IL322004A - Modeling defect probability from expected critical dimensions to improve accuracy of failure rate prediction - Google Patents

Description

2022P00393WO 1 MODELING DEFECT PROBABILITY FROM EXPECTED CRITICAL DIMENSIONS TO IMPROVE ACCURACY OF FAILURE RATE PREDICTION CROSS-REFERENCE TO RELATED APPLICATION [0001]This application claims priority of US application 63/441,761 which was filed on 27 January 2023 and which is incorporated herein in its entirety by reference.

FIELD [0002]The embodiments provided herein disclose a method of modeling defect probability from critical dimensions of a sample to improve failure rate prediction, and more particularly, a method of modeling defect probability using a sigmoid function to improve failure prediction.

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). An IC chip in a smart phone can be as small as a person’s thumbnail and may include over 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 may potentially result in problems with the final IC and may cause device failure. Therefore, in manufacturing processes of ICs, unfinished or finished circuit components are inspected to ensure that they are manufactured according to design and are free of defects. Inspection systems utilizing optical microscopes or charged particle (e.g., electron) beam microscopes, such as a scanning electron microscope (SEM) can be employed. As the physical sizes of IC components continue to shrink, accuracy and yield in IC inspection become increasingly important. High process yield and high wafer throughput can be impacted by the present of defects, especially if operator intervention is required for reviewing the defects. Therefore, modeling lithographic conditions that may accurately predict the formation of defects in ICs before fabrication occurs is desired.

SUMMARY [0004]The embodiments provided herein disclose a method of modeling defect probability from critical dimensions of sample to improve failure rate prediction, and more particularly, a method of parameterizing a sigmoid function to improve failure prediction. [0005]Some embodiments provide a non-transitory computer readable medium comprising a set of instructions that is executable by one or more processors of a computing device to cause the computing device to perform a method for modeling defect probability. The method comprises obtaining a critical dimension measurement and defect data from a sample, and determining a lower defect threshold limit and an upper defect threshold limit based on a sigmoid function wherein the lower defect threshold limit 2022P00393WO 2 and the upper defect threshold limit comprise a center value and a spread value. [0006]Other advantages of the present invention will become apparent from the following description taken in conjunction with the accompanying drawings wherein are set forth, by way of illustration and example, certain embodiments of the present invention.

