Method and system for perturbing mask pattern

Abstract

Described herein is a method and system for optimizing a source of a lithographic apparatus using a perturbed mask pattern. A mask pattern corresponding to a target pattern is obtained, and a contour of the mask pattern is extracted. A set of points are assigned along a contour and a random displacement of the contour is determined at each point. The mask pattern is perturbed by random displacements at each point to generate a perturbed mask pattern. The source is optimized by performing a source optimization, source mask optimization, or other such process using the perturbed mask pattern.

Description

METHOD AND SYSTEM FOR PERTURBING MASK PATTERNCROSS-REFERENCE TO RELATED APPLICATIONS

[0001] This application claims priority of US application 63 / 608,156 which was filed on 08 December 2023 and which is incorporated herein in its entirety by reference.TECHNICAL FIELD

[0002] The embodiments provided herein relate to semiconductor manufacturing, and more particularly to perturbing mask pattern for source or mask optimization.BACKGROUND

[0003] A lithographic apparatus is a machine that applies a desired pattern onto a target portion of a substrate. The lithographic apparatus can be used, for example, in the manufacture of integrated circuits (ICs). For example, an IC chip in a smart phone, can be as small as a person’s thumbnail, and may include over 2 billion transistors. Making an IC is a complex and time-consuming process, with circuit components in different layers and including hundreds of individual steps. Errors in even one step have the potential to result in problems with the final IC and can cause device failure. High process yield and high wafer throughput can be impacted by the presence of defects.BRIEF SUMMARY

[0004] In some embodiments, the techniques described herein relate to a method of source optimization (SO), the method including: obtaining a mask pattern; assigning a set of points along a contour of the mask pattern; determining a random displacement for each point of the contour; perturbing the mask pattern by random displacements to generate a perturbed mask pattern; and optimizing a source of a lithographic apparatus by performing a SO process using the perturbed mask pattern.

[0005] In some embodiments, the techniques described herein relate to a method of perturbing a mask pattern, the method including: obtaining a mask pattern; interpolating a contour of the mask pattern to obtain multiple interpolated vertices of the contour; determining a random displacement of the contour at each interpolated vertex; and perturbing the mask pattern by random displacements to generate a perturbed mask pattern.

[0006] In some embodiments, there is provided a non-transitory computer readable medium having instructions that, when executed by a computer, cause the computer to execute a method of any of the above embodiments.

[0007] In some embodiments, there is provided an apparatus includes a memory storing a set of instructions and a processor configured to execute the set of instructions to cause the apparatus to perform a method of any of the above embodiments.BRIEF DESCRIPTION OF THE DRAWINGS

[0008] Embodiments will now be described, by way of example only, with reference to the accompanying drawings in which:

[0009] Figure 1 illustrates a block diagram of various subsystems of a lithographic projection apparatus, according to an embodiment.

[0010] Figure 2 is a schematic diagram of a lithographic projection apparatus, according to an embodiment.

[0011] Figure 3 illustrates an exemplary flow chart for simulating lithography in a lithographic projection apparatus, according to an embodiment.

[0012] Figure 4 is a block diagram of random perturbation of a mask pattern using a given set of stochastic roughness parameters, consistent with various embodiments.

[0013] Figures 5A-5C illustrate a random signal generated using a power spectral density (PSD) function, consistent with various embodiments.

[0014] Figure 6 shows mask perturbations for the same line edge roughness (LER) and scaling exponent but different correlation lengths, consistent with various embodiments.

[0015] Figure 7 shows mask perturbations for the same correlation length and scaling exponent but different LER values, consistent with various embodiments.

[0016] Figure 8 shows perturbation of a mask pattern having polygonal patterns, consistent with various embodiments.

[0017] Figure 9 is a flow diagram of a method for perturbing mask pattern using a PSD function, consistent with various embodiments.

[0018] Figure 10 is a block diagram of an example computer system, according to an embodiment.

[0019] Embodiments will now be described in detail with reference to the drawings, which are provided as illustrative examples so as to enable those skilled in the art to practice the embodiments. Notably, the figures and examples below are not meant to limit the scope to a single embodiment, but other embodiments are possible by way of interchange of some or all of the described or illustrated elements. Wherever convenient, the same reference numbers will be used throughout the drawings to refer to same or like parts. Where certain elements of these embodiments can be partially or fully implemented using known components, only those portions of such known components that are necessary for an understanding of the embodiments will be described, and detailed descriptions of other portions of such known components will be omitted so as not to obscure the description of the embodiments. In the present specification, an embodiment showing a singular component should not be considered limiting; rather, the scope is intended to encompass other embodiments including a plurality of the same component, and vice-versa, unless explicitly stated otherwise herein. Moreover, applicants do not intend for any term in the specification or claims to be ascribed an uncommon or special meaning unless explicitly set forth as such. Further, the scope encompasses present and futureknown equivalents to the components referred to herein by way of illustration.DETAILED DESCRIPTION

[0020] A lithographic apparatus is a machine that applies a designed pattern onto a target portion of a substrate. This process of transferring the designed pattern to the substrate is called a patterning process. The patterning process can include a patterning step to transfer a pattern from a patterning device (such as a mask) to the substrate. Various variations (e.g., variations in the patterning process or the lithographic apparatus) can potentially limit lithography implementation for semiconductor high volume manufacturing (HVM). To find a source of a lithographic apparatus that is tolerant to stochastic mask variation, a randomized mask is generated and a source optimization process (e.g., source optimization (SO), source mask optimization (SMO), source mask lens optimization (SMLO) or such) is performed using the randomized mask to generate an optimized source that minimizes the impact of the stochastic mask variation. The conventional methods for simulating or generating a randomized mask pattern have drawbacks or limitations. For example, the randomization in the conventional methods is limited to applying bias per edge of a polygon or uniformly scaling the polygons. The randomized masks generated using the conventional methods do not accurately reflect the stochastic errors in the physically manufactured masks and therefore, may result in a source that is not optimized or is not tolerant to the stochastic errors in the actual manufactured mask. Further, the conventional methods of generating randomized masks do not facilitate the user to customize the randomization based on stochastic roughness parameters of the mask (e.g., local critical dimension uniformity (LCDU), line edge roughness (LER), correlation length, or scaling exponent, etc.), which may limit the extent to which the randomized mask mimics the physically manufactured mask. Furthermore, the randomization of mask patterns in conventional methods may work with polygon shaped features but not with freeform or curvilinear features. These and other drawbacks exist.

