Modeling defect probabilities from expected critical dimensions to improve the accuracy of failure rate predictions

Abstract

A method for modeling defect probability from critical dimensions of a sample to improve failure rate prediction, and more particularly, a method for modeling defect probability using a sigmoid function to improve failure prediction, is disclosed. The sigmoid function can be used to consider the range of critical dimensions that can accurately predict failure and defect formation in the sample.

Description

[Technical Field]

[0001] CROSS-REFERENCE TO RELATED APPLICATIONS

[0001] This application claims priority to U.S. Patent Application No. 63 / 441,761, filed January 27, 2023, which is incorporated herein by reference in its entirety.

[0002]

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

[0003] A lithography apparatus is a machine that applies a desired pattern to a target portion of a substrate. Lithography apparatuses can be used, for example, in the manufacture of integrated circuits (ICs). An IC chip in a smartphone can be as small as a human thumbnail and contain over two billion transistors. Fabricating an IC is a complex and time-consuming process involving hundreds of individual steps, involving different layers of circuit components. An error in even one step can potentially cause problems in the final IC, resulting in device failure. Therefore, in the IC manufacturing process, unfinished or completed circuit components are inspected to ensure that they are manufactured according to design and are free of defects. Inspection systems that utilize optical microscopes or charged particle (e.g., electron) beam microscopes, such as scanning electron microscopes (SEMs), can be employed. As the physical size of IC components continues to shrink, the accuracy and yield of IC inspection become increasingly important. The presence of defects can affect high process yields and high wafer throughput, especially if operator intervention is required to identify the defects. Therefore, it is desirable to model lithography conditions that can accurately predict the formation of defects in ICs before fabrication occurs. Summary of the Invention

[0004]

[0004] The embodiments provided herein disclose a method for modeling defect probability from critical dimensions of samples to improve failure rate prediction, and more particularly, a method for parameterizing a sigmoid function to improve failure prediction.

[0005] Some embodiments provide a non-transitory computer-readable medium including a set of instructions 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 including obtaining critical dimension measurements and defect data from a sample, and determining a lower defect threshold limit and an upper defect threshold limit based on a sigmoid function, the lower defect threshold limit and the upper defect threshold limit including a center value and a scatter value.

[0006]

[0006] Other advantages of the present invention will become apparent from the following description, taken in conjunction with the accompanying drawings, in which certain embodiments of the invention are shown by way of illustration and example.

[0007]

[0007] The above and other aspects of the present disclosure will become more apparent from the following description of exemplary embodiments taken in conjunction with the accompanying drawings. [Brief explanation of the drawings]

[0008] [Figure 1] FIG. 1 is a schematic diagram illustrating an exemplary lithographic projection assembly for fabricating ICs consistent with embodiments of the present disclosure. [Figure 2]

[0009] 1 is an exemplary conventional method for predicting the non-defective probability of an IC structure. [Figure 3]

[0010] 1 is an exemplary flowchart for predicting non-fault and defect probabilities of an IC structure using a sigmoid function, consistent with embodiments of the present disclosure. [Figure 4A]

[0011] FIG. 1 is a schematic block diagram illustrating a throughput of generating data consistent with embodiments of the present disclosure. [Figure 4B]

[0012] 1 is an exemplary focus-dose matrix illustrating a two-dimensional layout of lithography conditions for fabricating an IC structure consistent with embodiments of the present disclosure. [Figure 5A]

[0013] FIG. 1 is a schematic diagram illustrating an example sigmoid function for predicting non-defect probability and failure rate of an IC structure consistent with embodiments of the present disclosure. [Figure 5B] FIG. 1 is a schematic diagram illustrating an example sigmoid function for predicting non-defect probability and failure rate of an IC structure consistent with embodiments of the present disclosure. [Figure 5C] FIG. 1 is a schematic diagram illustrating an example sigmoid function for predicting non-defect probability and failure rate of an IC structure consistent with embodiments of the present disclosure. [Figure 5D] FIG. 1 is a schematic diagram illustrating an example sigmoid function for predicting non-defect probability and failure rate of an IC structure consistent with embodiments of the present disclosure. [Figure 5E] FIG. 1 is a schematic diagram illustrating an example sigmoid function for predicting non-defect probability and failure rate of an IC structure consistent with embodiments of the present disclosure. [Figure 6A]

[0014] 1 is an exemplary slope plot of calculated failure rate as a function of sigmoid function parameters consistent with embodiments of the present disclosure. [Figure 6B] 1 is an exemplary slope plot of calculated failure rates as a function of sigmoid function parameters consistent with embodiments of the present disclosure. [Figure 7]

[0015] 1 is a flowchart illustrating a first exemplary process for parameterizing variables of a sigmoid function to accurately predict the probability of non-defect and the probability of failure of an IC structure, consistent with embodiments of the present disclosure. [Figure 8A]

[0016] 7 is an exemplary data plot illustrating steps of a method 700 consistent with embodiments of the present disclosure. [Figure 8B]7 is an exemplary data plot illustrating steps of a method 700 consistent with embodiments of the present disclosure. [Figure 8C] 7 is an exemplary data plot illustrating steps of a method 700 consistent with embodiments of the present disclosure. [Figure 9]

[0017] 10 is a flowchart illustrating a second exemplary process for parameterizing variables of a sigmoid function to accurately predict the probability of non-defect and the probability of failure of an IC structure, consistent with embodiments of the present disclosure. [Figure 10A]

[0018] 9 is an exemplary data plot illustrating steps of a method 900 consistent with embodiments of the present disclosure. [Figure 10B] 9 is an exemplary data plot illustrating steps of a method 900 consistent with embodiments of the present disclosure. [Figure 10C] 9 is an exemplary data plot illustrating steps of a method 900 consistent with embodiments of the present disclosure. [Figure 11A]

