrate table WT, and a projection system PS. The patterning device table T may be constructed to support a patterning device (e.g., a mask or reticle) MA and may be connected to a first positioner PM configured to accurately position the patterning device. The substrate table (e.g., a wafer table) WT may be constructed to hold a substrate (e.g., a resist-coated wafer) W and may be connected to a second positioner PW configured to accurately position the substrate. The projection system (eg a reflective projection system) PS may be configured to project a pattern imparted to the radiation beam B by a patterning device MA onto a target portion C of the substrate W (eg comprising one or more dies).

[0117]

[0138] As this example shows, the LPA can be reflective (e.g., using a reflective patterning device). Note that since most materials are absorbing in the EUV wavelength range, the patterning device can have a multi-layer reflector including, for example, multiple stacks of molybdenum and silicon. In one example, a multi-stack reflector has 40 layer pairs of molybdenum and silicon, with each layer being a quarter wavelength thick. Even smaller wavelengths can be produced using x-ray lithography. Since most materials are absorbing at EUV and x-ray wavelengths, a thin piece of patterned absorbing material (e.g., TaN absorber on a multi-layer reflector) on the patterning device topography defines where features will or will not print (positive resist) or print (negative resist).

[0118]

[0139] The illuminator IL may receive an extreme ultraviolet radiation (EUV) beam from the source collector module SO. Methods of generating EUV radiation include, but are not necessarily limited to, converting a material into a plasma state having at least one element (e.g., xenon, lithium, or tin) with one or more emission lines in the EUV range. In one such method, often referred to as laser-produced plasma ("LPP"), the plasma may be generated by irradiating a fuel, such as a droplet, stream, or cluster of a material having a line-emitting element, with a laser beam. The source collector module SO may be part of an EUV radiation system that includes a laser (not shown in FIG. 17) that provides a laser beam that excites the fuel. The resulting plasma emits output radiation (e.g., EUV radiation), which is collected using a radiation collector disposed in the source collector module. The laser and the source collector module may be separate entities, for example, when a CO2 laser is used to provide the laser beam for fuel excitation. In such a case, the laser is not considered to form part of the lithographic apparatus, and the radiation beam may be passed from the laser to the source collector module using a beam delivery system including, for example, appropriate directing mirrors or beam expanders. In other examples, the source may be an integral part of the source collector module, for example when the source is a discharge produced plasma EUV generator, often referred to as a DPP source.

[0119]

[0140] The illuminator IL may include an adjuster for adjusting the angular intensity distribution of the radiation beam. Generally, at least the outer or inner radial extent (commonly referred to as σ-outer and σ-inner, respectively) of the intensity distribution in a pupil plane of the illuminator may be adjusted. In addition, the illuminator IL may include various other components, such as facetted field and pupil mirror devices. The illuminator may be used to condition the radiation beam to have a desired uniformity and intensity distribution in its cross-section.

[0120]

[0141] The radiation beam B may be incident on and patterned by a patterning device (e.g. mask) MA, which is held on a patterning device table T. After being reflected from the patterning device (e.g. mask) MA, the radiation beam B passes through a projection system PS, which focuses the beam onto a target portion C of the substrate W. The substrate table WT may be accurately moved (e.g. to position a different target portion C in the path of the radiation beam B) using a second positioner PW and a position sensor PS2 (e.g. an interferometric device, a linear encoder or a capacitive sensor). Similarly, the patterning device (e.g. mask) MA may be accurately positioned with respect to the path of the radiation beam B using a first positioner PM and another position sensor PS1. The patterning device (e.g. mask) MA and substrate W may be aligned using patterning device alignment marks M1, M2 and substrate alignment marks P1, P2.

[0121]

[0142] The depicted apparatus LPA can be used in at least one of the following modes: step mode, scan mode and stationary mode. In step mode, the patterning device table T and the substrate table WT remain essentially stationary and a pattern imparted to the radiation beam is projected onto the target portion C in its entirety at once (e.g. a single static exposure). The substrate table WT is then shifted in the X or Y direction so that a different target portion C can be exposed. In scan mode, the patterning device table T and the substrate table WT are scanned synchronously while a pattern imparted to the radiation beam is projected onto the target portion C (i.e. a single dynamic exposure). The velocity and direction of the substrate table WT relative to the patterning device table T may be determined by the de-magnification and image reversal characteristics of the projection system PS. In stationary mode, the patterning device table T remains essentially stationary holding a programmable patterning device and the substrate table WT is moved or scanned while a pattern imparted to the radiation beam is projected onto the target portion C. In this mode, a pulsed radiation source is typically used and the programmable patterning device is updated as required after each movement of the substrate table WT, or between successive radiation pulses during a scan. This mode of operation can be readily applied to maskless lithography utilising a programmable patterning device such as a programmable mirror array of the type referred to above.