BRIEF DESCRIPTION OF FIGURES [0007]The above and other aspects of the present disclosure will become more apparent from the description of exemplary embodiments, taken in conjunction with the accompanying drawings. [0008] FIG. 1is a schematic diagram illustrating an example lithographic projection assembly to fabricate an IC, consistent with embodiments of the present disclosure. [0009] FIG. 2is an example conventional method to predict non-defect probability in an IC structure. [0010] FIG. 3is an example flowchart of predicting IC structure non-failure and defect probability using a sigmoid function, consistent with embodiments of the present disclosure. [0011] FIG.4A is a schematic block diagram illustrating throughput to generate data, consistent with embodiments of the present disclosure. [0012] FIG.4B is an example focus-dose matrix illustrating a two-dimensional layout of lithographic conditions to fabricate an IC structure, consistent with embodiments of the present disclosure. [0013] FIGS. 5A, 5B, 5C, 5D, 5Eare schematic diagrams illustrating an example sigmoid function to predict non-defect probability and failure rate probability of an IC structure, consistent with embodiments of the present disclosure. [0014] FIGS. 6Aand 6Bare example gradient plots of calculated failure rates as a function of sigmoid function parameters, consistent with embodiments of the present disclosure. [0015] FIG. 7is a flowchart representing a first example process for parameterizing variables of a sigmoid function to accurately predict non-defect probability and failure rate probability for an IC structure, consistent with embodiments of the present disclosure. [0016] FIGS. 8A, 8B,and 8Care example data plots illustrating steps of method 700, consistent with embodiments of the present disclosure. [0017] FIG. 9is a flowchart representing a second example process for parameterizing variables of a sigmoid function to accurately predict non-defect probability and failure rate probability for an IC structure, consistent with embodiments of the present disclosure. [0018] FIGS. 10A, 10B,and 10Care example data plots illustrating steps of method 900, consistent with embodiments of the present disclosure. [0019] FIGS. 11A-11Iare example results illustrating how embodiments of the present disclosure provide improvements over conventional methods to model failure rate of an IC structure. [0020] FIGS. 12A, 12B,and 12Care example generated failure rate process windows illustrating how embodiments of the present disclosure provide improvements over conventional methods to model failure rate of an IC structure according to fabrication condition. 2022P00393WO 3 DETAILED DESCRIPTION [0021]Reference will now be made in detail to exemplary embodiments, examples of which are illustrated in the accompanying drawings. The following description refers to the accompanying drawings in which the same numbers in different drawings represent the same or similar elements unless otherwise represented. The implementations set forth in the following description of exemplary embodiments do not represent all implementations consistent with the invention. Instead, they are merely examples of apparatuses and methods consistent with aspects related to the invention as recited in the appended claims. [0022]The enhanced computing power of electronic devices, while reducing the physical size of the devices, can be accomplished by significantly increasing the packing density of circuit components such as transistors, capacitors, diodes, etc. on an IC chip. For example, an IC chip of a smart phone, which is the size of a thumbnail, may include over 2 billion transistors, the size of each transistor being less than l/1000th of a human hair. Thus, it is not surprising that semiconductor IC manufacturing is a complex and time-consuming process, with hundreds of individual steps. Errors in even one step have the potential to dramatically affect the functioning of the final product. Even one "killer defect" can cause device failure. The goal of the manufacturing process is to improve the overall yield of the process. For example, for a 50-step process to get to a 75% yield, each individual step must have a yield greater than 99.4%, and if the individual step yield is 95%, the overall process yield drops to 7%. [0023]While high process yield is desirable in an IC chip manufacturing facility, maintaining a high wafer throughput, defined as the number of wafers processed per hour, is also essential. High process yields and high wafer throughput can be impacted by the presence of defects, especially if operator intervention is required for reviewing the defects. Thus, high throughput detection and identification of micro and nano-sized defects by inspection tools (such as a charged particle beam inspection tool) is essential for maintaining high yields and low cost. Inspection of a wafer using an electron beam inspection tool may generate images of the wafer to measure IC structure dimensions. The measured dimensions may be compared to a reference structure absent any defects to determine the presence of defects in the imaged structure. However, inspection of ICs for defect detection is often a time- consuming process. It may be desirable to prevent defects from occurring during the fabrication stages instead of further refining IC inspection methods. Therefore, it is desired to improve the accuracy of IC failure rate predictive modeling during fabrication and generating more accurate process windows of fabrication conditions. [0024]ICs may be manufactured using lithography, which is a fabrication process involving creating complex circuit patterns drawn on a mask deposited onto a substrate. Lithography may be performed by a lithographic apparatus, which is a machine that applies a source of radiation (e.g., light or X-ray) onto a target portion of the substrate to form a desired pattern. The target portion of the substrate may be covered with a pattern device (e.g., mask) that may be either eliminated or developed after exposure to the radiation source. This process of transferring the desired pattern to the substrate is called a 2022P00393WO 4 patterning process. The patterning process may include a patterning step to transfer a pattern from a pattern device (e.g., a mask) to the substrate. There can also be one or more related pattern processing steps, such as mask development by a development apparatus, baking of the substrate using a bake tool, etching the pattern onto the substrate using an etch apparatus, or other chemical and physical processing steps involved in fabricating a pattern onto the substrate. Variations in experimental parameters (e.g., stochastic variations, errors, or noise due to an inspection tool or pattern processing tool) can potentially limit lithography implementation for high volume manufacturing (HVM), or process yield, of ICs and introduce defects into IC structures. [0025]In the manufacture of ICs using a lithographic apparatus, typically many lithographic patterning steps are performed, thereby forming functional features in successive layers on the substrate. A critical aspect of performance of the lithographic apparatus is therefore the ability to place the applied pattern correctly and accurately in relation to features laid down in previous layers. For this purpose, the substrate is provided with one or more sets of alignment marks. Each mark is a structure whose position can be measured later using, for example, an electron beam inspection tool. Defects may occur in which an applied pattern structure or pattern layer is incorrectly placed in relation to a reference mark, or when the fabrication conditions are suboptimal. A reference mark or layout define the desired structure, structure dimensions, and the distance between IC structures (such as gates, capacitors, etc.) or interconnect lines. This may ensure that the IC devices or lines do not interact with one another in an undesirable way. The structure limitations provided by the reference layouts are typically referred to as critical dimensions. A critical dimension of a circuit can be defined as the smallest width of a line or hole or the smallest space between two lines or two holes. Thus, the critical dimension determines the overall size and packing density of the designed IC. A goal in IC fabrication is to faithfully reproduce the original IC design on the substrate. If an error occurs during fabrication where the created IC design pattern does not match the reference design, this may result in a defect in the IC structure and render the IC inoperable. [0026]As mentioned above, high throughput of IC fabrication with low amounts of structural defects is desired. Therefore, the ability to accurately predict failure rates of IC structures according to fabrication conditions in lithography is desired. Conventional methods use measured critical dimensions of ICs formed under various lithographic conditions as input data to model a failure rate probability of the formed structures. A failure probability rate is related to a critical dimension of formed structures from various lithographic fabrication or patterning conditions (e.g., focus and dose, explained more below) that results in a small defect (e.g., a missing structure on an IC) or a large defect (e.g., a bridging structure on an IC). Other defects may be structures on the IC that have dimensions markedly different from a reference structure. To model the critical dimensions for defect prediction as a function of lithographic fabrication condition, conventional systems assume a constant critical dimension limit, which is a critical dimension at which failure may likely occur in an IC. For small defects, a critical dimension value below this limit can have a significant failure probability. For large defects, a critical 2022P00393WO 5 dimension above this limit can have a significant failure probability. However, assuming a constant value for the critical dimension limit may result in model inaccuracies. According to conventional methods, a measured critical dimension below the lower critical dimension limit or above the upper critical dimension limit may not actually result in defects. Therefore, conventional methods to predict IC failure and defect probability according to lithographic fabrication of ICs may not be accurate enough. [0027]Embodiments of the present disclosure provide a method to predict defects and failure according to lithographic fabrication condition of ICs using a sigmoid function. Some embodiments of the present disclosure may provide a wider range for a critical dimension limit and therefore provide a more robust prediction method. Moreover, some embodiments of the present disclosure may increase throughput of IC manufacturing and confidence in predicting lithographic fabrication conditions to manufacture defect-free ICs. Some embodiments of the present disclosure may also provide a method to parameterize the sigmoid function to more accurately determine the lithographic fabrication conditions necessary to minimize defects and failure in ICs. Some embodiments of the present disclosure may provide a range of critical dimensions for ICs that may more accurately predict defects compared to conventional methods, and therefore further increase throughput. [0028]Relative dimensions of components in drawings may be exaggerated for clarity. Within the following description of drawings the same or like reference numbers refer to the same or like components or entities, and only the differences with respect to the individual embodiments are described. 