[0021] Disclosed are embodiments for perturbing a mask pattern to generate a perturbed mask pattern that more accurately represents physically manufactured masks. In some embodiments, to mimic the stochastic variations in the physically manufactured masks, a mask pattern is perturbed by assigning points along a contour of the mask pattern and introducing random displacement at each point of the contour along the direction of its normal vector. For example, points or vertices of the contour of the mask pattern may be extracted (e.g., using known methods) and interpolated to generate vertices that are evenly spaced on the contour, and random displacements are introduced at each interpolated vertex in the direction of the normal vector of the interpolated vertex to generate the perturbed mask pattern. In some embodiments, the random displacements may be determined using a power spectral density (PSD) function, which can be tuned, customized, or controlled for desired stochastic roughness parameters of the mask (e.g., LCDU, LER, correlation length, or scaling exponent, etc.). For example, a random signal is generated using the PSD function in a frequency domain, converted to spatial domain using any of a known number of methods (e.g., fast-Fourier transform (FFT)), and the converted signalis then applied as random displacements at the interpolated vertices. The vertices of the contour may be interpolated in any number of ways. In some embodiments, even interpolation of the vertices is preferred as the FFT method assumes constant spacing. By randomizing the mask pattern based on interpolation of the vertices of contour and introducing controlled random displacements at the interpolated vertices, the perturbed mask pattern mimics the physical mask pattern more accurately than the perturbed mask patterns generated using conventional methods. Further, such a method of perturbation may also be applied to curvilinear patterns in addition to polygon patterns.

[0022] In the present disclosure, although specific reference may be made to the manufacture of ICs, it should be explicitly understood that the description herein has many other possible applications. For example, it may be employed in the manufacture of integrated optical systems, guidance and detection patterns for magnetic domain memories, liquid crystal display panels, thin film magnetic heads, etc. The skilled artisan will appreciate that, in the context of such alternative applications, any use of the terms “reticle”, “wafer” or “die” in this text should be considered as interchangeable with the more general terms “mask”, “substrate” and “target portion”, respectively.

[0023] In the present document, the terms “radiation” and “beam” are used to encompass all types of electromagnetic radiation, including ultraviolet radiation (e.g., with a wavelength of 365, 248, 193, 157 or 126 nm) and EUV (extreme ultra-violet radiation, e.g., having a wavelength in the range of about 5- 100 nm). In the present document, the term “radiation source” or “source” is used to encompass all types of sources of radiation, including laser sources, incandescent sources, etc. which may include treatment of the radiation between the radiation source and the target or other parts of the optics, including filtering, collimating, focusing, etc.

[0024] A patterning device can comprise, or can form, one or more design layouts. The design layout can be generated utilizing CAD (computer-aided design) programs. This process is often referred to as EDA (electronic design automation). Most CAD programs follow a set of predetermined design rules in order to create functional design layouts / patterning devices. These rules are set based processing and design limitations. For example, design rules define the space tolerance between devices (such as gates, capacitors, etc.) or interconnect lines, to ensure that the devices or lines do not interact with one another in an undesirable way. One or more of the design rule limitations may be referred to as a “critical dimension” (CD). A critical dimension of a device can be defined as the smallest width of a line or hole, or the smallest space between two lines or two holes. Thus, the CD regulates the overall size and density of the designed device. One of the goals in device fabrication is to faithfully reproduce the original design intent on the substrate (via the patterning device).

[0025] The term “mask” or “patterning device” as employed in this text may be broadly interpreted as referring to a generic patterning device that can be used to endow an incoming radiation beam with a patterned cross-section, corresponding to a pattern that is to be created in a target portion of the substrate. The term “light valve” can also be used in this context. Besides the classic mask (transmissive or reflective; binary, phase-shifting, hybrid, etc.), examples of other such patterning devices include aprogrammable mirror array. An example of such a device is a matrix-addressable surface having a viscoelastic control layer and a reflective surface. The basic principle behind such an apparatus is that (for example) addressed areas of the reflective surface reflect incident radiation as diffracted radiation, whereas unaddressed areas reflect incident radiation as undiffracted radiation. Using an appropriate filter, the said undiffracted radiation can be filtered out of the reflected beam, leaving only the diffracted radiation behind; in this manner, the beam becomes patterned according to the addressing pattern of the matrix-addressable surface. The required matrix addressing can be performed using suitable electronic means. Examples of other such patterning devices also include a programmable LCD array. An example of such a construction is given in U.S. Patent No. 5,229,872, which is incorporated herein by reference.

[0026] The term “projection optics” as used herein should be broadly interpreted as encompassing various types of optical systems, including refractive optics, reflective optics, apertures and catadioptric optics, for example. The term “projection optics” may also include components operating according to any of these design types for directing, shaping or controlling the projection beam of radiation, collectively or singularly. The term “projection optics” may include any optical component in the lithographic projection apparatus, no matter where the optical component is located on an optical path of the lithographic projection apparatus. Projection optics may include optical components for shaping, adjusting and / or projecting radiation from the source before the radiation passes the patterning device, and / or optical components for shaping, adjusting and / or projecting the radiation after the radiation passes the patterning device. The projection optics generally exclude the source and the patterning device.

[0027] Figure 1 illustrates a block diagram of various subsystems of a lithographic projection apparatus 10A, according to an embodiment. Major components are a radiation source 12A, which may be a deep-ultraviolet excimer laser source or other type of source including an extreme ultra violet (EUV) source (the lithographic projection apparatus itself need not have the radiation source), illumination optics which, e.g., define the partial coherence (denoted as sigma) and which may include optics 14A, 16Aa and 16Ab that shape radiation from the source 12A; a patterning device (or mask) 18A; and transmission optics 16Ac that project an image of the patterning device pattern onto a substrate plane 22A.

[0028] A pupil 20A can be included with transmission optics 16Ac. In some embodiments, there can be one or more pupils before and / or after mask 18 A. As described in further detail herein, pupil 20A can provide patterning of the light that ultimately reaches substrate plane 22A. An adjustable filter or aperture at the pupil plane of the projection optics may restrict the range of beam angles that impinge on the substrate plane 22A, where the largest possible angle defines the numerical aperture of the projection optics NA= n sin(0max), wherein n is the refractive index of the media between the substrate and the last element of the projection optics, and ©max is the largest angle of the beam exiting from the projection optics that can still impinge on the substrate plane 22A.

[0029] In a lithographic projection apparatus, a source provides illumination (i.e., radiation) to a patterning device and projection optics direct and shape the illumination, via the patterning device, onto a substrate. This is not to disclaim that the source does not itself provide patterning, directing, or shaping to the radiation or that patterning, directing, or shaping does not occur between the source and the projection optics. The projection optics may include at least some of the components 14A, 16Aa, 16Ab and 16Ac. An aerial image (Al) is the radiation intensity distribution at substrate level. A resist model can be used to calculate the resist image from the aerial image, an example of which can be found in U.S. Patent Application Publication No. US 2009-0157360, the disclosure of which is hereby incorporated by reference in its entirety. The resist model is related to properties of the resist layer (e.g., effects of chemical processes which occur during exposure, post-exposure bake (PEB) and development). Optical properties of the lithographic projection apparatus (e.g., properties of the illumination, the patterning device and the projection optics) dictate the aerial image and can be defined in an optical model. Since the patterning device used in the lithographic projection apparatus can be changed, it is desirable to separate the optical properties of the patterning device from the optical properties of the rest of the lithographic projection apparatus including at least the source and the projection optics. Details of techniques and models used to transform a design layout into various lithographic images (e.g., an aerial image, a resist image, etc.), apply OPC using those techniques and models and evaluate performance (e.g., in terms of process window) are described in U.S. Patent Application Publication Nos. US 2008-0301620, 2007-0050749, 2007-0031745, 2008-0309897, 2010- 0162197, and 2010-0180251, the disclosure of each which is hereby incorporated by reference in its entirety.