[0122]

[0143] FIG. 19 is a more detailed diagram of the lithographic projection apparatus shown in FIG. 19 (or FIG. 1). As shown in FIG. 19, the LPA may include a source collector module SO, an illumination system IL, and a projection system PS. The source collector module SO is configured such that a vacuum environment may be maintained within a closed structure 220 of the source collector module SO. The EUV radiation emitting plasma 210 may be formed by a discharge produced plasma source. The EUV radiation may be generated by a gas or vapor (e.g. Xe gas, Li vapor, or Sn vapor from which a high temperature plasma 210 is created to emit radiation in the EUV range of the electromagnetic spectrum). The high temperature plasma 210 is created, for example, by a discharge that produces an at least partially ionized plasma. A partial pressure of, for example, 10 Pa of Xe, Li, Sn vapor, or any other suitable gas or vapor may be required for efficient generation of radiation. In some embodiments, a plasma of excited tin (Sn) is provided to generate the EUV radiation.

[0123]

[0144] Radiation emitted by the high temperature plasma 210 passes from the source chamber 211 through an optional gas barrier or contaminant trap 230 (sometimes also called a contaminant barrier or foil trap) located in or behind the opening of the source chamber 211 and into the collector chamber 212. The contaminant trap 230 may include a channel structure. The contaminant trap 230 may also include a gas barrier, or a combination of a gas barrier and a channel structure. The contaminant trap or contaminant barrier trap 230 (shown below) also includes a channel structure. The collector chamber 212 may include a radiation collector CO, which may be a grazing incidence collector. The radiation collector CO has an upstream radiation collector side 251 and a downstream radiation collector side 252. Radiation traversing the collector CO may be reflected off a grating spectral filter 240 to be focused into a virtual source point IF along an optical axis indicated by line "O". The virtual source point IF is commonly called the intermediate focus, and the source collector module is positioned such that the intermediate focus IF is located at or near the opening 221 of the enclosure structure 220. The virtual source point IF is an image of the radiation emitting plasma 210.

[0124]

[0145] The radiation subsequently traverses the illumination system IL, which may include a faceted field mirror device 22 and a faceted pupil mirror device 24 arranged to provide a desired angular distribution of the radiation beam 21 at the patterning device MA, and a desired uniformity of the radiation intensity at the patterning device MA. Upon reflection of the radiation beam 21 at the patterning device MA held by the patterning device table 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 a substrate table WT. In general, more elements than shown may be present in the illumination optics unit IL and the projection system PS. A grating spectral filter 240 may optionally be present, for example depending on the type of lithographic apparatus. Furthermore, there may be more mirrors than shown in the drawings, for example 1 to 6 additional reflective elements than shown in FIG. 19 may be present in the projection system PS.

[0125]

[0146] 19 is depicted as a nested collector with grazing incidence reflectors 253, 254, and 255, just as an example of a collector (or collector mirror). The grazing incidence reflectors 253, 254, and 255 are arranged axially symmetrically about the optical axis O, and this type of collector system CO may be used in combination with a discharge produced plasma source, often referred to as a DPP source.

[0126]

[0147] 20 is a detailed view of the source collector module SO of the lithographic projection apparatus LPA (shown in the previous figure). The source collector module SO may be part of the LPA radiation system. The laser LA may be arranged to deposit laser energy into a fuel such as xenon (Xe), tin (Sn) or lithium (Li) to generate a highly ionized plasma 210 with an electron temperature of a few tens of eV. Energetic radiation generated during de-excitation and recombination of these ions is emitted from the plasma, collected by a near-normal incidence collector system CO and focused onto an opening 221 of a closing structure 220.