[0019] 10A-10C are exemplary results illustrating how embodiments of the present disclosure provide improvements over conventional methods for modeling failure rates of IC structures. [Figure 11B] 10 is an exemplary result illustrating how embodiments of the present disclosure provide improvements over conventional methods for modeling failure rates of IC structures. [Figure 11C] 10 is an exemplary result illustrating how embodiments of the present disclosure provide improvements over conventional methods for modeling failure rates of IC structures. [Figure 11D] 10 is an exemplary result illustrating how embodiments of the present disclosure provide improvements over conventional methods for modeling failure rates of IC structures. [Figure 11E] 10 is an exemplary result illustrating how embodiments of the present disclosure provide improvements over conventional methods for modeling failure rates of IC structures. [Figure 11F]10 is an exemplary result illustrating how embodiments of the present disclosure provide improvements over conventional methods for modeling failure rates of IC structures. [Figure 11G] 10 is an exemplary result illustrating how embodiments of the present disclosure provide improvements over conventional methods for modeling failure rates of IC structures. [Figure 11H] 10 is an exemplary result illustrating how embodiments of the present disclosure provide improvements over conventional methods for modeling failure rates of IC structures. [Figure 11I] 10 is an exemplary result illustrating how embodiments of the present disclosure provide improvements over conventional methods for modeling failure rates of IC structures. [Figure 12A]

[0020] 1 is a process window of exemplary generated failure rates illustrating how embodiments of the present disclosure provide improvements over conventional methods for modeling failure rates of IC structures according to fabrication conditions. [Figure 12B] 1 is an exemplary generated failure rate process window illustrating how embodiments of the present disclosure provide improvements over conventional methods for modeling failure rates of IC structures according to fabrication conditions. [Figure 12C] 1 is an exemplary generated failure rate process window illustrating how embodiments of the present disclosure provide improvements over conventional methods for modeling failure rates of IC structures according to fabrication conditions. DETAILED DESCRIPTION OF THE INVENTION

[0009]

[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, unless otherwise stated, like numbers in different drawings represent the same or similar elements. The implementations described in the following description of exemplary embodiments do not represent all implementations consistent with the present invention. Instead, they are merely examples of apparatus and methods consistent with aspects related to the present invention as recited in the appended claims.

[0010]

[0022] Increasing the computing power of electronic devices while shrinking their physical size can be achieved by significantly increasing the packing density of circuit components such as transistors, capacitors, and diodes on IC chips. For example, a smartphone IC chip can be about the size of a thumbnail yet contain over 2 billion transistors, each less than 1 / 1000 the size of a human hair. It is not surprising, therefore, that semiconductor IC manufacturing is a complex and time-consuming process involving hundreds of individual steps. An error in even one step can dramatically affect the functionality of the final product. Even a single "killer defect" can cause device failure. The goal of a manufacturing process is to improve the overall yield of the process. For example, to achieve a 75% yield for a 50-step process, each individual step must have a yield greater than 99.4%; if each individual step has a 95% yield, the overall process yield drops to 7%.

[0011]

[0023] In IC chip manufacturing facilities, high process yields are desirable, but maintaining high wafer throughput, defined as the number of wafers processed per hour, is also essential. The presence of defects can impact high process yields and high wafer throughput, especially if operator intervention is required to identify the defects. Therefore, to maintain high yields and low costs, it is essential to detect and identify micro- and nano-sized defects at high throughput using inspection tools (e.g., charged particle beam inspection tools). Inspection of wafers using electron beam inspection tools can generate images of the wafer to measure IC structure dimensions. The measured dimensions can be compared to defect-free reference structures to determine the presence of defects in the imaged structures. However, inspecting ICs for defect detection is often a time-consuming process. Instead of further improving IC inspection methods, it may be desirable to prevent defects from occurring during the fabrication stage. Therefore, it is desirable to improve the accuracy of IC failure rate prediction modeling during fabrication and to generate more accurate process windows for fabrication conditions.

[0012]

[0024] ICs can be manufactured using lithography, a fabrication process that involves creating intricate circuit patterns depicted on a mask that is deposited on a substrate. Lithography can be performed by a lithography apparatus, which is a machine that applies a radiation source (e.g., light or x-rays) onto a target portion of a substrate to form a desired pattern. The target portion of the substrate can be covered with a pattern device (e.g., a mask), which can be removed or developed after exposure to the radiation source. This process of transferring a desired pattern onto a substrate is called a patterning process. A patterning process can include a patterning step that transfers the pattern from the pattern device (e.g., mask) to the substrate. There can also be one or more associated pattern processing steps, such as developing a mask using a developer, baking the substrate using a bake tool, etching the pattern onto the substrate using an etcher, or other physical processing steps involved in creating the pattern on the substrate. Variations in experimental parameters (e.g., stochastic variations, errors, or noise due to inspection tools or pattern processing tools) can potentially limit lithography practices or process yields for high volume manufacturing (HVM) of ICs and introduce defects into IC structures.

[0013]

[0025] In the fabrication of ICs using a lithography apparatus, many lithographic patterning steps are typically performed to form functional features in successive layers on a substrate. Therefore, a key aspect of a lithography apparatus's performance is its ability to correctly and precisely align an applied pattern with respect to features located in previous layers. To this end, the substrate is provided with one or more sets of alignment marks. Each mark is a structure whose position can be subsequently measured, for example, using an electron beam inspection tool. Defects can occur when an applied pattern structure or pattern layer is misaligned with respect to a reference mark or when fabrication conditions are suboptimal. Reference marks or layouts define desired structures, structure dimensions, and distances between IC structures (e.g., gates, capacitors, etc.) or interconnects. This can ensure that IC devices or lines do not interact with each other in undesirable ways. The structural limitations imposed by the reference layout are typically referred to as critical dimensions. A critical dimension of a circuit can be defined as the minimum width of a line or hole or the minimum spacing between two lines or two holes. Therefore, critical dimensions determine the overall size and packing density of a designed IC. The goal of IC fabrication is to faithfully reproduce the original IC design on a substrate. During fabrication, if an error occurs such that the created IC design pattern does not match the reference design, this can result in a defective IC structure, rendering the IC inoperable.