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 database can include A or B, then, unless specifically stated otherwise or infeasible, the database can include A, or B, or A and B. As a second example, if it is stated that a database can include A, B, or C, then, unless specifically stated otherwise or infeasible, the database can include A, or B, or C, or A and B, or A and C, or B and C, or A and B and C. [0029]Reference is now made to FIG. 1,which is a schematic diagram illustrating an exemplary lithographic projection apparatus 100. Lithographic projection apparatus 100 may include a radiation source 101, which may be a deep-ultraviolet excimer laser source or other type of source including an extreme ultraviolet (EUV) source, and emits a radiation beam 108. Illumination optics which may include illumination optics components 102 and 103 that shape radiation from the radiation source 101; a patterning device 104; and transmission optics 105 that project an image of the patterning device pattern onto a substrate 106. An adjustable filter or aperture 107 at the pupil plane of the projection optics may restrict the range of beam angles that impinge on substrate 106, where the largest possible angle defines the numerical aperture of the projection optics NA=sin(Smax). The area or region on substrate 106 where radiation beam 108 impacts is called a target portion 109, in which the radiation beam impacts the top layer or mask (not shown). [0030]Illumination optics components 102 and 103 may direct and shape radiation beam 108 via patterning device 104 onto substrate 106 and may include any optical component that may alter the 2022P00393WO 6 wavefront of radiation beam 108. A resist layer on substrate 106 may be exposed and a radiation intensity distribution at substrate 106 (i.e., an aerial image) may be transferred to the resist layer. Optical properties of the lithographic projection apparatus (e.g., properties of the source, the patterning device, and the projection optics) dictate this process. The resist layer may be removed and the applied pattern from radiation beam 108 may then be applied to the substrate as discussed above. [0031]Although reference may be made in the present disclosure to ICs, it is appreciated that the present disclosure may be applicable to other possible applications or designs. For example, the present disclosure may be applied to integrated optical systems, magnetic domain memories, liquid-crystal display panels, thin-film magnetic heads, and other nanoscale structures. It is further appreciated that the terms "reticle", "wafer", or "die" may be used interchangeably with the terms "mask", "substrate" or "sample", and "target portion", respectively. [0032]Reference is now made to FIG. 2,which is an example conventional method to predict non- defect probability in an IC structure according to critical dimension distribution. It is appreciated that FIG. 2illustrates the example prediction according to a lithographic fabrication condition. The non- defect probability and critical dimension distribution in FIG. 2may vary depending on the fabrication condition applied to a sample. A lithographic fabrication condition may be a focus and dose condition, which is explained further below. FIG. 2illustrates an example model 210 of non-failure probability 201 for an IC structure modeled using a conventional stepwise function 203 method plotted against critical dimension 202. The critical dimension 202 value at which non-failure probability 201 jumps from a low to high value is a critical dimension limit 204. The conventional method of modeling non- failure probability 201 for an IC using a stepwise function assumes the IC structure may either contain a void or a bridge or is designed as desired. A critical dimension below critical dimension limit 2may likely result in a failed structure on an IC. Therefore, critical dimension limit 204 may represent an inflection point in predicting failure in an IC structure and is the critical dimension value at which a failure may likely occur when fabricating an IC at a fabrication condition. Reference is now made to an example model 220 of a critical dimension probability 205 of on IC plotted against critical dimension 202. The critical dimension probability 205 may be modeled using a Gaussian function or any other method for modeling probability distributions. To determine a non-defect probability of an IC structure, the stepwise function 203 modeling non-failure probability 201 of model 210 is multiplied by the polynomial function modeling critical dimension probability 205 of model 220. Reference is now made to model 230, where the product of model 210 and model 220 equals a non-defect probability 206, which may predict a small defect of an IC structure according to a lithographic fabrication parameter. As illustrated in model 230, the non-defect probability 206 transition at the critical dimension limit 2is modeled as a vertical line. Therefore, conventional methods represent a binary method of modeling defect formation where an IC structure with critical dimension 202 below critical dimension limit 2will likely contain defects and fail. It is appreciated that FIG. 2illustrates modeling non-defect probability for small defects in an IC structure, but is also applicable to modeling and predicting large 2022P00393WO 7 defects in an IC structure. For large defects, a critical dimension value above a critical dimension limit may likely result in failure for an IC structure. [0033]Conventional methods may model IC structure non-failure and defect probabilities too sharply. This may miss critical dimensions of IC structures above or below the critical dimension limit that may or may not result in a defect. For example, an IC structure with a critical dimension below critical dimension limit 204 in model 230 may not contain a defect when fabricated at a lithographic condition and therefore exhibit a non-defect probability 206 greater than zero. However, the conventional model would predict an IC structure with essentially a non-defect probability of zero (i.e., the IC structure will contain a defect) and therefore inaccurately predict IC structure formation at the lithographic fabrication condition. Therefore, conventional methods may unnecessarily limit the packing density of ICs. Similarly, an IC structure with a critical dimension above critical dimension limit 204 in model 2may contain a defect, while the conventional model may predict a defect-free IC structure. This would therefore lead to an inoperable IC that may not be identified until post-processing and therefore decrease process yield and wafer throughput. [0034]Reference is now made to FIG. 3,which is an example flowchart of IC structure non-failure and defect modeling using a sigmoid function for lithographic fabrication of IC structures, consistent with embodiments of the present disclosure. In step 301, input data for the model is acquired. Step 3may involve generating one or more IC structures on a sample under a lithographic fabrication condition, inspecting the one or more IC structures using an inspection tool, measuring a critical dimension of the one or more formed IC structures and measuring a failure rate of the one or more formed IC structures at the lithographic fabrication condition, and then modeling the critical dimension probability distribution. It is appreciated that step 301 may be performed for a plurality of lithographic fabrication conditions, wherein one or more IC structures are formed according to a plurality of lithographic fabrication conditions and critical dimensions, failure rate, and modeled critical dimension probability distributions are determined for each fabrication condition. The inspection tool (e.g., an optical microscope or a charged particle microscope) may be used to generate an image of the one or more IC structures. Failure rates of an IC structure in relation to a critical dimension may be measured as, for example, 1 part per million (ppm) or 1 part per billion (ppb) of a feature of the IC structure. The feature of the IC structure may be identified by inspecting a generated image of the formed IC structure using a processor. Failure rates may be measured by analyzing a feature (e.g., a pixel or region of the IC structure pattern) of a corresponding image of the formed IC structure. [0035]In step 302, a critical dimension limit for a fabrication condition is determined using a sigmoid function model. This may include determining a critical dimension limit for small and large defects of an IC structure. In step 303, a failure rate for an IC structure at a fabrication condition is calculated. The calculated failure rate is dependent on the sigmoid function parameters and measured failure rates and critical dimension values of the IC structure. In step 304, each sigmoid function modeling a critical dimension limit according to a fabrication condition is parameterized to determine a critical dimension 2022P00393WO 8 limit value for each fabrication condition used to generate the one or more IC structures. In step 305, a process window is generated to accurately predict successful formation of defect-free IC structures according to all fabrication conditions. Each step will be explained in more detail as follows. [0036]Reference is now made to FIG. 4A,which is an example block diagram for generating input data, consistent with embodiments of the present disclosure. Input data may be generated using two steps as illustrated in FIG. 4A.A lithographic projection apparatus 401 (such as lithographic projection apparatus 100 in FIG. 1)may be used to generate a focus-dose matrix using a plurality of focus and dose conditions for a radiation source (such as radiation source 101 in FIG. 1)onto a surface of a sample. As illustrated in FIG. 4B,a focus-dose matrix 410 may be formed on a target portion 420 of a substrate (such as target portion 109 in FIG. 1).The areas of the substrate within the grids of focus-dose matrix 410 may be exposed to a radiation source with varied focus 430 and dose 440. Focus 430 is an indication of the focal point of the radiation beam (such as radiation beam 108 in FIG. 1)onto the surface of the sample, and dose 440 is an indication of the energy of the radiation beam (such as radiation beam 1in FIG. 1)per area of the sample (e.g., mJ/cm2). Thus, each area outlined by a grid of focus-dose matrix 410 corresponds to a two-dimensional layout of a specific focus and dose of the radiation beam applying a pattern onto the sample. Focus-dose matrix 410 may be a grid layout. Each area within a grid of focus- dose matrix 410 may contain a same IC structure or pattern, or each area with a grid may contain a different IC structure or pattern. Referring back to FIG. 4A,a processor 403 with a memory may be communicatively connected to lithographic projection apparatus 401 to store the focus and dose conditions of the radiation beam corresponding to each grid area of a focus-dose matrix. An inspection tool 402 (e.g., an optical microscope or a charged particle beam microscope) may be used to measure critical dimensions of the structures formed in each grid area of the focus-dose matrix (410 in FIG. 4B) that was generated by lithographic projection apparatus 401. Each critical dimension measured for a structure within a grid of the focus-dose matrix may correspond to a focus and dose condition during the lithographic fabrication process. A processor 403 with a memory may be communicatively connected to electron inspection tool 402 to store the measured critical dimension values. [0037]Reference is now made to FIGS. 5A-E,which are examples of a sigmoid function used to predict non-defect probability of an IC structure formed according to a fabrication condition, consistent with embodiments of the present disclosure. The fabrication condition may be a lithographic focus, dose, or combined focus and dose condition and may correspond to a grid area in a focus-dose matrix as described above. Illustrated in FIG. 5A,a non-failure probability 501 for an IC structure may be modeled using a sigmoid function 503 as a function of critical dimension 502. The dotted line 5passing through the center of sigmoid function 503 represents a critical dimension center value 504 and may be referred to as zeta 1 (£1). Zeta l may also be referred to as a lower defect threshold limit. The horizontal distance from zeta l to the horizontal portions of sigmoid function 503 may represent a variability 505 of zeta 1 and may be referred to as gamma (yl) 1. Gamma l represents the variation of the critical dimension center value 504 and may account for error in predicting non-defect probability. 2022P00393WO 9 Gamma l may also be referred to as a spread value. Therefore, a critical dimension limit value for an IC structure fabricated at a focus and dose condition may be predicted using a sigmoid function to model non-failure probability, may be a critical dimension value within the region of variability 505, and may be dependent on zeta 1 and gamma 1. Sigmoid function 503 describing the non-failure probability for small critical dimensions may be represented as: = 1+e«1-x)/(YD Equation(!) id="p-38" id="p-38"