[0030] One aspect of understanding a lithographic process is understanding the interaction of the radiation and the patterning device. The electromagnetic field of the radiation after the radiation passes the patterning device may be determined from the electromagnetic field of the radiation before the radiation reaches the patterning device and a function that characterizes the interaction. This function may be referred to as the mask transmission function (which can be used to describe the interaction by a transmissive patterning device and / or a reflective patterning device).

[0031] The mask transmission function may have a variety of different forms. One form is binary. A binary mask transmission function has either of two values (e.g., zero and a positive constant) at any given location on the patterning device. A mask transmission function in the binary form may be referred to as a binary mask. Another form is continuous. Namely, the modulus of the transmittance (or reflectance) of the patterning device is a continuous function of the location on the patterning device. The phase of the transmittance (or reflectance) may also be a continuous function of the location on the patterning device. A mask transmission function in the continuous form may be referred to as a continuous tone mask or a continuous transmission mask (CTM). For example, the CTM may be represented as a pixelated image, where each pixel may be assigned a value between 0 and 1 (e.g., 0.1, 0.2, 0.3, etc.) instead of binary value of either 0 or 1. In an embodiment, CTM may be a pixelated grayscale image, where each pixel has values (e.g., within a range [-255, 255], normalized values within a range [0, 1] or [-1, 1] or other appropriate ranges).

[0032] The thin-mask approximation, also called the Kirchhoff boundary condition, is widely used to simplify the determination of the interaction of the radiation and the patterning device. The thin-mask approximation assumes that the thickness of the structures on the patterning device is very small compared with the wavelength and that the widths of the structures on the mask are very large compared with the wavelength. Therefore, the thin-mask approximation assumes the electromagnetic field after the patterning device is the multiplication of the incident electromagnetic field with the mask transmission function. However, as lithographic processes use radiation of shorter and shorter wavelengths, and the structures on the patterning device become smaller and smaller, the assumption of the thin-mask approximation can break down. For example, interaction of the radiation with the structures (e.g., edges between the top surface and a sidewall) because of their finite thicknesses (“mask 3D effect” or “M3D”) may become significant. Encompassing this scattering in the mask transmission function may enable the mask transmission function to better capture the interaction of the radiation with the patterning device. A mask transmission function under the thin-mask approximation may be referred to as a thin-mask transmission function. A mask transmission function encompassing M3D may be referred to as a M3D mask transmission function.

[0033] Figure 2 schematically depicts an exemplary lithographic projection apparatus whose illumination source could be optimized utilizing the methods described herein. The apparatus comprises:- an illumination system IL, to condition a beam B of radiation. In this particular case, the illumination system also comprises a radiation source SO;- a first object table (e.g., mask table, patterning device table or reticle stage) MT provided with a patterning device holder to hold a patterning device MA (e.g., a reticle), and connected to a first positioner to accurately position the patterning device with respect to item PS;- a second object table (substrate table or wafer stage) WT provided with a substrate holder to hold a substrate W (e.g., a resist-coated silicon wafer), and connected to a second positioner to accurately position the substrate with respect to item PS;- a projection system (“lens”) PS (e.g., a refractive, catoptric or catadioptric optical system) to image an irradiated portion of the patterning device MA onto a target portion C (e.g., comprising one or more dies) of the substrate W.

[0034] As depicted herein, the apparatus is of a transmissive type (i.e., has a transmissive mask). However, in general, it may also be of a reflective type, for example (with a reflective mask). Alternatively, the apparatus may employ another kind of patterning device as an alternative to the use of a classic mask; examples include a programmable mirror array or LCD matrix.

[0035] The source SO (e.g., a mercury lamp or excimer laser) produces a beam of radiation. This beam is fed into an illumination system (illuminator) IL, either directly or after having traversed conditioning means, such as a beam expander Ex, for example. The illuminator IL may comprise adjusting meansAD for setting the outer or inner radial extent (commonly referred to as o-outer and o-inner, respectively) of the intensity distribution in the beam. In addition, 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.

[0036] It should be noted with regard to Figure 2 that the source SO may be within the housing of the lithographic projection apparatus (as is often the case when the source SO is a mercury lamp, for example), but that it may also be remote from the lithographic projection apparatus, the radiation beam that it produces being led into the apparatus (e.g., with the aid of suitable directing mirrors); this latter scenario is often the case when the source SO is an excimer laser (e.g., based on KrF, ArF or F2 lasing).

[0037] The beam B subsequently intercepts the patterning device MA, which is held on a patterning device table MT. Having traversed the patterning device MA, the beam B passes through the lens PS, which focuses the beam B onto a target portion C of the substrate W. With the aid of the second positioning means (and interferometric measuring means IF), the substrate table WT can be moved accurately, e.g., so as to position different target portions C in the path of the beam B. Similarly, the first positioning means can be used to accurately 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, movement of the object tables MT, WT will be realized with the aid of a long-stroke module (coarse positioning) and a short-stroke module (fine positioning), which are not explicitly depicted in Figure 2. However, in the case of a wafer stepper (as opposed to a step- and-scan tool) the patterning device table MT may just be connected to a short stroke actuator, or may be fixed.

[0038] The depicted tool can be used in two different modes:- In step mode, the patterning device table MT is kept essentially stationary, and an entire patterning device image is projected in one go (i.e., a single “flash”) onto a target portion C. The substrate table WT is then shifted in the x or y directions 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 MT is movable in a given direction (the so-called “scan direction”, e.g., the y direction) with a speed v, so that the projection beam B is caused to scan over a patterning device image; concurrently, the substrate table WT is simultaneously moved in the same or opposite direction at a speed V = Mv, in which M is the magnification of the lens PS (typically, M = 1 / 4 or 1 / 5). In this manner, a relatively large target portion C can be exposed, without having to compromise on resolution.

[0039] Figure 3 illustrates an exemplary flow chart for simulating lithography in a lithographic projection apparatus, according to an embodiment. As will be appreciated, the models may represent a different patterning process and need not comprise all the models described below. A source model 300 represents optical characteristics (including radiation intensity distribution, bandwidth and / or phasedistribution) of the illumination of a patterning device. The source model 300 can represent the optical characteristics of the illumination that include, but not limited to, numerical aperture settings, illumination sigma (o) settings as well as any particular illumination shape (e.g., off-axis radiation shape such as annular, quadrupole, dipole, etc.), where o (or sigma) is outer radial extent of the illuminator.

[0040] A projection optics model 310 represents optical characteristics (including changes to the radiation intensity distribution and / or the phase distribution caused by the projection optics) of the projection optics. The projection optics model 310 can represent the optical characteristics of the projection optics, including aberration, distortion, one or more refractive indexes, one or more physical sizes, one or more physical dimensions, etc.