[0014]

[0026] As discussed above, high throughput for IC fabrication with a small number of structural defects is desirable. Therefore, the ability to accurately predict the failure rate of IC structures according to lithographic fabrication conditions is desirable. Conventional methods use measured critical dimensions of ICs formed under various lithographic conditions as input data to model the probability of failure rate of the formed structures. The failure rate probability is related to the critical dimensions of the formed structures from various lithographic fabrication or patterning conditions (e.g., focus and dose, as further described below) that result in small defects (e.g., missing structures on the IC) or large defects (e.g., bridge structures on the IC). Other defects may be structures on the IC that have dimensions significantly different from the reference structure. To model critical dimensions as a function of lithographic fabrication conditions for defect prediction, conventional systems assume a certain critical dimension limit, which is the critical dimension above which a failure may be more likely to occur in the IC. For small defects, critical dimension values ​​below this limit may have a large failure probability. For large defects, critical dimensions above this limit may have a large failure probability. However, assuming constant values ​​for the critical dimension limits can result in inaccuracies in the model. 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 a defect. Therefore, conventional methods for predicting IC failure and defect probabilities due to IC lithography fabrication may not be sufficiently accurate.

[0015]

[0027] Embodiments of the present disclosure provide a method for predicting defects and failures according to IC lithography fabrication conditions using a sigmoid function. Some embodiments of the present disclosure may provide a wider range for critical dimension limits, thus providing a more robust prediction method. Furthermore, some embodiments of the present disclosure may increase IC manufacturing throughput and the reliability of predicting lithography fabrication conditions for producing defect-free ICs. Some embodiments of the present disclosure may also provide a method for parameterizing the sigmoid function to more accurately determine the lithography fabrication conditions necessary to minimize IC defects and failures. 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, thereby further increasing throughput.

[0016]

[0028] Relative dimensions of components in the drawings may be exaggerated for clarity. Within the following description of the drawings, the same or similar reference numbers refer to the same or similar components or entities, and only differences with respect to individual embodiments are described. As used herein, unless otherwise stated, the term "or" encompasses all possible combinations unless infeasible. For example, if it is stated that a database may include A or B, the database may include A or B, or A and B, unless otherwise stated or infeasible. As a second example, if it is stated that a database may include A, B, or C, the database may include A, or B, or C, or A and B, or A and C, or B and C, or A, B, and C, unless otherwise stated or infeasible.

[0017]

[0029] Reference is now made to Figure 1, which is a schematic diagram of 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 radiation source, including an extreme ultraviolet (EUV) source, that emits a radiation beam 108. Illumination optics may include illumination optical components 102 and 103 that shape the radiation from source 101, a patterning device 104, and transmission optics 105 that projects an image of the patterning device pattern onto a substrate 106. An adjustable filter or aperture 107 in a pupil plane of the projection optics may limit the range of beam angles that impinge on substrate 106; the maximum possible angle is determined by the numerical aperture of the projection optics, NA=sin(θ max ) The area or region on the substrate 106 that is impinged by the radiation beam 108 is called the target portion 109, where the radiation beam impinges on a top layer or mask (not shown).

[0018]

[0030] Illumination optical components 102 and 103 may direct and shape radiation beam 108 onto substrate 106 via patterning device 104, and may include any optical component capable of modifying the wavefront of radiation beam 108. A resist layer on substrate 106 may be exposed, and the radiation intensity distribution (i.e., an aerial image) of substrate 106 may be transferred into the resist layer. The optical properties of the lithographic projection apparatus (e.g., the properties of the radiation source, patterning device, and projection optics) dictate this process. The resist layer may be removed, after which the pattern applied from radiation beam 108 may be applied to the substrate as described above.

[0019]

[0031] Although ICs may be referenced in this disclosure, it should be understood that the disclosure may be applicable to other possible applications or designs. For example, the disclosure may be applied to integrated optical systems, magnetic domain memories, liquid crystal display panels, thin film magnetic heads, and other nanoscale structures. It should further be understood that the terms "reticle," "wafer," or "die" may be used interchangeably with the terms "mask," "substrate," or "sample," and "target portion," respectively.

[0020]

[0032] Referring now to FIG. 2, FIG. 2 illustrates an exemplary conventional method for predicting the non-failure probability of an IC structure according to a critical dimension distribution. It should be understood that FIG. 2 illustrates an exemplary prediction based on lithography fabrication conditions. The non-failure probability and critical dimension distribution in FIG. 2 may vary depending on the fabrication conditions applied to the sample. The lithography fabrication conditions may be focus and dose conditions, which are further described below. FIG. 2 illustrates an exemplary model 210 of the non-failure probability 201 of an IC structure modeled using a conventional stepwise function 203 method, plotted against the critical dimension 202. The critical dimension 202 where the non-failure probability 201 rises sharply from a low value to a high value is the critical dimension limit 204. The conventional method for modeling the non-failure probability 201 of an IC using a stepwise function assumes that the IC structure may contain voids or bridges or is designed as needed. A critical dimension that falls below the critical dimension limit 204 may be more likely to result in a failed structure on the IC. Thus, critical dimension limit 204 may represent an inflection point in predicting failure of an IC structure, being the critical dimension value at which failure may occur when fabricating an IC under certain fabrication conditions. Reference is now made to an exemplary model 220 of an IC's critical dimension probability 205 plotted against critical dimension 202. Critical dimension probability 205 may be modeled using a Gaussian function or any other method for modeling a probability distribution. To determine the non-defect probability of an IC structure, stepwise function 203, which models non-failure probability 201 of model 210, is multiplied by a polynomial function, which models critical dimension probability 205 of model 220. Referring now to model 230, the product of model 210 and model 220 equals non-defect probability 206, which may predict minor defects in an IC structure according to lithographic fabrication parameters. As shown in model 230, the transition of non-defect probability 206 at critical dimension limit 204 is modeled as a vertical line.Thus, the conventional method represents a binary method of modeling defect formation, where an IC structure having a critical dimension 202 less than the critical dimension limit 204 is likely to contain a defect and fail. While Figure 2 illustrates modeling the non-defect probability for small defects in an IC structure, it should be understood that it is also applicable to modeling and predicting large defects in an IC structure. For large defects, critical dimension values ​​above the critical dimension limit may be more likely to result in failure for the IC structure.