[0127]

[0148] An embodiment of the present disclosure may take the form of a computer program comprising one or more sequences of machine-readable instructions describing the methods as disclosed herein, or a data storage medium (e.g., a semiconductor memory, a magnetic or optical disk) having such a computer program stored thereon. Further, the machine-readable instructions may be embodied in two or more computer programs. The two or more computer programs may be stored on one or more different memories and / or data storage media.

[0128]

[0149] Any controllers described herein, each or in combination, are operable when one or more computer programs are read by one or more computer processors located in at least one component of the lithographic apparatus. The controllers, each or in combination, may have any suitable configuration for receiving, processing, and transmitting signals. The one or more processors are configured to communicate with at least one of the controllers. For example, each controller may include one or more processors for executing a computer program including machine-readable instructions of the above-mentioned methods. The controllers may include a data storage medium for storing such computer programs, and / or hardware to accommodate such a medium. Thus, the controllers may operate according to the machine-readable instructions of one or more computer programs.

[0129]

[0150] Although specific reference may be made in this document to the use of metrology apparatus in the manufacture of ICs, it should be understood that the metrology apparatus and processes described herein may have other applications (such as the manufacture of integrated optical systems, guidance and detection patterns for magnetic domain memories, flat panel displays, liquid crystal displays (LCDs), thin film magnetic heads, etc.). Those skilled in the art will understand that any use of the terms "wafer" or "die" herein in the context of such alternative applications may be considered synonymous with the more general terms "substrate" or "target portion", respectively. Substrates referred to herein may be processed before or after exposure in, for example, a track (a tool that typically applies a layer of resist to a substrate and develops the exposed resist), a metrology tool, and / or one or more of a variety of other tools. Where applicable, what is disclosed herein may be applied to such and other substrate processing tools. Furthermore, a substrate may be processed more than once, for example to produce a multi-layer IC, such that the term "substrate" as used herein may also refer to a substrate that already includes multiple processing layers.

[0130]

[0151] Although specific reference may be made to the use of embodiments of the present disclosure in the context of optical lithography, it will be understood that the present disclosure may be used in other applications (e.g., nanoimplant lithography) and is not limited to optical lithography, where the context permits, In the case of nanoimplant lithography, the patterning device is an imprint template or mold.

[0131]

[0152] The concepts disclosed herein may be simulated or mathematically modeled on any common imaging system for imaging sub-wavelength features and may be particularly useful with emerging imaging technologies that can produce shorter and shorter wavelengths. Emerging technologies already in use include EUV (extreme ultraviolet), DUV lithography, which can produce 193 nm wavelengths using ArF lasers and even 157 nm wavelengths using fluorine lasers. EUV lithography can also produce wavelengths in the range of 20-5 nm by using synchrotrons or by bombarding materials (solids or plasmas) with high energy electrons to produce photons in this range.

[0132]

[0153] Although the concepts disclosed herein may be used for imaging on substrates such as silicon wafers, it should be understood that the disclosed concepts may be used in any type of lithographic imaging system (e.g., those used for imaging on substrates other than silicon wafers). In addition, combinations and subcombinations of the disclosed elements may include separate embodiments. For example, determining the enhanced MRC criteria, as described herein, may include its own separate embodiment, or may be included with one or more other embodiments that also include performing the actual review.

[0133]

[0154] The above description is intended to be illustrative and not limiting. Thus, it will be apparent to one skilled in the art that modifications as described may be made without departing from the scope of the claims set forth below.

[0134]