[0041] The patterning device / design layout model module 320 captures how the design features are laid out in the pattern of the patterning device and may include a representation of detailed physical properties of the patterning device, as described, for example, in U.S. Patent No. 7,587,704, which is incorporated by reference in its entirety. In an embodiment, the patterning device / design layout model module 320 represents optical characteristics (including changes to the radiation intensity distribution and / or the phase distribution caused by a given design layout) of a design layout (e.g., a device design layout corresponding to a feature of an integrated circuit, a memory, an electronic device, etc.), which is the representation of an arrangement of features on or formed by the patterning device. Since the patterning device used in the lithographic projection apparatus can be changed, it is desirable to separate the optical properties of the patterning device from the optical properties of the rest of the lithographic projection apparatus including at least the illumination and the projection optics. The objective of the simulation is often to accurately predict, for example, edge placements and CDs, which can then be compared against the device design. The device design is generally defined as the pre-OPC patterning device layout, and will be provided in a standardized digital file format such as GDSII or OASIS.

[0042] An aerial image 330 can be simulated from the source model 300, the projection optics model 310 and the patterning device / design layout model module 320. An aerial image (Al) is the radiation intensity distribution at substrate level. Optical properties of the lithographic projection apparatus (e.g., properties of the illumination, the patterning device, and the projection optics) dictate the aerial image.

[0043] A resist layer on a substrate is exposed by the aerial image and the aerial image is transferred to the resist layer as a latent “resist image” (RI) therein. The resist image (RI) can be defined as a spatial distribution of solubility of the resist in the resist layer. A resist image 350 can be simulated from the aerial image 330 using a resist model 340. The resist model can be used to calculate the resist image from the aerial image, an example of which can be found in U.S. Patent Application No. 8,200,468, the disclosure of which is hereby incorporated by reference in its entirety. The resist model 340 typically describes the effects of chemical processes which occur during resist exposure, post exposure bake (PEB) and development, in order to predict, for example, contours of resist features formed on the substrate and so it typically related only to such properties of the resist layer (e.g., effects of chemical processes which occur during exposure, post-exposure bake and development). In an embodiment, theoptical properties of the resist layer, e.g., refractive index, film thickness, propagation, and polarization effects — may be captured as part of the projection optics model 310.

[0044] So, in general, the connection between the optical and the resist model is a simulated aerial image intensity within the resist layer, which arises from the projection of radiation onto the substrate, refraction at the resist interface and multiple reflections in the resist film stack. The radiation intensity distribution (aerial image intensity) is turned into a latent “resist image” by absorption of incident energy, which is further modified by diffusion processes and various loading effects. Efficient simulation methods that are fast enough for full-chip applications approximate the realistic 3- dimensional intensity distribution in the resist stack by a 3-dimensional aerial (and resist) image.

[0045] In an embodiment, the resist image 350 can be used as an input to a post-pattern transfer process model module 360. The post-pattern transfer process model module 360 defines performance of one or more post-resist development processes (e.g., etch, development, etc.).

[0046] Simulation of the patterning process can, for example, predict contours, CDs, edge placement (e.g., edge placement error), etc. in the resist and / or etched image. Thus, the objective of the simulation is to accurately predict, for example, edge placement, and / or aerial image intensity slope, and / or CD, etc. of the printed pattern. These values can be compared against an intended design to, e.g., correct the patterning process, identify where a defect is predicted to occur, etc. The intended design is generally defined as a pre-OPC design layout which can be provided in a standardized digital file format such as GDSII or OASIS or other file format.

[0047] Thus, the model formulation describes most, if not all, of the known physics and chemistry of the overall process, and each of the model parameters desirably corresponds to a distinct physical or chemical effect. The model formulation thus sets an upper bound on how well the model can be used to simulate the overall manufacturing process.

[0048] The following paragraphs describe a system and a method for perturbing a mask pattern to generate a perturbed mask pattern that more accurately reflects or mimics physically manufactured masks. A mask pattern may be perturbed by assigning points along a contour of the mask pattern (e.g., using interpolation) and introducing random displacement at each interpolated point of the contour along the direction of its normal vector. For example, a random signal is generated using a power spectral density (PSD) function in a frequency domain, converted to spatial domain using any of a known number of methods (e.g., FFT), and the converted signal is then applied as random displacements at the interpolated points. The random displacements may be controlled by tuning, customizing, or controlling the PSD function for desired stochastic roughness parameters of the mask (e.g., ECDU, EER, correlation length, or scaling component, etc.). By randomizing the mask pattern based on interpolation of points of contour and introducing controlled random displacements at the interpolated points, the perturbed mask pattern may represent the physical mask pattern more accurately. Further, such a method of perturbation may also be applied to curvilinear patterns in addition to polygon patterns.

[0049] Figure 4 is a block diagram of random perturbation of a mask pattern using a given set of stochastic roughness parameters, consistent with various embodiments. A mask pattern design corresponding to a design layout is obtained. The mask pattern design may have a number of mask patterns, such as a mask pattern 402 (e.g., a contact hole). The mask patterns may be post-optical proximity correction (OPC) mask patterns, that is, patterns that are corrected using OPC process. The mask pattern 402 may be processed to extract a contour of the mask pattern and obtain (e.g., using known methods) a set of points or vertices 406 along the contour. For example, the set of vertices 406 of the contour may include a number of vertices such as a first vertex 421a, a second vertex 431a and a third vertex 441a. The set of vertices 406 is interpolated to obtain a set of interpolated vertices 408. The set of vertices 406 may be interpolated in any number of ways. In some embodiments, even interpolation is performed such that interpolated vertices 408 are constantly or evenly spaced. For example, interpolating the vertices 42 la-44 la generates the corresponding interpolated vertices - a first interpolated vertex 421b, a second interpolated vertex 431b and a third interpolated vertex 441b, respectively, that are evenly spaced from each other, as illustrated in the set of interpolated vertices 408. That is, the distance between the interpolated vertices is constant. In some embodiments, after the interpolation, the coordinates of some of the vertices may change. For example, if the coordinates of a first vertex 421a is (xl,yl), after interpolation, the coordinates of the first vertex 421a may change to (x'l, y'l), which are the coordinates of the first interpolated vertex 421b.

[0050] The random displacements may be applied to the interpolated vertices in the direction of the normal vector of the vertices. A normal vector direction is determined for each of the interpolated vertices 408. In some embodiments, a normal is an object (e.g., a line, ray, or vector) that is perpendicular to a given object. For example, the normal line to a plane curve at a given point is the line perpendicular to the tangent line to the curve at the point. In some embodiments, the normal vector of the interpolated vertices may be determined based on derivatives of the interpolated vertices. For example, consider that the coordinates of the interpolated vertices 421b, 431b and 441b are (x'l, y'l), (x'2, y'2), and (x'3, y'3), respectively. The local derivative may be determined as a difference between the coordinates of two consecutive interpolated vertices. The local derivative at interpolated vertex 421b may be represented as: dx = [x'2 — x'l] dy = [y'2 — y'l]... Eq. (1)

[0051] In some embodiments, a local normal vector, N, may be perpendicular to the derivative and therefore, its slope may be equal to the negative reciprocal of the derivative. The local normal vector may be represented as:N = [Nx, Ny] = [~dy, dx]... Eq. (2) which may be normalized, n, as: n = [—dy, dx] / N... Eq. (3)

[0052] In the example of Figure 4, the direction of local normal vector of one of the interpolated vertices is away from (or extends externally to) the mask pattern, as indicated by the first direction 410, and the direction of local normal vector of another one of the interpolated vertices is into (or extends internally to) the mask pattern 402, as indicated by the second direction 412. Similarly, the local normal vector may be calculated for each of the interpolated vertices 408.