[0021]

[0033] Conventional methods may model the non-failure and defect probabilities of IC structures with excessively steep slopes. This may overlook critical dimensions of IC structures that are above or below the critical dimension limit, where defects may or may not occur. For example, an IC structure with a critical dimension below the critical dimension limit 204 in model 230 may not contain defects when fabricated under certain lithography conditions and thus may exhibit a non-defect probability 206 greater than zero. However, conventional models predict IC structures with essentially zero non-defect probability (i.e., the IC structure will contain defects), thus inaccurately predicting IC structure formation under those lithography fabrication conditions. Thus, conventional methods may unnecessarily limit IC packing density. Similarly, IC structures with critical dimensions above the critical dimension limit 204 in model 230 may contain defects, while conventional models may predict defect-free IC structures. This may result in inoperable ICs that may not be identified until later processing, resulting in reduced process yield and wafer throughput.

[0022]

[0034] Referring now to FIG. 3, FIG. 3 is an exemplary flowchart for non-fault and defect modeling of IC structures 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 obtained. Step 301 may include generating one or more IC structures on a sample under lithographic fabrication conditions, inspecting the one or more IC structures using an inspection tool, measuring critical dimensions of the one or more formed IC structures, and measuring failure rates of the one or more formed IC structures at the lithographic fabrication conditions, and then modeling a critical dimension probability distribution. It should be understood that step 301 may be performed for multiple lithographic fabrication conditions, with one or more IC structures formed according to multiple lithographic fabrication conditions and critical dimensions, and a failure rate and modeled critical dimension probability distribution determined for each fabrication condition. An inspection tool (e.g., an optical microscope or a charged particle microscope) may be used to generate images of the one or more IC structures. The failure rate of an IC structure for a critical dimension may be measured, for example, in parts per million (ppm) or parts per billion (ppb) of features of the IC structure. The features of the IC structure may be identified by inspecting a generated image of the formed IC structure using a processor. The failure rate may be measured by analyzing features (e.g., pixels or regions of an IC structure pattern) of the corresponding image of the formed IC structure.

[0023]

[0035] In step 302, critical dimension limits for a fabrication condition are determined using a sigmoid function model. This may include determining critical dimension limits for small and large defects in the IC structure. In step 303, a failure rate for the IC structure at the fabrication condition is calculated. The calculated failure rate depends on the sigmoid function parameters and the measured failure rate and critical dimension values ​​of the IC structure. In step 304, each sigmoid function that models the critical dimension limits according to the fabrication condition is parameterized to determine critical dimension limit values ​​for each fabrication condition used to produce one or more IC structures. In step 305, a process window is generated to accurately predict the success of forming defect-free IC structures according to all fabrication conditions. Each step is described in more detail below.

[0024]

[0036] Reference is now made to Figure 4A, which is an exemplary block diagram for generating input data consistent with embodiments of the present disclosure. The input data may be generated using two steps as shown in Figure 4A. A lithographic projection apparatus 401 (such as lithographic projection apparatus 100 of Figure 1) may be used to generate a focus-dose matrix on the surface of a sample using multiple focus and dose conditions for a radiation source (such as radiation source 101 of Figure 1). As shown in Figure 4B, a focus-dose matrix 410 may be formed on a target portion 420 of a substrate (such as target portion 109 of Figure 1). Areas of the substrate within the grid of focus-dose matrix 410 may be exposed to the radiation source at varying focus 430 and dose 440. Focus 430 is a measurement of the focal point of the radiation beam (such as radiation beam 108 in FIG. 1) on the surface of the sample, and dose 440 is a measurement of the energy of the radiation beam (such as radiation beam 108 in FIG. 1) per area of ​​the sample (e.g., mJ / cm 2). In this manner, each area delineated by the grid of focus-dose matrix 410 corresponds to a two-dimensional layout of a particular focus and dose of the radiation beam that applies a pattern on the sample. Focus-dose matrix 410 may be a grid layout. Each area within the grid of focus-dose matrix 410 may include the same IC structure or pattern, or each area with the grid may include a different IC structure or pattern. Returning to FIG. 4A , a processor 403 having a memory may be communicatively connected to lithographic projection apparatus 401 to store focus and dose conditions of the radiation beam that correspond to each grid area of ​​the focus-dose matrix. Inspection tool 402 (e.g., an optical microscope or a charged particle beam microscope) may be used to measure critical dimensions of structures formed in each grid area of ​​the focus-dose matrix (410 in FIG. 4B) 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 a lithographic fabrication process. A processor 403 having a memory may be communicatively connected to the electronic inspection tool 402 for storing the measured critical dimension values.