[0155] Further embodiments according to the invention are described in the following numbered clauses: 1. A non-transitory computer readable medium storing computer program instructions that, when executed by a computer, cause the computer to perform operations including obtaining image data of features on a substrate, analyzing the image data in Fourier space, determining amplitude and phase based on the analysis, and determining overlay of the features based on the amplitude and phase. 2. The non-transitory computer-readable medium of clause 1, wherein the features include a first feature on a first layer of the substrate and a second feature on a second layer of the substrate formed on the first layer. 3. The non-transitory computer-readable medium of clause 2, wherein the image data includes an image of a substrate formed by scanning an electron beam in one or more directions. 4. The non-transitory computer-readable medium of any one of clauses 1 to 3, wherein the image data is captured via electron microscopy. 5. The non-transitory computer-readable medium of clause 4, wherein a scanning electron microscope (SEM) is used to perform the electron microscopy to capture an SEM image, and the image data includes SEM image data representative of the SEM image. 6. The non-transitory computer-readable medium of any one of clauses 1 to 5, wherein analyzing the image data in Fourier space includes generating an amplitude image and a phase image in response to application of a Fast Fourier Transform (FFT) to the image data, wherein the amplitude is determined based on the amplitude image and the phase is determined based on the phase image. 7. The non-transitory computer-readable medium of clause 6, wherein the image data includes images of a substrate, a first layer formed on the substrate, and a second layer formed on the first layer, and the amplitude image depicts amplitudes of features in Fourier space, the features being formed on at least one of the first layer or the second layer, and determining the amplitudes includes determining a first set of features located on the first layer from the features based on the amplitudes, and determining a second set of features located on the second layer from the features based on the amplitudes, and the overlay is determined based on the first set of features and the second set of features. 8. The non-transitory computer-readable medium of clause 7, wherein determining the phase includes determining a phase of each corresponding first feature of the first set of features in Fourier space for each feature of the first set of features based on the phase image to obtain a set of phases, and the overlay is determined based on the set of phases. 9. The non-transitory computer-readable medium of any one of clauses 1 to 8, wherein the image data includes an image of a substrate having a first feature on a first layer and a second feature on a second layer, the second layer being formed on the first layer, and determining the overlay includes providing the phase as an input to a model and obtaining an estimated overlay between the first feature and the second feature from the model. 10. The non-transitory computer-readable medium of clause 9, wherein the model is trained by using a plurality of images of a substrate including stacked features, each image of the plurality of images including metadata indicating an overlay of the stacked features, each of the stacked features including a first set of features on a first layer of the respective substrate and a second set of features on a second layer of the respective substrate. 11. The non-transitory computer-readable medium of any one of clauses 1 to 10, wherein the image data includes an image of a substrate including a first layer and a second layer formed on the first layer, and analyzing the image data in Fourier space includes generating an amplitude image and a phase image representing the image data in Fourier space in response to application of a Fast Fourier Transform (FFT) to the image data, assigning peaks in the amplitude image to features, each of the peaks being assigned to one of a first feature included in the first layer or a second feature included in the second layer, and the phase of each feature of the first features being interpolated based on the respective peaks. 12. The non-transitory computer-readable medium of any one of clauses 1 to 11, wherein the image data is acquired by imaging the substrate using a metrology system. 13. A metrology system including a computing system configured to execute computer program instructions to accomplish operations including acquiring image data of features on a substrate, analyzing the image data in Fourier space, determining amplitude and phase based on the analysis, and determining overlay based on the amplitude and phase. 14. The metrology system of clause 13, wherein the image data includes an image of a substrate, and the substrate is patterned using a patterning process to form features including a first feature on a first layer of the substrate and a second feature on a second layer of the substrate, the second layer being formed on the first layer. 15. The metrology system of claim 14, wherein the image data includes an image of the substrate formed by scanning the electron beam in one or more directions. 16. The measurement system of any one of clauses 13 to 15, wherein the image data is captured via electron microscopy. 17. The metrology system of clause 16, wherein a scanning electron microscope (SEM) is used to perform the electron microscopy to capture an SEM image, and the image data includes SEM image data representative of the SEM image. 18. A measurement system as described in any one of clauses 13 to 17, wherein analyzing the image data in Fourier space includes generating an amplitude image and a phase image in response to application of a Fast Fourier Transform (FFT) to the image data, and the amplitude is determined based on the amplitude image and the phase is determined based on the phase image. 19. The measurement system of clause 18, wherein the image data includes images of the substrate, a first layer formed on the substrate, and a second layer formed on the first layer, the amplitude image depicts amplitude in Fourier space of the features, the features being formed on at least one of the first layer or the second layer, and determining the amplitude includes determining a first set of features located on the first layer from the features based on the amplitude, and determining a second set of features located on the second layer from the features based on the amplitude, and the overlay is determined based on the first set of features and the second set of features. 20. The measurement system of clause 19, wherein determining the phase includes determining a phase of each corresponding first feature of the first set of features in Fourier space for each feature of the first set of features based on the phase image to obtain a set of phases, and the overlay is determined based on the set of phases. 