[0053] The mask pattern 402 may be perturbed by introducing random displacements at each of the interpolated vertices 408. The random displacements are determined based on a random signal 415 that is generated using a PSD function. Figures 5A-5C illustrate a random signal generated using a PSD function, consistent with various embodiments. For example, Figure 5A shows a first frequency domain signal 502 for a low-frequency noise and the corresponding first random signal 512 (e.g., random displacements) in spatial domain generated using the PSD function. In a second example, Figure 5B shows a second frequency domain signal 504 for a mid-frequency noise and the corresponding second random signal 514 (e.g., random displacements) in the spatial domain generated using the PSD function. In a third example, Figure 5C shows a third frequency domain signal 506 for a high-frequency noise and the corresponding third random signal 516 (e.g., random displacements) in the spatial domain generated using the PSD function.

[0054] The random signal 415 may be controlled or tuned to match a specific stochastic roughness of the mask, which may be represented using one or more parameters such as EER, correlation length or a scaling exponent. In some embodiments, the LCDU refers to stochastic-induced variation in CD of a group of the patterns. In some embodiments, LER is defined as a deviation of a feature edge from an ideal shape. In some embodiments, the correlation length may be a measure of the constraint between height displacements of neighboring points of the surface of a mask, or a measure of the typical length scale over which surface roughness or fluctuations occur. In some embodiments, the user may input one or more of the above stochastic roughness parameters, or they may be derived from the mask pattern design (e.g., from the GDS files). In some embodiments, the PSD function used to generate the random signal 415 may be represented as:PSD( ) = 1 / 1 + (27T / O2H+1... Eq. (4) where is frequency, f is correlation length, and H is scaling exponent. The random signal generated in a frequency domain is converted to spatial domain using any of a known number of methods. For example, an FFT method is applied on an output of the PSD function in the frequency domain to obtain signal value in spatial domain. In some embodiments, even interpolation is preferred as one of the interpolation methods as the FFT assumes the signal to evenly spaced. The conversion of the random signal from frequency domain to spatial domain may be represented as:PSD(x) = FFT (PSD (J))... Eq. (5) where PSD (x) is the output of PSD function in the spatial domain. The PSD function in the spatial domain may be normalized (e.g., based on EER). The normalization may be represented as:PSDnorm(x) = PSD(x) * LER * (l / aPSD(x)')... Eq. (6) where PSDnorm(x) is normalized PSD in spatial domain (e.g., random signal or random displacement), and LER is the line edge roughness.

[0055] The random signal 415 (e.g., random displacements) may be generated as an array of values, S, (e.g., using Eq. (6)) which may be represented as:S = [S0,SnS2,S3... ] ... Eq. (7)

[0056] The mask pattern 402 may be perturbed by introducing random displacement at each of the interpolated vertices 408 in the direction of the normal vector. For example, the contour of the mask pattern 402 may be perturbed at the interpolated vertex, vt, by an amount Stin the direction of the normal vector n at the corresponding interpolated vertex. The perturbation will result in new coordinates for the corresponding interpolated vertex, vt, on the contour 418 resulting in the contour 418 of the mask pattern 402 being displaced from its original coordinates. The perturbation is performed for each of the interpolated vertices 408, which may result in the contour 418 of the mask pattern 402 being displaced from its original coordinates at least at some of the interpolated vertices 408, to generate a perturbed mask pattern 452. The perturbation may be represented mathematically as:... Eq. (8) where (x_perturbedpj y_perturbedp] ) are coordinates of perturbed vertex, vt, of the perturbed mask pattern, (x , y ) are coordinates of the interpolated vertex, vt, is random displacement for vertex, vt, and n is the normal vector for the vertex.

[0057] The above perturbation may be performed for some or all patterns in the mask pattern design. For example, as illustrated in the perturbed mask pattern design 460, patterns in a given portion of the mask pattern design are perturbed (e.g., perturbed mask pattern 452 and perturbed mask pattern 454). In some embodiments, each mask pattern may be perturbed using a different random signal while the stochastic roughness parameters are the same across the random signals. That is, the random signal used to perturb a mask pattern may be determined independently for each mask pattern to be perturbed. For example, the mask pattern 402 may be perturbed using a first random signal to generate the perturbed mask pattern 452, and another mask pattern in the mask pattern design may be perturbed using a second random signal different from the first random signal to generate the perturbed mask pattern 454.

[0058] As described above, while the random signal 415 is generated in a random manner, the randomization may be customized, controlled, or tuned to match specific stochastic roughness of the mask. Figures 6 and 7 show mask perturbations for different stochastic roughness parameters.

[0059] Figure 6 shows mask perturbations for the same EER and scaling exponent but different correlation lengths, consistent with various embodiments. For example, Figure 6 shows three different perturbations 602-606 of a mask pattern using the same LER and scaling exponent but decreasing correlation lengths. For example, a first perturbation 602 is generated using a correlation length of a first value (e.g., “4”nm), a second perturbation 604 is generated using a correlation length of a second value lesser than the first value (e.g., “l”nm), and a third perturbation 606 is generated using a correlation length of a third value lesser than the second value (e.g., “0.2”nm). Note that the frequency of random displacements increases as the correlation length decreases.

[0060] Figure 7 shows mask perturbations for the same correlation length and scaling exponent but different LER values, consistent with various embodiments. For example, Figure 7 shows three different perturbations 702-706 of a mask pattern using the same correlation length and scaling exponent but decreasing LER. For example, a first perturbation 702 is generated using an LER of a first value (e.g., “4”nm), a second perturbation 704 is generated using an LER of a second value lesser than the first value (e.g., “2”nm), and a third perturbation 706 is generated using an LER of a third value lesser than the second value (e.g., “l”nm). Note that the magnitude of the random displacement decreases as the LER decreases.

[0061] Figure 8 shows perturbation of a mask pattern having polygonal patterns, consistent with various embodiments. The above perturbation method may be applied to any type of mask patterns, such as curvilinear mask patterns (e.g., contact holes illustrated in Figures 4 and 6-7) or polygonal patterns (e.g., line space patterns, rectangular, square patterns, or other similar patterns). For example, as illustrated in Figure 8, polygonal features such as a first polygonal pattern 802 and a second polygonal pattern 804 may be perturbed (e.g., using random signal 415 as described at least with reference to Figure 4) to generate a first perturbed polygonal pattern 812 and a second perturbed polygonal pattern 814. In some embodiments, each polygonal mask pattern in the mask pattern design may be perturbed using a different random signal. For example, the first polygonal pattern 802 and the second polygonal pattern 804 may be perturbed using different random signals.

[0062] Figure 9 is a flow diagram of a method for perturbing mask pattern using a PSD function, consistent with various embodiments. The method of Figure 9 is described at least with reference to Figure 4 above.

[0063] At process P902, a mask pattern is obtained. For example, the mask pattern 402 is obtained from a mask pattern design corresponding to a design layout. A mask pattern design may include a number of mask patterns corresponding to target patterns in the design layout. The mask patterns may be post-OPC mask patterns.