[0038]In Equation 1, S(x) represents a non-failure probability of an IC structure and x represents a critical dimension of the IC structure, in which the IC structure is formed according to a fabrication condition. The measured non-failure probability of an IC structure within a grid area of the focus-dose matrix is matched with the measured critical dimension values of the IC structure (step 301 in FIG. 3) to generate sigmoid function 303. 1؛ is the zeta l value that represents the center value of sigmoid function 303 and yl is the gamma 1 value and represents a variability of the zeta l value. [0039] FIG. 5Billustrates a critical dimension probability 506 for an IC structure as a function of critical dimension 502 according to a fabrication condition. Critical dimension probability 506 may be modeled by, for example, applying a Gaussian distribution, or any other method for modeling probability distributions, to the measured critical dimension values of the IC structure according to a fabrication condition. The product of sigmoid function 503 modeling non-failure probability 501 and the Gaussian function modeling critical dimension probability 506 may yield a non-defect probability 507 (solid line) for an IC structure as illustrated in FIG. 5C. FIG. 5Cincludes a non-defect probability 508 (dotted line) calculated using a stepwise function according to conventional methods for comparison. Instead of the drastic shift in non-defect probability predicted by conventional methods at a critical dimension limit, embodiments of the present disclosure provide a wider range of defect prediction. For example, to the left of the horizontal portion of non-defect probability 508, non-defect probability 507 of the present disclosure predicts a defect-free IC structure may form according to a fabrication condition, whereas the conventional method predicts no defect-free IC structure will form. Conversely, non-defect probability 507 may predict a lower probability of a small defect in the formed IC structure compared to the conventional method non-defect probability 508 and therefore account for missed predictions using conventional methods. Therefore, FIG. 5Cillustrates a sigmoid function that may provide a wider range of predicting small defects in an IC structure compared to conventional methods using a stepwise function, as described above. Furthermore, embodiments of the present disclosure may more accurately predict defects at critical dimensions above or below a critical dimension limit predicted by conventional methods. The sigmoidal function may therefore result in more robust defect modeling and IC structure failure predictions. [0040] FIGS. 5A, 5B, 5Cillustrate IC structure non-failure and non-defect probability for small 2022P00393WO 10 critical dimension, or missing, defects. It is appreciated that a sigmoid function to model IC structure non-failure and non-defect probability for large critical dimension, or bridging defects, may be used, and is illustrated in FIG. 5D.As described above, sigmoid function 513 may be generated by measured non-failure probabilities and measured critical dimensions of an IC structure formed according to a fabrication condition. Sigmoid function 513 may have a center value 514, which may be referred to as zeta 2 ((2). As described above, zeta 2 may be referred to as an upper defect threshold limit. Furthermore, the horizontal distance between center value 514 and the horizontal portions of sigmoid function 513 is a variability 515 of center value 514. Variability 515 may be referred to as gamma 2 (y2) or a spread value. Therefore, a critical dimension limit value of a large defect for an IC structure fabricated according to a fabrication condition may be determined using sigmoid function 513 to model non-failure probability 501, may be a critical dimension value within the region of variability 515, and may be dependent on zeta 2 and gamma 2. Sigmoid function 513 describing the non-failure probability for large critical dimensions may be represented as: ( 2 ) 2،2 ) Equation )؛ ،, + 1 ־ 1 = id="p-41" id="p-41"