21. A measurement system as described in any one of clauses 13 to 20, wherein the features include a first feature formed on a first layer of the substrate and a second feature formed on a second layer of the substrate, the second layer being formed on the first layer, and determining the overlay includes providing the phase as an input to a model, and obtaining an estimated overlay between an instance of the first feature and an instance of the second feature from the model. 22. The metrology system of clause 21, wherein the model is trained by using a plurality of images of a substrate including stacked features, each image of the plurality of images including metadata indicating an overlay of the stacked features, each of the stacked features including a first set of features on a first layer of the respective substrate and a second set of features on a second layer of the respective substrate. 23. A measurement system as described in any one of clauses 13 to 22, wherein the image data includes an image of a substrate including a first layer and a second layer formed on the first layer, and analyzing the image data in Fourier space includes generating an amplitude image and a phase image representing the image data in Fourier space in response to application of a Fast Fourier Transform (FFT) to the image data, assigning peaks in the amplitude image to features, each of the peaks being assigned to one of a first feature included in the first layer or a second feature included in the second layer, and the phase of each of the first features being interpolated based on the respective peaks. 24. A method implemented by one or more processors executing computer program instructions stored in memory, the method including acquiring image data of features on a substrate, analyzing the image data in Fourier space, determining amplitude and phase based on the analysis, and determining overlay based on the amplitude and phase. 25. The method of clause 24, wherein the features include a first feature on a first layer of the substrate and a second feature on a second layer of the substrate, the second layer being formed on the first layer. 26. The method of clause 25, wherein the image data includes an image of the substrate formed by scanning the electron beam in one or more directions. 27. The method of any one of clauses 24 to 26, wherein the image data is captured via electron microscopy. 28. The method of claim 27, wherein a scanning electron microscope (SEM) is used to perform the electron microscopy to capture an SEM image, and the image data includes SEM image data representative of the SEM image. 29. A method according to any one of clauses 24 to 28, wherein analyzing the image data in Fourier space includes generating an amplitude image and a phase image in response to application of a Fast Fourier Transform (FFT) to the image data, the amplitude being determined based on the amplitude image and the phase being determined based on the phase image. 30. The method of clause 29, wherein the image data includes images of a substrate, a first layer formed on the substrate, and a second layer formed on the first layer, the amplitude image depicting amplitudes of features in Fourier space, the features being formed on at least one of the first layer or the second layer, determining the amplitudes includes determining a first set of features located on the first layer from the features based on the amplitudes, and determining a second set of features located on the second layer from the features based on the amplitudes, and the overlay is determined based on the first set of features and the second set of features. 31. The method of claim 30, wherein determining the phase includes determining a phase of each corresponding first feature of the first set of features in Fourier space for each feature of the first set of features based on the phase image to obtain a set of phases, and the overlay is determined based on the set of phases. 32. The method of any one of clauses 24 to 31, wherein the image data includes an image of a substrate having a first feature on a first layer and a second feature on a second layer, the second layer being formed on the first layer, and determining the overlay includes providing the phase as an input to a model, and obtaining an estimated overlay between the first feature and the second feature from the model. 33. The method of claim 32, wherein the model is trained by using a plurality of images of a substrate including stacked features, each image of the plurality of images including metadata indicating an overlay of stacked features, each of the stacked features including a first set of features on a first layer of the respective substrate and a second set of features on a second layer of the respective substrate. 34. A method according to any one of clauses 24 to 33, wherein the image data includes an image of a substrate including a first layer and a second layer formed on the first layer, and analyzing the image data in Fourier space includes generating an amplitude image and a phase image representing the image data in Fourier space in response to application of a Fast Fourier Transform (FFT) to the image data, and assigning peaks in the amplitude image to features, each of the peaks being assigned to one of a first feature included in the first layer or a second feature included in the second layer, and the phase of each of the first features being interpolated based on the respective peaks. 35. The method of any one of clauses 24 to 34, wherein the image data is acquired by imaging the substrate by using a metrology system. 36. A computer program product comprising a computer non-transitory readable medium having instructions recorded thereon, the instructions, when executed by a computer, implementing a method according to any one of clauses 24 to 35. 37. A system including a hardware processor system and a non-transitory computer readable storage medium configured to store machine readable instructions, the machine readable instructions, when executed, causing the hardware processor system to perform a method according to any one of clauses 24 to 35. 38. A metrology apparatus for measuring an object in a patterning process, the metrology apparatus being configured to perform a method according to any one of clauses 24 to 35. 39. A system comprising a metrology device configured to provide a beam of radiation onto a surface of an object and to detect radiation redirected by structures on the surface of the object, and a computer program product as described in clause 36. 40. The system of clause 39 further comprising a lithographic apparatus including a support structure configured to hold a patterning device to modulate a radiation beam, and projection optics configured to project the modulated radiation beam onto a radiation sensitive substrate, wherein the object is a substrate, and the lithographic apparatus is configured to control settings of the lithographic apparatus based on information obtained by using a metrology device and a computer program product.