[0064] At process P904, a set of points (or vertices) are assigned along a contour of the mask pattern. In some embodiments, a contour of the mask pattern is extracted (e.g., using known methods) to obtain multiple points along the contour, which are then interpolated to obtain the set of points. For example, the contour 418 of the mask pattern 402 is extracted to obtain a set of vertices 406 of the contour. The set of vertices 406 is interpolated to obtain a corresponding set of interpolated vertices 408. The set of vertices 406 may be interpolated in any number of ways. In some embodiments, even interpolation is performed such that the vertices 406 are constantly or evenly spaced to generate the interpolated vertices 408. The interpolation may change coordinates (e.g., (x, y) coordinates) of some or all of the vertices of the contour 418.

[0065] At process P906, a random displacement is determined for each of the interpolated vertices of the contour. In some embodiments, a random signal is generated using a PSD function, and the values of the random signal are used as the random displacement for perturbing the mask pattern. In some embodiments, the PSD function is tuned to match specific stochastic roughness parameter of the mask, such as at least one of LCDU, LER, correlation length or scaling exponent. The stochastic roughness parameter may be specified by a user or obtained in other ways (e.g., derived from a GSD file having the mask pattern design). As an example, the random signal 415 may be generated using the PSD function (e.g., Eq. (6)) for specific stochastic roughness parameters.

[0066] At process P908, the mask pattern may be perturbed by introducing the random displacements at the interpolated vertices of the mask pattern. For example, the mask pattern 402 may be perturbed using the random signal 415 by introducing a random displacement at each of the interpolated vertices408 to generate a perturbed mask pattern 452 (e.g., using Eq. (8)). In some embodiments, a random displacement is introduced at an interpolated vertex in the direction of a normal vector of the interpolated vertex. The normal vector of a vertex may be determined using any of a known number of methods. For example, the normal vector direction of an interpolated vertex may be determined based on local derivative at the interpolated vertex. The random displacement may be applied to each interpolated vertex of the contour in corresponding normal vector direction to generate the perturbed mask pattern 452.

[0067] In some embodiments, the above process (e.g., processes P902-P908) may be repeated for some or all mask patterns in the mask pattern design to obtain a perturbed mask pattern design 460, which represents a physically manufactured mask of the mask pattern design. In some embodiments, by perturbing the mask pattern using the above method, the perturbed mask pattern design 460 mimics a physically manufactured mask more accurately than the ones generated using conventional randomization methods.

[0068] At process P910, the perturbed mask pattern may be used for various applications. In some embodiments, in a first application, the perturbed mask pattern may be used in optimizing a source (e.g., source profile or pupil profile) of the lithographic apparatus to reduce the impact of the stochastic mask variation. For example, the perturbed mask pattern may be input to known source optimization methods, such as SO, SMO, SMEO or other similar methods, that optimize the source. In some embodiments, by using the perturbed mask pattern design 460 that is a more accurate representation of a physically manufactured mask, the source is optimized more accurately, and the impact of the stochastic mask variation is reduced significantly. In some embodiments, since the perturbed mask pattern is representative of mask variations in the physically manufactured mask, the perturbed mask pattern may be used in determining whether the mask variations in the physically manufactured mask causes a wafer pattern (e.g., a pattern corresponding to the perturbed mask pattern printed on a wafer) to be defective. Accordingly, in a second application, the perturbed mask pattern may be used in determining a probability of the mask variations in the physically manufactured mask causing a corresponding wafer pattern to be defective. For example, the perturbed mask pattern may be input to an OPC verification process to verify OPC patterns (e.g., perturbed mask patterns). The OPC verification process compares the simulated image of a wafer pattern corresponding to the perturbed mask pattern (e.g., contours of the wafer pattern such as aerial, resist or etch contours) with a target pattern in a design layout to identify any possible defects in the mask pattern. If the wafer pattern is found to be defective, then appropriate corrections can be made to the mask pattern (non-perturbed) until the wafer pattern is determined as non-defective.

[0069] Figure 10 is a block diagram that illustrates a computer system 100 which can assist in implementing various methods and systems disclosed herein. The computer system 100 may be used to implement any of the entities, components, modules, or services depicted in the examples of the figures (and any other entities, components, modules, or services described in this specification). Thecomputer system 100 may be programmed to execute computer program instructions to perform functions, methods, flows, or services (e.g., of any of the entities, components, or modules) described herein. The computer system 100 may be programmed to execute computer program instructions by at least one of software, hardware, or firmware.

[0070] Computer system 100 includes a bus 102 or other communication mechanism for communicating information, and a processor 104 (or multiple processors 104 and 105) coupled with bus 102 for processing information. Computer system 100 also includes a main memory 106, such as a random access memory (RAM) or other dynamic storage device, coupled to bus 102 for storing information and instructions to be executed by processor 104. Main memory 106 also may be used for storing temporary variables or other intermediate information during execution of instructions to be executed by processor 104. Computer system 100 further includes a read only memory (ROM) 108 or other static storage device coupled to bus 102 for storing static information and instructions for processor 104. A storage device 110, such as a magnetic disk or optical disk, is provided and coupled to bus 102 for storing information and instructions.

[0071] Computer system 100 may be coupled via bus 102 to a display 112, such as a cathode ray tube (CRT) or flat panel or touch panel display for displaying information to a computer user. An input device 114, including alphanumeric and other keys, is coupled to bus 102 for communicating information and command selections to processor 104. Another type of user input device is cursor control 116, such as a mouse, a trackball, or cursor direction keys for communicating direction information and command selections to processor 104 and for controlling cursor movement on display 112. This input device typically has two degrees of freedom in two axes, a first axis (e.g., x) and a second axis (e.g., y), that allows the device to specify positions in a plane. A touch panel (screen) display may also be used as an input device.

[0072] According to one embodiment, portions of one or more methods described herein may be performed by computer system 100 in response to processor 104 executing one or more sequences of one or more instructions contained in main memory 106. Such instructions may be read into main memory 106 from another computer-readable medium, such as storage device 110. Execution of the sequences of instructions contained in main memory 106 causes processor 104 to perform the process steps described herein. One or more processors in a multi-processing arrangement may also be employed to execute the sequences of instructions contained in main memory 106. In an alternative embodiment, hard-wired circuitry may be used in place of or in combination with software instructions. Thus, the description herein is not limited to any specific combination of hardware circuitry and software.

[0073] The term “computer-readable medium” as used herein refers to any medium that participates in providing instructions to processor 104 for execution. Such a medium may take many forms, including but not limited to, non-volatile media, volatile media, and transmission media. Non-volatile media include, for example, optical or magnetic disks, such as storage device 110. Volatile media includedynamic memory, such as main memory 106. Transmission media include coaxial cables, copper wire and fiber optics, including the wires that comprise bus 102. Transmission media can 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, hard disk, magnetic tape, any other magnetic medium, a CD-ROM, DVD, any other optical medium, punch cards, paper tape, any other physical medium with patterns of holes, a RAM, a PROM, and EPROM, a FLASH-EPROM, any other memory chip or cartridge, a carrier wave as described hereinafter, or any other medium from which a computer can read.