[0041]In Equation 2, S(x) represents a large defect non-failure probability for an IC structure and x represents a critical dimension of the IC structure, in which the IC structure is formed according to a fabrication condition. The measured non-failure probability of an IC structure within a grid area of the focus-dose matrix is matched with the measured critical dimension values of the IC structure (step 3in FIG. 3)to generate sigmoid function 513. 2؛ is the zeta 2 value that represents the center value of sigmoid function 513 and y2 is the gamma 2 value and represents a variability, or spread, of the zeta value. [0042] FIG. 5Eillustrates a non-defect probability 517 (solid line) to predict large defects of an IC structure, which may be calculated as discussed above. FIG. 5Ealso illustrates a non-defect probability 518 (dotted line) when conventional methods are applied to predict large defects. As discussed above, embodiments of the present disclosure may predict a defect-free IC structure with a critical dimension above the large critical dimension limit value predicted by conventional non-defect probability 5whereas conventional methods would predict a large defect. Moreover, embodiments of the present disclosure may predict a lower non-defect probability 517 of a defect-free IC structure at a critical dimension below the critical dimension limit predicted conventional methods. As discussed above, using a sigmoid function may accurately predict the presence of IC structure defects below the critical dimension limit value and the presence of defect-free IC structure above the critical dimension limit value. [0043]It is appreciated that other forms of sigmoid functions besides those described in Equations and 2 may be used to model non-failure probability of critical dimension limits. Other sigmoid functions 2022P00393WO 11 may be used, which include, but are not limited to, a hyperbolic tangent function or arctangent function formula. [0044]Based on the measured critical dimension values and measured non-failure probabilities of an IC structure fabricated at a given fabrication condition, a failure rate can be estimated. A failure rate (FR) for either small or large critical dimension defects may be calculated as: FR = 1 — J S (x) X PDF Equation (3) id="p-45" id="p-45"