[0135]

[0156] As used herein, the terms "radiation" and "beam" encompass all types of electromagnetic radiation, including ultraviolet (UV) radiation (e.g., having wavelengths of 365, 355, 248, 193, 157, or 126 nm) and extreme ultraviolet (EUV) radiation (e.g., having wavelengths within the range of about 5 to 20 nm), as well as particle beams such as ion beams or electron beams.

[0136]

[0157] The term "lens", where the context allows, may refer to any one or combination of various types of optical components, including refractive, reflective, magnetic, electromagnetic and electrostatic optical components.

[0137]

[0158] References herein to crossing or passing a threshold include having a value less than or equal to a particular value, having a value greater than or equal to a particular value, For example, they may include those that are ranked (eg, via sorting) higher or lower than others based on a parameter, and so on.

[0138]

[0159] References herein to correcting an error or correcting an error include eliminating the error or reducing the error to within a tolerance range.

[0139]

[0160] The terms "optimizing" and "optimization" as used herein refer to or mean adjusting a lithography apparatus, patterning process, etc. such that the results and / or process of a lithography or patterning process have more desirable characteristics (higher accuracy of projection of a design layout on a substrate, larger process window, etc.). Thus, the terms "optimizing" and "optimization" as used herein refer to or mean the process of identifying one or more values ​​of one or more variables that provide an improvement (e.g., a local optimum) in at least one important metric compared to one or more values ​​of an initial set of those one or more variables. "Optimal" and other synonyms should be interpreted accordingly. In one embodiment, the optimization process may be applied iteratively to provide further improvements in one or more metrics.

[0140]

[0161] In a system optimization process, a performance index of a system or process may be expressed as a cost function. The optimization process amounts to a process of finding a set of parameters (design variables) of a system or process that optimizes (e.g., minimizes or maximizes) the cost function. The cost function may have any suitable form depending on the goal of the optimization. For example, the cost function may be a weighted root mean square (RMS) of deviations of some features (e.g., evaluation points) of the system or process from intended values ​​(e.g., ideal values) of these features. The cost function may also be the maximum of these deviations (i.e., worst-case deviation). The term "evaluation point" in this specification should be interpreted broadly to include any feature of a system or process. The design variables of a system may be constrained to specified ranges and / or may be interdependent for practicality of the implementation of the system or process. In the case of a lithographic apparatus or patterning process, the constraints often relate to physical properties and characteristics of the hardware, such as tunability ranges and / or patterning device manufacturability design rules, and the evaluation points may include not only physical points on the resist image on the substrate, but also non-physical features such as dose and focus.

[0141]

[0162] While specific embodiments of the present disclosure have been described above, it will be understood that the present disclosure may be practiced otherwise than as described. For example, the present disclosure may take the form of a computer program comprising one or more sequences of machine-readable instructions describing the methods as disclosed above, or a data storage medium (e.g., a semiconductor memory, a magnetic or optical disk) having such a computer program stored thereon.

[0142]

[0163] Although the block diagram depicts the illustrated components as discrete functional blocks, some embodiments are not limited to a system in which the functionality described herein is organized as shown. The functionality provided by each of the components may be provided by software or hardware modules organized in a different manner than currently depicted, for example, such software or hardware may be intermixed, combined, replicated, disaggregated, distributed (e.g., in a data center or geographically), or organized in a different manner. The functionality described herein may be provided by one or more processors of one or more computers executing code stored on a tangible, non-transitory machine-readable medium. In some cases, a third-party content delivery network may host some or all of the information carried on the network, in which case, to the extent information (e.g., content) is said to be provided or otherwise provided, the information may be provided by sending instructions to retrieve the information from the content delivery network.