[0025]

[0037] 5A-5E, which illustrate examples of sigmoid functions used to predict the defect-free probability of an IC structure formed according to fabrication conditions consistent with embodiments of the present disclosure. The fabrication conditions may be lithographic focus conditions, dose conditions, or combined focus and dose conditions, and may correspond to the grid areas of the focus-dose matrix described above. As shown in FIG. 5A, the defect-free probability 501 of an IC structure may be modeled using a sigmoid function 503 as a function of a critical dimension 502. A dashed line 504 passing through the center of the sigmoid function 503 represents the critical dimension center value 504 and may be referred to as Zeta 1 (ζ1). Zeta 1 may also be referred to as the defect threshold lower limit. The horizontal distance from Zeta 1 to the horizontal portion of the sigmoid function 503 may represent the variability 505 of Zeta 1 and may be referred to as Gamma 1 (γ1). Gamma 1 represents the variability of the critical dimension center value 504 and may account for the error in predicting the non-failure probability. Gamma 1 may also be referred to as the scatter value. Thus, the critical dimension limit value of an IC structure fabricated at a certain focus and dose condition may be predicted using a sigmoid function to model the non-failure probability, which may be the critical dimension value within the region of variability 505 and may depend on Zeta 1 and Gamma 1. The sigmoid function 503 describing the non-failure probability for a small critical dimension is

number

[0026]

[0038] In Equation 1, S(x) represents the non-failure probability of an IC structure, x represents the critical dimension of the IC structure, and the IC structure is formed according to the fabrication conditions. The measured non-failure probability of the IC structure within the grid area of ​​the focus-dose matrix is ​​matched with the measured critical dimension values ​​of the IC structure to generate a sigmoid function 303 (step 301 in FIG. 3). ζ1 is the zeta 1 value representing the center value of the sigmoid function 303, and γ1 is the gamma 1 value representing the variability of the zeta 1 value.

[0027]

[0039] FIG. 5B shows the critical dimension probability 506 of an IC structure as a function of the critical dimension 502 due to fabrication conditions. The critical dimension probability 506 may be modeled, for example, by applying a Gaussian distribution or any other method for modeling a probability distribution to the measured critical dimension values ​​of the IC structure due to fabrication conditions. The product of a sigmoid function 503 modeling the non-failure probability 501 and a Gaussian function modeling the critical dimension probability 506 may yield the non-defect probability 507 (solid line) of the IC structure shown in FIG. 5C. For comparison, FIG. 5C includes the non-defect probability 508 (dashed line) calculated using a stepwise function according to a conventional method. Instead of the dramatic change in non-defect probability predicted by conventional methods at the critical dimension limit, embodiments of the present disclosure provide a wider range of defect predictions. For example, to the left of the horizontal portion of non-defect probability 508, the non-defect probability 507 of the present disclosure predicts that a defect-free IC structure may be formed depending on the fabrication conditions, while the conventional method predicts that a defect-free IC structure will be formed. Conversely, the non-defect probability 507 may predict a lower probability of small defects in the formed IC structure compared to the non-defect probability 508 of the conventional method, thereby accounting for predictions that would be missed using the conventional method. Thus, FIG. 5C illustrates a sigmoid function that may result in a wider range of predictions of small defects in IC structures compared to the conventional method using a stepwise function, as described above. Furthermore, embodiments of the present disclosure may predict defects in critical dimensions that are above or below the critical dimension limit more accurately than predicted by the conventional method. Thus, the sigmoid function may result in more robust defect modeling and IC structure failure prediction.

[0028]

[0040] 5A, 5B, and 5C illustrate the non-failure and non-defect probabilities of an IC structure for small critical dimension or missing defects. It should be understood that a sigmoid function can be used to model the non-failure and non-defect probabilities of an IC structure for large critical dimension or bridging defects, as shown in FIG. 5D. As described above, the sigmoid function 513 can be generated by the measured non-failure probabilities and measured critical dimensions of an IC structure formed according to fabrication conditions. The sigmoid function 513 can have a center value 514, which can be referred to as zeta-2 (ζ2). As described above, zeta-2 can be referred to as the upper defect threshold. Furthermore, the horizontal distance between the center value 514 and the horizontal portion of the sigmoid function 513 is the variability 515 of the center value 514. The variability 515 can also be referred to as gamma-2 (γ2) or a scatter value. Thus, the critical dimension limit value for large defects in IC structures fabricated according to the fabrication conditions may be determined using a sigmoid function 513 to model the non-failure probability 501, which may be a critical dimension value within a region of variability 515 and may depend on zeta-2 and gamma-2. The sigmoid function 513 describing the non-failure probability for large critical dimensions is:

number

[0029]

[0041] In Equation 2, S(x) represents the non-failure probability of a major defect in an IC structure, x represents a critical dimension of the IC structure, and the IC structure is formed according to fabrication conditions. The measured non-failure probability of the IC structure within the grid area of ​​the focus-dose matrix is ​​matched with the measured critical dimension values ​​of the IC structure to generate a sigmoid function 513 (step 301 of FIG. 3). ζ2 is the zeta-2 value that represents the center value of the sigmoid function 513, and γ2 is the gamma-2 value that represents the variability or spread of the zeta-2 values.

[0030]

[0042] FIG. 5E shows the non-defect probability 517 (solid line) for predicting major defects in an IC structure, which may be calculated as described above. FIG. 5E also shows the non-defect probability 518 (dashed line) when a conventional method is applied to predict major defects. As described above, embodiments of the present disclosure may predict defect-free IC structures with critical dimensions above the large critical dimension limit predicted by the conventional non-defect probability 518, while the conventional method predicts major defects. Furthermore, embodiments of the present disclosure may predict a lower non-defect probability 517 for IC structures that are free of defects in critical dimensions below the critical dimension limit predicted by the conventional method. As described above, the use of a sigmoid function may accurately predict the presence of IC structure defects below the critical dimension limit and the presence of defect-free IC structures above the critical dimension limit.