[0074] Various forms of computer readable media may be involved in carrying one or more sequences of one or more instructions to processor 104 for execution. For example, the instructions may initially be borne on a magnetic disk of a remote computer. The remote computer can load the instructions into its dynamic memory and send the instructions over a telephone line using a modem. A modem local to computer system 100 can receive the data on the telephone line and use an infrared transmitter to convert the data to an infrared signal. An infrared detector coupled to bus 102 can receive the data carried in the infrared signal and place the data on bus 102. Bus 102 carries the data to main memory 106, from which processor 104 retrieves and executes the instructions. The instructions received by main memory 106 may optionally be stored on storage device 110 either before or after execution by processor 104.

[0075] Computer system 100 also preferably includes a communication interface 118 coupled to bus 102. Communication interface 118 provides a two-way data communication coupling to a network link 120 that is connected to a local network 122. For example, communication interface 118 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, communication interface 118 may be a local area network (LAN) card to provide a data communication connection to a compatible LAN. Wireless links may also be implemented. In any such implementation, communication interface 118 sends and receives electrical, electromagnetic, or optical signals that carry digital data streams representing various types of information.

[0076] Network link 120 typically provides data communication through one or more networks to other data devices. For example, network link 120 may provide a connection through local network 122 to a host computer 124 or to data equipment operated by an Internet Service Provider (ISP) 126. ISP 126 in turn provides data communication services through the worldwide packet data communication network, now commonly referred to as the “Internet” 128. Local network 122 and Internet 128 both use electrical, electromagnetic, or optical signals that carry digital data streams. The signals through the various networks and the signals on network link 120 and through communication interface 118, which carry the digital data to and from computer system 100, are exemplary forms of carrier waves transporting the information.

[0077] Computer system 100 can send messages and receive data, including program code, through the network(s), network link 120, and communication interface 118. In the Internet example, a server130 might transmit a requested code for an application program through Internet 128, ISP 126, local network 122 and communication interface 118. One such downloaded application may provide for the illumination optimization of the embodiment, for example. The received code may be executed by processor 104 as it is received, or stored in storage device 110, or other non-volatile storage for later execution. In this manner, computer system 100 may obtain application code in the form of a carrier wave.

[0078] While the concepts disclosed herein may be used for imaging on a substrate such as a silicon wafer, it shall be understood that the disclosed concepts may be used with any type of lithographic imaging systems, e.g., those used for imaging on substrates other than silicon wafers.

[0079] The terms “optimizing” and “optimization” as used herein refers to or means adjusting a patterning apparatus (e.g., a lithography apparatus), a patterning process, etc. such that results and / or processes have more desirable characteristics, such as higher accuracy of projection of a design pattern on a substrate, a larger process window, etc. Thus, the term “optimizing” and “optimization” as used herein refers to or means a process that identifies one or more values for one or more parameters that provide an improvement, e.g., a local optimum, in at least one relevant metric, compared to an initial set of one or more values for those one or more parameters. "Optimum" and other related terms should be construed accordingly. In an embodiment, optimization steps can be applied iteratively to provide further improvements in one or more metrics.

[0080] Aspects of the invention can be implemented in any convenient form. For example, an embodiment may be implemented by one or more appropriate computer programs which may be carried on an appropriate carrier medium which may be a tangible carrier medium (e.g., a disk) or an intangible carrier medium (e.g., a communications signal). Embodiments of the invention may be implemented using suitable apparatus which may specifically take the form of a programmable computer running a computer program arranged to implement a method as described herein. Thus, embodiments of the disclosure may be implemented in hardware, firmware, software, or any combination thereof. Embodiments of the disclosure may also be implemented as instructions stored on a machine-readable medium, which may be read and executed by one or more processors. A machine -readable medium may include any mechanism for storing or transmitting information in a form readable by a machine (e.g., a computing device). For example, a machine-readable medium may include read only memory (ROM); random access memory (RAM); magnetic disk storage media; optical storage media; flash memory devices; electrical, optical, acoustical, or other forms of propagated signals (e.g., carrier waves, infrared signals, digital signals, etc.), and others. Further, firmware, software, routines, instructions may be described herein as performing certain actions. However, it should be appreciated that such descriptions are merely for convenience and that such actions in fact result from computing devices, processors, controllers, or other devices executing the firmware, software, routines, instructions, etc.

[0081] In block diagrams, illustrated components are depicted as discrete functional blocks, but embodiments are not limited to systems in which the functionality described herein is organized asillustrated. The functionality provided by each of the components may be provided by software or hardware modules that are differently organized than is presently depicted, for example such software or hardware may be intermingled, conjoined, replicated, broken up, distributed (e.g., within a data center or geographically), or otherwise differently organized. 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, third party content delivery networks may host some or all of the information conveyed over networks, in which case, to the extent information (e.g., content) is said to be supplied or otherwise provided, the information may be provided by sending instructions to retrieve that information from a content delivery network.

[0082] Unless specifically stated otherwise, as apparent from the discussion, it is appreciated that throughout this specification discussions utilizing terms such as “processing,” “computing,” “calculating,” “determining” or the like refer to actions or processes of a specific apparatus, such as a special purpose computer or a similar special purpose electronic processing / computing device.

[0083] Embodiments of the present disclosure can be further described by the following clauses.1. A method of source optimization (SO), the method comprising: obtaining a mask pattern; assigning a set of points along a contour of the mask pattern; determining a random displacement for each point of the contour; perturbing the mask pattern by random displacements to generate a perturbed mask pattern; and optimizing a source of a lithographic apparatus by performing a SO process using the perturbed mask pattern.2. The method of clause 1 , wherein the random displacement is determined using a power spectral density (PSD) function in a frequency domain.3. The method of clause 2 further comprising: obtaining the random displacement in a spatial domain by applying a fast-Fourier transform method on an output of the PSD function.4. The method of clause 2, wherein the PSD function is tuned based on a stochastic roughness of the contour.5. The method of clause 4, wherein the stochastic roughness is represented using one or more of a local critical dimension uniformity (LCDU) parameter, a line edge roughness (LER) parameter, a correlation length parameter, or a scaling component.6. The method of clause 1, wherein assigning the set of points includes: interpolating the contour to obtain multiple interpolated points of the contour.7. The method of clause 6, wherein perturbing the mask pattern by the random displacements includes: determining a direction of a normal vector for each interpolated point, and perturbing the mask pattern by the random displacements at the interpolated points in the respective direction of the normal vector.8. The method of clause 7, wherein the normal vector is determined based on derivatives of the interpolated points of the contour.9. The method of clause 6, wherein interpolating the contour includes performing even interpolation to obtain evenly spaced interpolated points.10. The method of clause 1, wherein assigning the set of points includes: obtaining a set of vertices of the contour, and performing even interpolation on the set of vertices to obtain a set of interpolated vertices as the set of points, wherein a distance between any two interpolated vertices in the set of interpolated vertices is the same.11. The method of clause 1 , wherein the mask pattern is a curvilinear pattern or a polygon.12. The method of clause 1 further comprising: performing an optical proximity correction (OPC) verification on the perturbed mask pattern to determine a probability that mask variations cause a pattern printed on a substrate to be defective, wherein the perturbed mask pattern is representative of the mask variations in a manufactured mask.13. A method of perturbing a mask pattern, the method comprising: obtaining a mask pattern; interpolating a contour of the mask pattern to obtain multiple interpolated vertices of the contour; determining a random displacement of the contour at each interpolated vertex; and perturbing the mask pattern by random displacements to generate a perturbed mask pattern.14. The method of clause 13 further comprising: optimizing a source of a lithographic apparatus by performing a SO process using the perturbed mask pattern.15. The method of clause 13, wherein the random displacement is determined using a power spectral density (PSD) function in a frequency domain.16. The method of clause 15 further comprising: obtaining the random displacement in a spatial domain by applying a fast-Fourier transform method on an output of the PSD function.17. The method of clause 15, wherein the PSD function is tuned based on a stochastic roughness of the contour.18. The method of clause 17, wherein the stochastic roughness is represented using one or more of a local critical dimension uniformity (LCDU) parameter, a line edge roughness (LER) parameter, a correlation length parameter, or a scaling component.19. The method of clause 13, wherein perturbing the mask pattern by the random displacements includes: computing a normal vector for each interpolated vertex, and perturbing the mask pattern by the random displacement at each of the interpolated vertices in a respective direction of the normal vector.20. The method of clause 19, wherein the normal vector is determined based on local derivative of the interpolated vertices of the contour, which is determined as a difference between the coordinates of two consecutive interpolated vertices.21. The method of clause 13, wherein interpolating the contour includes: obtaining a set of vertices of the contour, and performing even interpolation on the set of vertices to obtain the interpolated vertices, wherein a distance between any two interpolated vertices is the same.22. The method of clause 13, wherein interpolating the contour includes performing even interpolation to obtain evenly spaced interpolated vertices.23. The method of clause 13, wherein the mask pattern is a curvilinear pattern or a polygon.24. The method of clause 13 further comprising: performing an OPC verification on the perturbed mask pattern to determine a probability that mask variations cause a pattern printed on a substrate to be defective, wherein the perturbed mask pattern is representative of the mask variations in a manufactured mask.25. An apparatus, the apparatus comprising: a memory storing a set of instructions; and a processor configured to execute the set of instructions to cause the apparatus to perform a method of any of the above clauses.26. A non-transitory computer-readable medium having instructions recorded thereon, the instructions when executed by a computer implementing the method of any of the above clauses.