[0143]

[0164] Unless specifically stated otherwise, as is apparent from the discussion, it will be recognized from the discussion throughout this specification that use of the terms "processing," "computing," "calculating," "determining," etc. refers to acts or processes of a particular apparatus, such as a special purpose computer or similar dedicated electronic processing / computing device.

[0144]

[0165] The reader should appreciate that this application describes several inventions. Rather than separating those inventions into multiple separate patent applications, the applicants have grouped these inventions into a single document because their related subject matter lends itself to economy in the filing process. However, the individual advantages and aspects of such inventions should not be compounded. While in some cases some embodiments address all of the deficiencies noted herein, it should be understood that the invention is independently useful, and that some embodiments address only a subset of such problems or offer other unmentioned advantages that will become apparent to those skilled in the art upon review of this disclosure. Due to cost constraints, the inventions disclosed herein may not be claimed now, and may be claimed in a later application, such as in a continuation application, or by amending the claims herein. Similarly, due to space constraints, neither the Abstract nor the Summary of the Invention section of this specification should be taken as a comprehensive listing of such inventions or as including all aspects of such inventions.

[0145]

[0166] It should be understood that the specification and accompanying drawings are not intended to limit the invention to the particular forms disclosed, but on the contrary, the intention is to cover all modifications, equivalents, and alternatives falling within the spirit and scope of the invention as defined by the appended claims.

[0146]

[0167] Modifications and alternative embodiments of various aspects of the invention will become apparent to those skilled in the art in view of this specification. Thus, this specification and the accompanying drawings should be interpreted as merely illustrative and are intended to teach those skilled in the art the general manner of carrying out the invention. It should be understood that the forms of the invention shown and described herein should be taken as examples of some embodiments. As will become apparent to those skilled in the art after having the benefit of this specification of the invention, some elements and materials may be substituted for those shown and described herein, some parts and processes may be reversed or omitted, some features may be utilized independently, and some embodiments or features of some embodiments may be combined and fitted together. As described in the following claims, some changes may be made in the elements described herein without departing from the spirit and scope of the invention. The headings used herein are for organizational purposes only and are not meant to be used to limit the scope of the specification.

[0147]

[0168] As used throughout this application, the word "may" is used in a permissive sense (i.e., meaning having the possibility of) rather than a mandatory sense (i.e., meaning must). The words "include," "includes," and conjugations thereof, etc., mean including but not limited to. As used throughout this application, indefinite and definite articles in the singular form include plural references unless the content expressly dictates otherwise. Thus, for example, a reference to "an" element includes a combination of two or more elements, notwithstanding the use of other terms and phrases of one or more elements, such as "one or more." The term "or" is non-exclusive (i.e., includes both "and" and "or") unless otherwise indicated. Terms describing conditional relationships (e.g., "depending on X, Y," "when X, Y," "if X, Y," "in the case of X, Y," etc.) encompass the following causal relations: the antecedent is a necessary causal condition, the antecedent is a sufficient causal condition, or the antecedent is a contributory causal condition of the outcome: for example, "state X occurs upon obtaining condition Y" is a general expression of "X occurs only when Y" and "X occurs when Y and Z." Such conditional relationships are not limited to outcomes that immediately follow the antecedent obtaining, since some outcomes can be delayed, and in conditional statements the antecedent is linked to the outcome (e.g., the antecedent is related to the likelihood of a subsequent occurrence). A statement that "a plurality of attributes or features are mapped to a plurality of objects (e.g., one or more processors performing steps A, B, C, and D)" includes both such attributes or features mapped to all such objects and a subset of attributes or features mapped to a subset of attributes or features (e.g., both all processors that each perform steps AD and the case where processor 1 performs step A, processor 2 performs step B and part of step C, and processor 3 performs step C and part of step D), unless otherwise indicated. Furthermore, unless otherwise indicated, a statement that one value or action is "based on" another condition or value includes both instances in which the condition or value is the only factor and instances in which the condition or value is one factor among multiple factors.Unless otherwise indicated, a statement that "each" instance of a collection has a certain property should not be read to exclude cases where some otherwise identical or similar members of the larger collection do not have that property; i.e., "each" does not necessarily mean "every."

[0148]

[0169] To the extent that any U.S. patents, U.S. patent applications, or other materials (e.g., articles) are incorporated by reference, the statements of such U.S. patents, U.S. patent applications, and other materials are only incorporated by reference to the extent that no conflict exists between such materials and the statements and drawings set forth herein. In the event of such a conflict, any such conflicting statements within the U.S. patents, U.S. patent applications, and other materials as incorporated by reference are not specifically incorporated herein by reference.