[0031]

[0043] It should be understood that other forms of the sigmoid function may be used to model the non-failure probability of the critical dimension limits other than those described in Equations 1 and 2. Other sigmoid functions may be used, including, but not limited to, forms of the hyperbolic tangent function or the arctangent function.

[0032]

[0044] Based on the measured critical dimension values ​​and the measured non-failure probability of an IC structure fabricated under given fabrication conditions, a failure rate can be estimated. The failure rate (FR) for either small or large critical dimension defects is: FR=1-∫S(x)×PDF Equation (3) It can be calculated as:

[0033]

[0045] Here, the PDF term may represent a critical dimension probability distribution function determined by the measured critical dimensions of the IC structure at the fabrication conditions. The above-mentioned S(x) may be expressed as a sigmoid function for small or large defects (such as sigmoid function 503 in FIG. 5A or sigmoid function 513 in FIG. 5D). However, because S(x) includes 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 shown in FIG. 6, which is an example slope plot of the calculated failure rate 601 as a function of zeta and gamma for IC structures fabricated according to the fabrication conditions. FIG. 6A shows the calculated failure rate for small defects 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. 6B shows the calculated failure rate 601 for large defects as a function of zeta 604 (e.g., zeta 2 in FIG. 5 ) and gamma 605 (e.g., gamma 2 in FIG. 5 ). Measured failure rates 610 and 620 for IC structures are plotted in FIGS. 6A and 6B , respectively, showing ranges of zeta and gamma values ​​that may equate to measured failure rates. The failure rates for IC structures may be determined as described above for each IC structure within a grid area of ​​the focus-dose matrix. The measured failures may be missing defects, bridging defects, or defects observed in IC structures that may have different dimensions when compared to a reference structure. Any number of zeta and gamma values ​​for a sigmoid function (such as sigmoid function 503 or sigmoid function 513 in FIG. 5) may not identify the relevant fabrication conditions necessary to predict defect-free IC structures because the calculated and measured failure rates of the IC structures may be equal. Therefore, it may be desirable to parameterize the zeta and gamma values ​​of the sigmoid function to more accurately predict IC defects and the fabrication conditions that produce defect-free ICs.

[0034]

[0046] Two methods for parameterizing the sigmoid function can be used to determine the zeta and gamma values ​​required to predict small or large defects in an IC structure based on the critical dimension distribution and the measured failure rate of the IC structure at multiple focus / dose conditions.

[0035]

[0047] FIG. 7 shows a flow diagram of method 700, which may be a first method for parameterizing a sigmoid function based on critical dimension distributions and measured failure rates of IC structures fabricated under 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 of FIG. 4A). For method 700, the scatter values ​​(such as gamma 1 and gamma 2 described above) for the sigmoid function (such as sigmoid function 503 or sigmoid function 513 of FIG. 5) may be assumed to be constant for all combined focus and dose conditions, while the center values ​​(such as zeta 1 and zeta 2 described above) may vary for each focus condition (i.e., may be focus-dependent). Thus, one or more dose conditions may be considered for each focus condition. Method 700 may be performed simultaneously for a sigmoid function modeling small defects (e.g., sigmoid function 503 in FIG. 5A ) and a sigmoid function modeling large defects (e.g., sigmoid function 513 in FIG. 5D ). In step 701, a first scatter value is assigned to be constant according to a fabrication condition. The fabrication condition may be a focus condition, a dose condition, or a combined focus and dose condition. The first scatter value is assumed to be constant for each structure created in a grid area of ​​a focus-dose matrix (focus-dose matrix 410 in FIG. 4B ) on a sample during lithographic fabrication.

[0036]

[0048] In step 702, a first center value corresponding to a first measured failure rate of the IC structure under first fabrication conditions is calculated. The first fabrication conditions may be a first combined focus and dose condition. The first center value may be calculated by solving Equation 3 with the known measured failure rate, the critical dimension probability distribution of the IC structure under the first fabrication conditions, and the first scatter value.

[0037]

[0049] In step 703, a second center value corresponding to a second measured failure rate of the IC structure due to a second fabrication condition is calculated. The second center value is calculated as described above, but by solving Equation 3 using the second measured failure rate and the critical dimension probability distribution due to the second fabrication condition. In step 704, second scatter values ​​are assigned to be independent according to fabrication condition, and in step 705, steps 702 and 703 are repeated to calculate third and fourth center values ​​due to third and fourth fabrication conditions, respectively.

[0038]

[0050] In step 706, an optimized central value is calculated. The optimized central value may be determined by calculating the root mean square error (RMS) for the first and second central values ​​at the first scatter value, and for the third and fourth central values ​​at the second scatter value. The value of the smallest RMS error may identify the optimized central value. In step 707, an optimized scatter value is determined. The optimized scatter value may be the scatter value corresponding to the optimized central value with the smallest RMS error value. In step 708, a process window of IC structure failure rates may be generated for each fabrication condition using the optimized central value and the optimized scatter value. The process window may be generated by fitting the optimized central value as a function of fabrication conditions and applying the optimized scatter value and the modeled critical dimension probability distribution to calculate a predicted failure rate as a function of fabrication conditions. The fabrication conditions may be focus conditions, dose conditions, or a combined focus and dose condition.

[0039]

[0051] In the optimization process, the figure of merit may be expressed as a cost function. The optimization process may involve finding a set of parameters of the system that minimizes the cost function. The cost function may have any suitable form depending on the goal of the optimization. For example, the cost function may be the root mean square error (RMSE) or standard deviation of certain characteristics (e.g., evaluation points) of the system relative to intended values ​​of those characteristics. The cost function may also be the maximum of these deviations (e.g., the worst deviation). The term "evaluation point" herein should be interpreted broadly to include any characteristic of the system. Exemplary evaluation points included in the present disclosure may include fabrication conditions, such as focus and dose parameters of the radiation beam.