[0045]Where the PDF term may represent a critical dimension probability distribution function determined by the measured critical dimensions of the IC structure at the fabrication condition. S(x) as discussed above may be represented as a sigmoid function for small or large defects (such as sigmoid function 503 or sigmoid function 513 in FIGS. 5Aor 5D).However, since S(x) contains two variables, zeta (zeta 1 or zeta 2) and gamma (gamma 1 or gamma 2), there may be multiple zeta and gamma values that may equal the same estimated failure rate. This is illustrated in FIG. 6,which are example gradient plots of calculated failure rates 601 as a function of zeta and gamma for an IC structure fabricated according to a fabrication condition. FIG. 6Aillustrates the calculated failure rates for a small defect as a function of zeta 602 (e.g., zeta 1 in FIG. 5)and gamma 603 (e.g., gamma 1 in FIG. 5)and FIG. 6Billustrates the calculated failure rates 601 for a large defect as a function of zeta 604 (e.g., zeta 2 in FIG. 5)and gamma 605 (e.g., gamma 2 in FIG. 5).In both FIGS. 6Aand 6B,the measured failure rates 610 and 620, respectively, for the IC structure are plotted and indicate that a range of zeta and gamma values may equal the measured failure rate. The failure rate for an IC structure may be determined as described above for each IC structure in a grid area of a focus-dose matrix. A measured failure may be an observed defect in the IC structure that may be a missing defect, a bridging defect, or when compared to a reference structure may have a different dimension. Since any number of zeta and gamma values for a sigmoid function (such as sigmoid function 503 or sigmoid function 513 of FIG. 5)may satisfy the condition of the calculated and measured failure rate of an IC structure being equal, it may not be possible to identify the associated fabrication conditions necessary to predict defect-free IC structures. Therefore, it may be desired to parameterize the zeta and gamma values of the sigmoid function to more accurately predict the fabrication conditions that create defects in ICs and defect-free ICs. [0046]Two methods may be used to parameterize the sigmoid function to determine zeta and gamma values necessary to predict small or large defects in an IC structure based on the critical dimension distributions and measured failure rates of an IC structure at multiple focus/dose conditions. [0047] FIG. 7illustrates a flow diagram of method 700, which may be a first method to parameterize a sigmoid function based on critical dimension distributions and measured failure rates of an IC structure fabricated at multiple fabrication conditions. The fabrication conditions may be combined focus and dose conditions. Method 700 may be performed by a computing device (e.g., processor 403 2022P00393WO 12 in FIG. 4A).For method 700, the spread values (such as gamma 1 and gamma 2 described above) for the sigmoid function (such as sigmoid function 503 or sigmoid function 513 in FIG. 5)may be assumed to be constant for all combined focus and dose conditions and the center values (such as zeta l and zeta described above) may be varied with each focus condition (i.e., may be focus-dependent). Thus, one or more dose conditions per focus condition may be considered. Method 700 may occur concurrently for a sigmoid function modeling small defects (e.g., sigmoid function 503 in FIG. 5A)and for a sigmoid function modeling large defects (e.g., sigmoid function 513 in FIG. 5D).In step 701, a first spread value is assigned to be constant according to a fabrication condition. The fabrication condition may be a focus, dose, or combined focus and dose condition. The first spread value is assumed to be constant for each structure created in a grid area of a focus-dose matrix on the sample (such as focus-dose matrix 410 in FIG. 4B)during lithography fabrication. [0048]In step 702, a first center value corresponding to a first measured failure rate of an IC structure according to a first fabrication condition is calculated. The first fabrication condition may be a first combined focus and dose condition. The first center value may be calculated by solving Equation with a known measured failure rate, critical dimension probability distribution of the IC structure according to the first fabrication condition, and the first spread value. [0049]In step 703, a second center value corresponding to a second measured failure rate of the IC structure according to a second fabrication condition is calculated. The second center value is calculated by solving Equation 3 as described above but using the second measure failure rate and critical dimension probability distribution according to the second fabrication condition. In step 704, a second spread value is assigned to be independent according to a fabrication condition and in step 705, steps 702 and 703 are repeated to calculate a third center value and a fourth center value according to a third and fourth fabrication condition, respectively. [0050]In step 706, an optimized center value is calculated. The optimized center value may be determined by calculating a root mean square error for the first and second center values at the first spread value, and for the third and fourth center values at the second spread value. A lowest root mean square error value may identify the optimized center value. In step 707, an optimized spread value is determined. The optimized spread value may be a spread value corresponding to the optimized center value containing the lowest root mean square error value. In step 708, a process window of IC structure failure rate may be generated for each fabrication condition using the optimized center value and the optimized spread value. The process window may be generated by fitting the optimized center value as a function of a fabrication condition and applying the optimized spread value and modeled critical dimension probability distribution to calculate a predicted failure rate as a function of fabrication condition. The fabrication condition may be focus, dose, or a combined focus and dose condition. [0051]In an optimization process, a figure of merit can be represented as a cost function. The optimization process may include finding a set of parameters of the system that minimizes the cost function. The cost function can have any suitable form depending on the goal of the optimization. For 2022P00393WO 13 example, the cost function can be a root mean square error (RMSE) or a standard deviation of certain characteristics (e.g., evaluation points) of the system with respect to the intended values of these characteristics. The cost function can also be the maximum of these deviations (e.g., worst deviation). The term "evaluation points" herein should be interpreted broadly to include any characteristics of the system. Example evaluation points included in this present disclosure may include fabrication conditions, such as, for example, focus and dose parameters for a radiation beam. [0052]Reference is now made to FIGS. 8A-C,which illustrates portions of method 700 using sample data. FIG. 8Aillustrates a RMSE value 801 of a calculated center value plotted against a spread value 802. FIG. 8Amay illustrate steps 701-707 where a first RMSE value 801_1 of a calculated center value and a second RMSE value 801_2 of a calculated center value root are plotted at a first spread value 802_l and second spread value 802_l, respectively. FIG. 8Aillustrates that second RMSE value 801_is the lowest RMSE value. Therefore, a center value to corresponding to RMSE value 801_2 is the optimized center value and the corresponding spread value 802_2 is the optimized spread value according to method 700. FIGS. 8Band 8Cillustrate a fit of a modeled center value 803 as a function of focus 804 and illustrate a portion of step 708 of method 700. Specifically, FIG. 8Cillustrates an optimal fit 805 of the modeled center values at a given focus condition 804 when the optimized spread value 802_2 and the corresponding center value are selected, whereas FIG. 8Billustrates a suboptimal fit 806 as a function of focus condition 804 when an unoptimized spread value 802_l and corresponding center value are selected. FIG. 8Ctherefore illustrates the most accurate prediction of an optimized center value for a sigmoid function modeling small defect and failure rate in an IC structure as a lithographic focus condition is varied, when method 700 is performed. The sample data illustrated in FIGS. 8B-Cand the steps of method 700 in FIG. 7may also apply to parameterizing a sigmoid function for large defect predictions. [0053] FIG. 9illustrates a flow diagram of method 900, which may be a second method to parameterize a sigmoid function predicting non-failure probability and defect probability of an IC structure fabricated according to multiple fabrication conditions. The focus conditions may be a focus, dose, or combined focus and dose condition. Method 900 may be performed by a computing device (e.g., processor 403 in FIG. 4A).For method 900, the spread and center values of a sigmoid function are assumed to both be focus-dependent. Therefore, a set of dose conditions is evaluated at each focus condition to parameterize the center and spread values. As described above, method 900 may occur concurrently for a sigmoid function modeling small defects (e.g., sigmoid function 503 in FIG. 5A) and for a sigmoid function modeling large defects (e.g., sigmoid function 513 in FIG. 5D).In step 901, a first spread value for a first set of fabrication conditions is assigned. The fabrication conditions may be a set of combined focus and dose conditions where a first focus condition is constant for each fabrication condition in the set. In step 902, a first set of center values corresponding to each fabrication condition of the first set of fabrication conditions is calculated. The calculation for step 902 is like that described above (e.g., step 702 in method 700), except multiple center values are calculated using 2022P00393WO 14 Equation 3 with multiple measured failure rates and modeled critical dimension probability distributions, and the first spread value. In step 903, a second spread value is assigned for the first set of fabrication conditions. In step 904, a second set of center values corresponding to each fabrication condition of the first set of fabrication condition is calculated. In step 905, an optimized spread value for the first set of fabrication conditions is determined. The optimized spread value may be determined by calculating a RMSE value for the first and second set of calculated center values and selecting a lowest RMSE value. The optimized spread value is determined by then calculating the standard deviation of the set of calculated center values that corresponds to the lowest RMSE value. In step 906, an optimized spread value for a second set of fabrication conditions is determined. The optimized spread value for the second set of fabrication conditions is calculated by repeating steps 901-905, but instead with a second focus condition that is constant for each fabrication condition in the second set of fabrication conditions. In step 907, an optimized center value for the first and second set of fabrication conditions is determined. This may be performed by fitting the optimized spread value for the first and second set of fabrication conditions as a function of focus. The optimized center value may be calculated using the fitted spread value for each fabrication condition along with the measured failure rates and critical dimension probabilities. In step 908, a process window of IC structure failure rate according to a fabrication condition may be generated using the optimized center value and the optimized spread value. [0054]Reference is now made to FIGS. 10A-C,which illustrates portions of method 900 using sample data. FIG. 10Aillustrates a dataset of calculated center values 1001 at different fabrication conditions (e.g., focus) 1002 generated according to steps 901 and 902 in method 900. The fabrication conditions may be different combinations of at least three dose conditions for a given focus condition. The outlined box 1003 indicates a set of center values that were calculated using an assigned spread value as described above in step 902 and using Equation 3, and the fitted line 1004 indicates a fit of all calculated center values at all fabrication conditions as a function of focus. FIG. 10Billustrates a calculated optimized spread value 1005 for each focus condition 1002 and illustrates the fit 1006 of optimized spread values as a function of focus according to steps 905 and 906 of method 900. As seen in FIG. 10B,an optimized spread value for each focus condition is determined (e.g., determined by calculating standard deviation of calculated zeta values in FIG. 10Aand selecting a lowest value) and fit as a function of focus. The outlined square 1007 in FIG. 10Bcorresponds to the optimized spread value for the calculated center values in outlined box 1003 of FIG. 10A.In FIG. 10C,the optimized spread value for each focus condition 1002 is used to calculate an optimized center value 1008 for each focus condition using Equation 3 as described above. The fitted optimized center values 1009 as a function of focus exhibit a better correlation compared to fitted line 1004 illustrated in FIG. 10A. [0055]Reference is now made to Table 1, which displays results of three datasets modeled using a conventional stepwise function to model failure rate on IC structures compared to a sigmoid function (e.g., sigmoid model 1 and sigmoid model 2). The sigmoid model 1 corresponds to method 700 and the sigmoid model 2 corresponds to method 900. Two metrics may be used to quantify an improvement in 2022P00393WO 15 modeling failure rate: a correlation of an R2 value of measured to modeled failure rate, and a change (delta) in failure rate between the modeled and measured failure from a verification datapoint. A larger R2 value and a lower delta failure rate value may indicate better model performance. As seen in Table 1, embodiments of the present disclosure exhibit higher R2 values and lower delta values between modeled and measured failure rate at the verification datapoint compared to the conventional method.In dataset 1 and dataset 2, embodiments of the present disclosure provide improved modeling of large and small defects, provide improved modeling of large defects in dataset 3.