[0084] The reader should appreciate that the present application describes several inventions. Rather than separating those inventions into multiple isolated patent applications, these inventions have been grouped into a single document because their related subject matter lends itself to economies in the application process. But the distinct advantages and aspects of such inventions should not be conflated. In some cases, embodiments address all of the deficiencies noted herein, but it should be understood that the inventions are independently useful, and some embodiments address only a subset of such problems or offer other, unmentioned benefits that will be apparent to those of skill in the art reviewing the present disclosure. Due to cost constraints, some inventions disclosed herein may not be presently claimed and may be claimed in later filings, such as continuation applications or by amending the present claims. Similarly, due to space constraints, neither the Abstract nor the Summary sections of the present document should be taken as containing a comprehensive listing of all such inventions or all aspects of such inventions.

[0085] It should be understood that the description and the drawings are not intended to limit the present disclosure to the particular form disclosed, but to the contrary, the intention is to cover all modifications, equivalents, and alternatives falling within the spirit and scope of the inventions as defined by the appended claims.

[0086] Modifications and alternative embodiments of various aspects of the inventions will be apparent to those skilled in the art in view of this description. Accordingly, this description and the drawings are to be construed as illustrative only and are for the purpose of teaching those skilled in the art the general manner of carrying out the inventions. It is to be understood that the forms of the inventions shown and described herein are to be taken as examples of embodiments. Elements and materials may be substituted for those illustrated and described herein, parts and processes may be reversed or omitted, certain features may be utilized independently, and embodiments or features of embodiments may be combined, all as would be apparent to one skilled in the art after having the benefit of this description. Changes may be made in the elements described herein without departing from the spirit and scope of the invention as described in the following claims. Headings used herein are for organizational purposes only and are not meant to be used to limit the scope of the description.

[0087] As used herein, unless specifically stated otherwise, the term “or” encompasses all possible combinations, except where infeasible. For example, if it is stated that a component includes A or B, then, unless specifically stated otherwise or infeasible, the component may include A, or B, or A and B. As a second example, if it is stated that a component includes A, B, or C, then, unless specifically stated otherwise or infeasible, the component may include A, or B, or C, or A and B, or A and C, or B and C, or A and B and C. Expressions such as “at least one of’ do not necessarily modify an entirety of a following list and do not necessarily modify each member of the list, such that “at least one of A, B, and C” should be understood as including only one of A, only one of B, only one of C, or any combination of A, B, and C. The phrase “one of A and B” or “any one of A and B” shall be interpreted in the broadest sense to include one of A, or one of B.

[0088] The descriptions herein are intended to be illustrative, not limiting. Thus, it will be apparent to one skilled in the art that modifications may be made as described without departing from the scope of the claims set out below.

Claims

CLAIMS1. A method of source optimization (SO), the method comprising: obtaining a mask pattern; assigning a set of points along a contour of the mask pattern; determining a random displacement for each point of the contour; perturbing the mask pattern by random displacements to generate a perturbed mask pattern; and optimizing a source of a lithographic apparatus by performing a SO process using the perturbed mask pattern.

2. The method of claim 1 , wherein the random displacement is determined using a power spectral density (PSD) function in a frequency domain.

3. The method of claim 2 further comprising: obtaining the random displacement in a spatial domain by applying a fast-Fourier transform method on an output of the PSD function.

4. The method of claim 2, wherein the PSD function is tuned based on a stochastic roughness of the contour.

5. The method of claim 4, wherein the stochastic roughness is represented using one or more of a local critical dimension uniformity (LCDU) parameter, a line edge roughness (LER) parameter, a correlation length parameter, or a scaling component.

6. The method of claim 1, wherein assigning the set of points includes: interpolating the contour to obtain multiple interpolated points of the contour.

7. The method of claim 6, wherein perturbing the mask pattern by the random displacements includes: determining a direction of a normal vector for each interpolated point, and perturbing the mask pattern by the random displacements at the interpolated points in the respective direction of the normal vector.

8. The method of claim 7, wherein the normal vector is determined based on derivatives of the interpolated points of the contour.

9. The method of claim 6, wherein interpolating the contour includes performing even interpolation to obtain evenly spaced interpolated points.

10. The method of claim 1, wherein assigning the set of points includes: obtaining a set of vertices of the contour, and performing even interpolation on the set of vertices to obtain a set of interpolated vertices as the set of points, wherein a distance between any two interpolated vertices in the set of interpolated vertices is the same.

11. The method of claim 1 , wherein the mask pattern is a curvilinear pattern or a polygon.

12. The method of claim 1 further comprising: performing an optical proximity correction (OPC) verification on the perturbed mask pattern to determine a probability that mask variations cause a pattern printed on a substrate to be defective, wherein the perturbed mask pattern is representative of the mask variations in a manufactured mask.

13. An apparatus, the apparatus comprising: a memory storing a set of instructions; and a processor configured to execute the set of instructions to cause the apparatus to perform a method of any of the above claims.

14. A non-transitory computer-readable medium having instructions recorded thereon, the instructions when executed by a computer implementing the method of any of the above claims.

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