[0040]

[0052] 8A-8C, which illustrate a portion of method 700 using sample data. FIG. 8A illustrates a calculated RMSE value 801 of the central values ​​plotted against a scatter value 802. FIG. 8A may illustrate steps 701-707. A first RMSE value 801_1 of the calculated central values ​​and a second RMSE value 801_2 of the calculated central value roots are plotted against a first scatter value 802_1 and a second scatter value 802_2, respectively. FIG. 8A illustrates that the second RMSE value 801_2 is the smallest RMSE value. Thus, the central value corresponding to RMSE value 801_2 is the optimized central value, and the corresponding scatter value 802_2 is the optimized scatter value according to method 700. FIGS. 8B and 8C illustrate the fit of the modeled central value 803 as a function of focus 804, illustrating a portion of step 708 of method 700. Specifically, Figure 8C shows the best fit 805 of the modeled center value at a given focus condition 804 when optimized scatter value 802_2 and corresponding center value are selected, and Figure 8B shows the suboptimal fit 806 as a function of focus condition 804 when non-optimized scatter value 802_1 and corresponding center value are selected. Thus, Figure 8C shows the most accurate prediction of the optimized center value for a sigmoid function that models small defect and failure rates of IC structures as lithography focus conditions change when method 700 is performed. The sample data shown in Figures 8B-8C and the steps of method 700 of Figure 7 may also be applied to parameterizing a sigmoid function for predicting large defects.

[0041]

[0053] FIG. 9 shows a flow diagram of method 900, which may be a second method for parameterizing a sigmoid function to predict non-failure and defect probabilities for IC structures fabricated according to multiple fabrication conditions. The focus condition may be a focus condition, a dose condition, or a combined focus and dose condition. Method 900 may be performed by a computing device (e.g., processor 403 of FIG. 4A). For method 900, the scatter value and center value of the sigmoid function are both assumed to be focus-dependent. Therefore, a set of dose conditions is evaluated for each focus condition to parameterize the center value and scatter value. As described above, method 900 may be performed simultaneously for a sigmoid function modeling small defects (e.g., sigmoid function 503 of FIG. 5A) and a sigmoid function modeling large defects (e.g., sigmoid function 513 of FIG. 5D). In step 901, a first scatter value for a first set of fabrication conditions is assigned. The fabrication conditions may be a set of combined focus and dose conditions, with a first focus condition being constant for each fabrication condition within the set. In step 902, a first set of median values ​​corresponding to each fabrication condition in the first set of fabrication conditions is calculated. The calculation in step 902 is similar to that described above (e.g., step 702 of method 700), except that the plurality of median values ​​is calculated using Equation 3 with the plurality of measured failure rates and modeled critical dimension probability distributions and the first scatter value. In step 903, a second scatter value is assigned to the first set of fabrication conditions. In step 904, a second set of median values ​​corresponding to each fabrication condition in the first set of fabrication conditions is calculated. In step 905, an optimized scatter value for the first set of fabrication conditions is determined. The optimized scatter value may be determined by calculating RMSE values ​​for the first and second sets of calculated median values ​​and selecting the smallest RMSE value. An optimized scatter value is then determined by calculating the standard deviation of the set of calculated center values ​​corresponding to the smallest RMSE value. In step 906, an optimized scatter value for the second set of production conditions is determined.Optimized scatter values ​​for a second set of fabrication conditions are calculated by repeating steps 901-905, but instead at a second focus condition that is constant for each fabrication condition in the second set of fabrication conditions. In step 907, optimized center values ​​for the first and second sets of fabrication conditions are determined. This may be done by fitting the optimized scatter values ​​for the first and second sets of fabrication conditions as a function of focus. The optimized center values ​​may be calculated using the fitted scatter values ​​for each fabrication condition along with the measured failure rates and critical dimension probabilities. In step 908, a process window for IC structure failure rates by fabrication condition may be generated using the optimized center values ​​and optimized scatter values.

[0042]

[0054] 10A-10C, which illustrate a portion of method 900 using sample data. FIG. 10A shows a data set of calculated center values ​​1001 at different fabrication conditions (e.g., focus) 1002 generated according to steps 901 and 902 of method 900. The fabrication conditions can be different combinations of at least three dose conditions for a given focus condition. The boxed box 1003 shows the set of center values ​​calculated using Equation 3 using the scatter values ​​assigned as described above in step 902, and the fitted line 1004 shows the fit of all calculated center values ​​at all fabrication conditions as a function of focus. FIG. 10B shows the calculated optimized scatter values ​​1005 for each focus condition 1002 and the fit 1006 of the optimized scatter values ​​as a function of focus according to steps 905 and 906 of method 900. As shown in FIG. 10B, an optimized scatter value for each focus condition is determined (e.g., by calculating the standard deviation of the calculated zeta values ​​of FIG. 10A and selecting the minimum value) and fitted as a function of focus. The lined rectangle 1007 in FIG. 10B corresponds to the optimized scatter value relative to the calculated center value within the lined box 1003 in FIG. 10A. In FIG. 10C, the optimized scatter 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 optimized center value 1009 fitted as a function of focus shows better correlation compared to the fitted line 1004 shown in FIG. 10A.

[0043]

[0055] Referring now to Table 1, Table 1 displays the results of three data sets modeled using a conventional stepwise function to model the failure rate of an IC structure compared to a sigmoid function (e.g., Sigmoid Model 1 and Sigmoid Model 2). Sigmoid Model 1 corresponds to Method 700, and Sigmoid Model 2 corresponds to Method 900. Two metrics are used: the measured R versus the modeled failure rate. 2The correlation of values ​​and the change in failure rate (delta) between modeled and measured failures from validation data points can be used to quantify the improvement in failure rate modeling. 2 A larger value of R and a lower delta failure rate may indicate better model performance. As shown in Table 1, the embodiments of the present disclosure significantly reduce R compared to conventional methods. 2 The larger the value, the lower the delta between the modeled and measured failure rates at the validation data points. For Dataset 1 and Dataset 2, embodiments of the present disclosure provide improved modeling of large and small defects, and for Dataset 3, improved modeling of large defects.