Dataset Pattern type Metric Conventional methodSigmoid model 1Sigmoid model 2Finding 1 Staggered pillarFailure rate correlationR2 0.698 0.897 0.909 Sigmoid model and improve modeling small and large defects DeltaFailure rate verification .76 0.90 0.03 2 Staggered pillarFailure rate correlation R2 0.726 0.925 0.931 Sigmoid model and improve modeling small and large defects Delta failure rate verification 4.59 0.64 0.48 3 Diagonal arrayFailure rate correlationR2 0.732 0.856 0.892 Sigmoid model and improve modeling large defects Delta failure rate verification N/A id="p-56" id="p-56"

[0056]Table 1. Summary of model accuracy metrics for conventional method and embodiments of thepresent disclosure. 2022P00393WO 16 id="p-57" id="p-57"

[0057]Reference is now made to FIGs. HA-I,which are example results illustrating how embodiments of the present disclosure provide improvements over conventional methods to model failure rate of an IC structure created according to a fabrication condition for dataset I presented in Table 1. FIGS. 11A-Cillustrate results for the conventional method, FIGS. 11D-Fillustrate results for sigmoid model 1, and FIGS. 11G-Iillustrate results for sigmoid model 2. FIGS. 11A, 11D,and 11G illustrate modeling results for a modeled lower center value 1101 plotted against a focus condition 1102. For FIGS. HDand 11G,this may be a zeta l value as described above. FIGS. 11B, 11E,and HH illustrate modeling results for a modeled upper center value 1103 plotted against a focus condition 1102. For FIGS. HEand HFthis may be a zeta 2 value as described above. The modeled center values in FIGS. HD, HE, 11G,and HHdisplay an improved fit to focus condition 1102 compared to modeled center values in FIGS. HAand HB.This indicates the sigmoid model 1 and the sigmoid model 2 can better predict a center value as a function of a fabrication condition compared to conventional methods. FIGS. HC, HF,and HIillustrate a modeled failure rate 1104 for each method plotted against a measured failure rate 1105. The fitted lines displayed in FIGS. HC, HF,and HIindicate a correlation R2 value (i.e., the R2 value is the slope of a fitted line). It is clearly observed that sigmoid model I (FIG. 11F)and sigmoid model 2 (FIG. HI)demonstrate a larger R2 value compared to the conventional model (FIG. HC).Furthermore, a delta failure rate verification point 1106, 1107, and 1108 for the conventional method, sigmoid model 1, and sigmoid model 2, respectively. Sigmoid model 1 and sigmoid model 2 exhibit delta failure rate verification points 1107 and 1108 that are much closer to the fitted lines in FIGS. HFand HIcompared to delta failure rate verification point 1106 for the conventional method (FIG. HC).Therefore, sigmoid model 1 and sigmoid model 2 both improve predictive performance of center value modeling and failure rate modeling compared to the conventional model for dataset 1. [0058]Reference is now made to FIGS. 12A-C,which are example generated failure rate process windows illustrating how embodiments of the present disclosure provide improvements over conventional methods to model failure rate of an IC structure according to fabrication condition. The process windows were generated to display a failure rate probability for a focus 1201 and dose 12condition applied to fabricate the IC structures. The gradient scale bar on the right indicates a failure rate exponent value 1203 (i.e., 10N where N is determined by the greyscale intensity in each process window). Each process window illustrates boundary lines indicating a boundary where the 10g(failure rate) is equal to failure rate exponent value 1203 (from now on referred to as N). FIG. 12A(e.g., process window for conventional method) illustrates three boundary lines, where boundary line A corresponds to an N value of -6, boundary line B corresponds to an N value of -5, and boundary line C corresponds to an N value of -4. FIG. 12B(e.g., process window for method 700) illustrates boundary line B and boundary line C. FIG. 12C(e.g., process window for method 900) illustrates boundary line A, B, and C. Boundary line C may represent a boundary condition of a cut off failure rate of 104 that is satisfactory for IC fabrication. Fabrication conditions with a predicted failure rate outside of boundary line C (e.g., 2022P00393WO 17 failure rate above 104־) may be considered unsatisfactory. FIG. 12Aindicates a greater area within boundary line A and greater variation in failure rate, where the failure rate may vary from 10-12 to 10-6. This may be because using a stepwise function may miss defects as discussed earlier and may over- predict defect-free IC formation at a given fabrication condition. Furthermore, FIG. 12Aindicates smaller distances between each boundary line A, B, and C compared to the boundary lines in FIGS. 12Band 12C.This indicates that a parameterized sigmoid function consistent with embodiments of the present disclosure may more consistently predict a failure rate of an IC structure at a fabrication condition within this region of the process window. [0059]A non-transitory computer readable medium may be provided that may store instructions for a processor of a lithographic projection apparatus (e.g., lithographic projection apparatus 100 of FIG. 1) and inspection tool (e.g., inspection tool 402 of FIG. 4A)to determine critical dimension measurements and defect data of a sample, determine lower and upper center values, determine a center value and spread value for a sigmoid function, method 700 of FIG. 7,method 900 of FIG. 9,and other executable functions relating to the predicting defect probability and failure probability a sigmoid function and parameterizing the sigmoid function. Common forms of non-transitory media include, for example, a floppy disk, a flexible disk, hard disk, solid state drive, magnetic tape, or any other magnetic data storage medium, a Compact Disc Read Only Memory (CD-ROM), any other optical data storage medium, any physical medium with patterns of holes, a Random Access Memory (RAM), a Programmable Read Only Memory (PROM), and Erasable Programmable Read Only Memory (EPROM), a FLASH-EPROM or any other flash memory, Non-Volatile Random Access Memory (NVRAM), a cache, a register, any other memory chip or cartridge, and networked versions of the same. [0060]The embodiments may further be described using the following clauses:1. A method of modeling defect probability, the method comprising:obtaining a critical dimension measurement and defect data from a sample; anddetermining a lower defect threshold limit and an upper defect threshold limit based on a sigmoid function wherein the lower defect threshold limit and the upper defect threshold limit comprise a center value and a spread value.2. The method of clause 1, wherein the sample is a focus-dose matrix wafer.3. The method of clause 1, wherein the lower defect threshold limit corresponds to smaller defects.4. The method of clause 1, wherein the upper defect threshold limit corresponds to larger defects.5. The method of clause 1, further comprising:assigning a first constant spread value;calculating a first center value according to a first fabrication condition of the sample and a second center value according to a second fabrication condition of the sample;assigning a second constant spread value;calculating a third center value according to a third fabrication condition of the sample and a fourth center value according to a fourth fabrication condition of the sample; 2022P00393WO 18 determining an optimized center value and an optimized spread value; andgenerating a process window based on the optimized center value and the optimized spread value.6. The method of clause 5, wherein determining the optimized center value is by selecting a lowest root mean square error of the first center value, the second center, the third center value, and the fourth center value.ר. The method of clause 5, wherein the fabrication condition is a lithographic patterning condition.8. The method of clause 7, wherein the fabrication condition is a focus, dose, or combined focus and dose condition.9. The method of clause 5, wherein the optimized center value is fit as a function of focus only.10. The method of clause 5, wherein the optimized spread value is assumed to be constant for all fabrication conditions.11. The method of clause 1, further comprising:assigning a first spread value according to a first set of fabrication conditions and calculating a first set of center values;assigning a second spread value according to the first set of fabrication conditions and calculating a second set of center values;calculating an optimized spread value according to the first set of fabrication conditions;assigning a first spread value according to a second set of fabrication conditions and calculating a first set of center values;assigning a second spread value according to the second set of fabrication conditions and calculating a second set of center values;calculating an optimized spread value according to the second set of fabrication conditions;fitting the optimized spread values according to the first set of fabrication conditions and second set of fabrication conditions to calculate an optimized center value; andgenerating a process window based on the optimized spread value according to the first and second set of fabrication conditions and the optimized center value.12. The method of clause 11, wherein calculating the optimized spread value according to the first set of fabrication conditions is by minimizing a root mean square error of the first set of center values and the second set of center values.13. The method of clause 11, wherein the first set of fabrication condition comprises a first constant focus and at least three different dose conditions.14. The method of clause 11, wherein the second set of fabrication conditions comprises a second constant focus and at least three different dose conditions.15. The method of clause 11, wherein the optimized spread value according to the first set of fabrication conditions and the second set of fabrication conditions are fit as a function of focus.16. The method of clause 11, wherein the optimized center value is fit as a function of focus.17. A method to parameterize non-failure probability based on a sigmoid function, the method 2022P00393WO 19 comprising:assigning a first constant spread value;calculating a first center value according to a first fabrication condition of the sample and a second center value according to a second fabrication condition of the sample;assigning a second constant spread value;calculating a third center value according to a third fabrication condition of the sample and a fourth center value according to a fourth fabrication condition of the sample;determining an optimized center value and an optimized spread value; andgenerating a process window based on the optimized center value and the optimized spread value.18. The method of clause 17, wherein determining the optimized center value is by selecting a lowest root mean square error of the first center value, the second center, the third center value, and the fourth center value.19. The method of clause 17, wherein the fabrication condition is a lithographic patterning condition.20. The method of clause 19, wherein the fabrication condition is a focus, dose, or combined focus and dose condition.21. The method of clause 17, wherein the optimized center value is fit as a function of focus only.22. The method of clause 17, wherein the optimized spread value is assumed to be constant for all fabrication conditions.23. A method to parameterize non-failure probability based on a sigmoid function, the method comprising:assigning a first spread value according to a first set of fabrication conditions and calculating a first set of center values;assigning a second spread value according to the first set of fabrication conditions and calculating a second set of center values;calculating an optimized spread value according to the first set of fabrication conditions;assigning a first spread value according to a second set of fabrication conditions and calculating a first set of center values;assigning a second spread value according to the second set of fabrication conditions and calculating a second set of center values;calculating an optimized spread value according to the second set of fabrication conditions;fitting the optimized spread values according to the first set of fabrication conditions and second set of fabrication conditions to calculate an optimized center value; andgenerating a process window based on the optimized spread value according to the first and second set of fabrication conditions and the optimized center value.24. The method of clause 23, wherein calculating the optimized spread value according to the first set of fabrication conditions is by minimizing a root mean square error of the first set of center values and the second set of center values. 2022P00393WO 20 . The method of clause 23, wherein the first set of fabrication condition comprises a first constant focus and at least three different dose conditions.26. The method of clause 23, wherein the second set of fabrication conditions comprises a second constant focus and at least three different dose conditions.27. The method of clause 23, wherein the optimized spread value according to the first set of fabrication conditions and the second set of fabrication conditions are fit as a function of focus.28. The method of clause 23, wherein the optimized center value is fit as a function of focus.29. A non-transitory computer readable medium comprising a set of instructions that is executable by one or more processors of a computing device to cause the computing device to perform a method for modeling defect probability, the method comprising:obtaining a critical dimension measurement and defect data from a sample; anddetermining a lower defect threshold limit and an upper defect threshold limit based on a sigmoid function wherein the lower defect threshold limit and the upper defect threshold limit comprise a center value and a spread value.30. The non-transitory computer readable medium of clause 29, wherein the sample is a focus-dose matrix wafer.31. The non-transitory computer readable medium of clause 29, wherein the lower defect threshold limit corresponds to smaller defects.32. The non-transitory computer readable medium of clause 29, wherein the upper defect threshold limit corresponds to larger defects.33. The non-transitory computer readable medium of clause 29, further comprising:assigning a first constant spread value;calculating a first center value according to a first fabrication condition of the sample and a second center value according to a second fabrication condition of the sample;assigning a second constant spread value;calculating a third center value according to a third fabrication condition of the sample and a fourth center value according to a fourth fabrication condition of the sample;determining an optimized center value and an optimized spread value; andgenerating a process window based on the optimized center value and the optimized spread value.34. The non-transitory computer readable medium of clause 33, wherein determining the optimized center value is by selecting a lowest root mean square error of the first center value, the second center, the third center value, and the fourth center value.35. The non-transitory computer readable medium of clause 33, wherein the fabrication condition is a lithographic patterning condition.36. The non-transitory computer readable medium of clause 35, wherein the fabrication condition is a focus, dose, or combined focus and dose condition.37. The non-transitory computer readable medium of clause 33, wherein the optimized center value is 2022P00393WO 21 fit as a function of focus only.38. The non-transitory computer readable medium of clause 33, wherein the optimized spread value is assumed to be constant for all fabrication conditions.39. The non-transitory computer readable medium of clause 29, further comprising:assigning a first spread value according to a first set of fabrication conditions and calculating a first set of center values;assigning a second spread value according to the first set of fabrication conditions and calculating a second set of center values;calculating an optimized spread value according to the first set of fabrication conditions;assigning a first spread value according to a second set of fabrication conditions and calculating a first set of center values;assigning a second spread value according to the second set of fabrication conditions and calculating a second set of center values;calculating an optimized spread value according to the second set of fabrication conditions;fitting the optimized spread values according to the first set of fabrication conditions and second set of fabrication conditions to calculate an optimized center value; andgenerating a process window based on the optimized spread value according to the first and second set of fabrication conditions and the optimized center value.40. The non-transitory computer readable medium of clause 39, wherein calculating the optimized spread value according to the first set of fabrication conditions is by minimizing a root mean square error of the first set of center values and the second set of center values.41. The non-transitory computer readable medium of clause 39, wherein the first set of fabrication condition comprises a first constant focus and at least three different dose conditions.42. The non-transitory computer readable medium of clause 39, wherein the second set of fabrication conditions comprises a second constant focus and at least three different dose conditions.43. The non-transitory computer readable medium of clause 39, wherein the optimized spread value according to the first set of fabrication conditions and the second set of fabrication conditions are fit as a function of focus.44. The non-transitory computer readable medium of clause 39, wherein the optimized center value is fit as a function of focus.45. A non-transitory computer readable medium comprising a set of instructions that is executable by one or more processors of a computing device to cause the computing device to perform a method for parameterizing non-failure probability based on a sigmoid function, the method comprising: assigning a first constant spread value;calculating a first center value according to a first fabrication condition of the sample and a second center value according to a second fabrication condition of the sample;assigning a second constant spread value; 2022P00393WO 22 calculating a third center value according to a third fabrication condition of the sample and a fourth center value according to a fourth fabrication condition of the sample;determining an optimized center value and an optimized spread value; andgenerating a process window based on the optimized center value and the optimized spread value.46. The non-transitory computer readable medium of clause 45, wherein determining the optimized center value is by selecting a lowest root mean square error of the first center value, the second center, the third center value, and the fourth center value.47. The non-transitory computer readable medium of clause 45, wherein the fabrication condition is a lithographic patterning condition.48. The non-transitory computer readable medium of clause 47, wherein the fabrication condition is a focus, dose, or combined focus and dose condition.49. The non-transitory computer readable medium of clause 45, wherein the optimized center value is fit as a function of focus only.50. The non-transitory computer readable medium of clause 45, wherein the optimized spread value is assumed to be constant for all fabrication conditions.51. A non-transitory computer readable medium comprising a set of instructions that is executable by one or more processors of a computing device to cause the computing device to perform a method for parameterizing non-failure probability based on a sigmoid function, the method comprising: assigning a first spread value according to a first set of fabrication conditions and calculating a first set of center values;assigning a second spread value according to the first set of fabrication conditions and calculating a second set of center values;calculating an optimized spread value according to the first set of fabrication conditions;assigning a first spread value according to a second set of fabrication conditions and calculating a first set of center values;assigning a second spread value according to the second set of fabrication conditions and calculating a second set of center values;calculating an optimized spread value according to the second set of fabrication conditions;fitting the optimized spread values according to the first set of fabrication conditions and second set of fabrication conditions to calculate an optimized center value; andgenerating a process window based on the optimized spread value according to the first and second set of fabrication conditions and the optimized center value.52. The non-transitory computer readable medium of clause 51, wherein calculating the optimized spread value according to the first set of fabrication conditions is by minimizing a root mean square error of the first set of center values and the second set of center values.53. The non-transitory computer readable medium of clauses 51, wherein the first set of fabrication condition comprises a first constant focus and at least three different dose conditions. 2022P00393WO 23 54. The non-transitory computer readable medium of clause 51, wherein the second set of fabrication conditions comprises a second constant focus and at least three different dose conditions.55. The non-transitory computer readable medium of clause 51, wherein the optimized spread value according to the first set of fabrication conditions and the second set of fabrication conditions are fit as a function of focus.56. The non-transitory computer readable medium of clause 51, wherein the optimized center value is fit as a function of focus. [0061]It will be appreciated that the embodiments of the present disclosure are not limited to the exact construction that has been described above and illustrated in the accompanying drawings, and that various modifications and changes may be made without departing from the scope thereof. The present disclosure has been described in connection with various embodiments, other embodiments of the invention will be apparent to those skilled in the art from consideration of the specification and practice of the invention disclosed herein. It is intended that the specification and examples be considered as exemplary only, with a true scope and spirit of the invention being indicated by the following claims.