[0149]

[0170] The above description is intended to be illustrative and not limiting. Thus, it will be apparent to one of ordinary skill in the art that modifications may be made to the disclosure as set forth without departing from the scope of the claims set forth below.

Claims

1. When executed by a computer, the computer acquiring image data of features on the substrate; analyzing the image data in Fourier space; determining an amplitude and a phase based on said analysis; and determining an overlay of the features based on the amplitude and the phase; Computer program instructions for performing operations including A non-transitory computer readable medium storing:

2. the features include a first feature on a first layer of the substrate and a second feature on a second layer of the substrate; The non-transitory computer-readable medium of claim 1 , wherein the second layer is formed on the first layer.

3. The non-transitory computer-readable medium of claim 2 , wherein the image data comprises an image of the substrate formed by scanning an electron beam in one or more directions.

4. The non-transitory computer-readable medium of claim 1 , wherein the image data is captured via electron microscopy.

5. a scanning electron microscope (SEM) is used to perform the electron microscopy to capture an SEM image; The non-transitory computer-readable medium of claim 4 , wherein the image data comprises SEM image data representative of the SEM image.

6. Analyzing the image data in Fourier space includes generating a magnitude image and a phase image in response to applying a Fast Fourier Transform (FFT) to the image data; The amplitude is determined based on the amplitude image; The non-transitory computer readable medium of claim 1 , wherein the phase is determined based on the phase image.

7. the image data includes an image of the substrate, a first layer formed on the substrate, and a second layer formed on the first layer; the amplitude image depicts the amplitude of the feature in Fourier space; the feature is formed on at least one of the first layer or the second layer; Determining the amplitude includes: determining a first set of features located on the first layer from the features based on the amplitudes; and determining a second set of features located on the second layer from the features based on the amplitudes; The non-transitory computer-readable medium of claim 6 , wherein the overlay is determined based on the first set of features and the second set of features.

8. determining the phase includes determining a phase of each corresponding first feature of the first set of features in Fourier space for each feature of the first set of features based on the phase image to obtain a set of phases; The non-transitory computer-readable medium of claim 7 , wherein the overlay is determined based on the set of phases.

9. the image data includes an image of the substrate having a first feature on a first layer and a second feature on a second layer; the second layer is formed on the first layer; Determining the overlay includes: providing said phase as an input to a model; and obtaining an estimated overlay between the first feature and the second feature from the model; The non-transitory computer-readable medium of claim 1 , comprising:

10. the model is trained using a plurality of images of a substrate including stacked features; each image of the plurality of images includes metadata indicating an overlay of the stacking feature; 10. The non-transitory computer-readable medium of claim 9, wherein each of the stacked features includes a first set of features on a first layer of a respective substrate and a second set of features on a second layer of the respective substrate.

11. the image data includes an image of a substrate including a first layer and a second layer formed on the first layer; Analyzing the image data in Fourier space includes: generating amplitude and phase images representing the image in Fourier space in response to applying a Fast Fourier Transform (FFT) to the image data; assigning peaks in the magnitude image to the features; each of the peaks is assigned to one of a first feature included in the first layer or a second feature included in the second layer; The non-transitory computer-readable medium of claim 1 , wherein a phase of each feature of the first features is interpolated based on a respective peak.

12. The non-transitory computer readable medium of claim 1 , wherein the image data is obtained by imaging the substrate by using a metrology system.

13. acquiring image data of features on the substrate; analyzing the image data in Fourier space; determining an amplitude and a phase based on said analysis; and determining overlay based on said amplitude and said phase; Computer program instructions for performing operations including: a computing system configured to execute Measurement system.

14. the image data includes an image of the substrate; the substrate is patterned using a patterning process to form features including a first feature on a first layer of the substrate and a second feature on a second layer of the substrate; The metrology system of claim 13 , wherein the second layer is formed on the first layer.

15. 1. A method implemented by one or more processors executing computer program instructions stored in a memory, comprising: acquiring image data of features on the substrate; analyzing the image data in Fourier space; determining an amplitude and a phase based on said analysis; and determining overlay based on said amplitude and said phase; A method comprising:

Images