Claims (15)

2022P00393WO 24 CLAIMS

1. A non-transitory computer readable medium comprising a set of instructions that is executable by one or more processors of a computing device to cause the computing device to perform a method for modeling defect probability, the method comprising:obtaining a critical dimension measurement and defect data from a sample; anddetermining a lower defect threshold limit and an upper defect threshold limit based on a sigmoid function wherein the lower defect threshold limit and the upper defect threshold limit comprise a center value and a spread value.

2. The non-transitory computer readable medium of claim 1, wherein the sample is a focus-dose matrix wafer.

3. The non-transitory computer readable medium of claim 1, wherein the lower defect threshold limit corresponds to smaller defects.

4. The non-transitory computer readable medium of claim 1, wherein the upper defect threshold limit corresponds to larger defects.

5. The non-transitory computer readable medium of claim 1, further comprising: assigning a first constant spread value;calculating a first center value according to a first fabrication condition of the sample and a second center value according to a second fabrication condition of the sample;assigning a second constant spread value;calculating a third center value according to a third fabrication condition of the sample and a fourth center value according to a fourth fabrication condition of the sample;determining an optimized center value and an optimized spread value; andgenerating a process window based on the optimized center value and the optimized spread value.

6. The non-transitory computer readable medium of claim 5, wherein determining the optimized center value is by selecting a lowest root mean square error of the first center value, the second center, the third center value, and the fourth center value. ר.

7. The non-transitory computer readable medium of claim 5, wherein the fabrication condition is a lithographic patterning condition. 2022P00393WO 25

8. The non-transitory computer readable medium of claim 7, wherein the fabrication condition is a focus, dose, or combined focus and dose condition.

9. The non-transitory computer readable medium of claim 5, wherein the optimized center value is fit as a function of focus only.

10. The non-transitory computer readable medium of claim 5, wherein the optimized spread value is assumed to be constant for all fabrication conditions.

11. The non-transitory computer readable medium of claim 1, further comprising: assigning a first spread value according to a first set of fabrication conditions and calculating a first set of center values;assigning a second spread value according to the first set of fabrication conditions and calculating a second set of center values;calculating an optimized spread value according to the first set of fabrication conditions;assigning a first spread value according to a second set of fabrication conditions and calculating a first set of center values;assigning a second spread value according to the second set of fabrication conditions and calculating a second set of center values;calculating an optimized spread value according to the second set of fabrication conditions;fitting the optimized spread values according to the first set of fabrication conditions and second set of fabrication conditions to calculate an optimized center value; andgenerating a process window based on the optimized spread value according to the first and second set of fabrication conditions and the optimized center value.

12. The non-transitory computer readable medium of claim 11, wherein calculating the optimized spread value according to the first set of fabrication conditions is by minimizing a root mean square error of the first set of center values and the second set of center values.

13. The non-transitory computer readable medium of claim 11, wherein the first set of fabrication condition comprises a first constant focus and at least three different dose conditions.

14. The non-transitory computer readable medium of claim 11, wherein the second set of fabrication conditions comprises a second constant focus and at least three different dose conditions.

15. The non-transitory computer readable medium of claim 11, wherein the optimized spreadvalue according to the first set of fabrication conditions and the second set of fabrication conditions 2022P00393WO 26 are fit as a function of focus.