[0044] [Table 1]

[0045]

[0056] Table 1. Summary of model accuracy metrics for conventional methods and embodiments of the present disclosure.

[0046]

[0057] 11A-11I, which are exemplary results illustrating how embodiments of the present disclosure provide improvements over conventional methods for modeling failure rates of IC structures made according to the fabrication conditions for Dataset 1 presented in Table 1. FIGS. 11A-11C show results for the conventional method, FIGS. 11D-11F show results for Sigmoid Model 1, and FIGS. 11G-11I show results for Sigmoid Model 2. FIGS. 11A, 11D, and 11G show modeling results for a modeled lower center value 1101 plotted against focus condition 1102. In the case of FIGS. 11D and 11G, this may be the Zeta 1 value discussed above. FIGS. 11B, 11E, and 11H show modeling results for a modeled upper center value 1103 plotted against focus condition 1102. In the case of FIGS. 11E and 11F, this may be the Zeta 2 value discussed above. The modeled center values ​​in FIGS. 11D, 11E, 11G, and 11H display an improved fit to the focus conditions 1102 compared to the modeled center values ​​in FIGS. 11A and 11B. This indicates that Sigmoid Model 1 and Sigmoid Model 2 can better predict center values ​​as a function of fabrication conditions compared to conventional methods. FIGS. 11C, 11F, and 11I show the modeled failure rates 1104 for each method plotted against the measured failure rates 1105. The fitted lines displayed in FIGS. 11C, 11F, and 11I show the correlation R 2 value (i.e., R 2 (The value is the slope of the fitted line.) Sigmoidal Model 1 (FIG. 11F) and Sigmoidal Model 2 (FIG. 11I) have larger R compared to the conventional model (FIG. 11C). 211F and 11I. Furthermore, delta failure rate validation points 1106, 1107, and 1108 for the conventional method, Sigmoid Model 1, and Sigmoid Model 2, respectively. Sigmoid Model 1 and Sigmoid Model 2 show delta failure rate validation points 1107 and 1108 that are closer to the fit lines in FIGS. 11F and 11I compared to delta failure rate validation point 1106 for the conventional method (FIG. 11C). Thus, both Sigmoid Model 1 and Sigmoid Model 2 improve the predictive performance of median modeling and failure rate modeling compared to the conventional models for Data Set 1.

[0047]

[0058] 12A-12C, which are exemplary generated failure rate process windows illustrating how embodiments of the present disclosure provide improvements over conventional methods for modeling the failure rate of an IC structure according to fabrication conditions. The process windows were generated to display the probability of failure rate for focus 1201 and dose 1202 conditions applied to fabricate the IC structure. The gradation scale bar on the right indicates the failure rate exponent value 1203 (i.e., 10 N where N is determined by the grayscale intensity within each process window. Each process window shows a boundary line indicating the boundary where log(failure rate) equals a failure rate index value 1203 (hereafter referred to as N). FIG. 12A (e.g., process window for a conventional method) shows three boundary lines, 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 (process window for method 700) shows boundary lines B and C. FIG. 12C (process window for method 900) shows boundary lines A, B, and C. Boundary line C indicates the boundary where 100% of the process window is sufficient for IC fabrication. -4 The predicted failure rate (10 -4 12A shows that the area within boundary line A is larger and the variation in failure rate is greater, with the failure rate exceeding 10-12 ~10 -6 This may be because the use of a stepwise function may miss defects, as previously discussed, and may over-predict defect-free IC formation at given fabrication conditions. Furthermore, FIG. 12A shows a smaller distance between each of the boundaries A, B, and C compared to the boundaries in FIGS. 12B and 12C. This indicates that a parameterized sigmoid function consistent with embodiments of the present disclosure may more consistently predict failure rates of IC structures at fabrication conditions within this region of the process window.

[0048]

[0059] A non-transitory computer-readable medium may be provided that can store instructions for processors of a lithographic projection apparatus (e.g., lithographic projection apparatus 100 of FIG. 1) and an inspection tool (e.g., inspection tool 402 of FIG. 4A) for determining critical dimension measurements and defect data of a sample, determining lower and upper center values, determining center and scatter values ​​for a sigmoid function, method 700 of FIG. 7, method 900 of FIG. 9, and other executable functions related to predicting defect probability and failure probability sigmoid functions and parameterizing sigmoid functions. Common forms of non-transitory media include, for example, floppy disks, flexible disks, hard disks, solid state drives, magnetic tape or any other magnetic data storage medium, compact disk read-only memory (CD-ROM), any other optical data storage medium, any physical medium with a pattern of holes, random access memory (RAM), programmable read-only memory (PROM) and erasable programmable read-only memory (EPROM), flash EPROM or any other flash memory, non-volatile random access memory (NVRAM), cache, registers, any other memory chip or cartridge, and network-connected versions thereof.

[0049]

[0060] The embodiments may be further described using the following clauses. 1. A method for modeling defect probability, comprising: obtaining critical dimension measurements and defect data from the sample; determining a lower defect threshold limit and an upper defect threshold limit based on a sigmoid function, the lower defect threshold limit and the upper defect threshold limit including a center value and a scatter value; A method comprising: 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 corresponds to smaller defects. 4. The method of clause 1, wherein the defect threshold upper limit corresponds to larger defects. 5. Assigning a first constant scatter value; Calculating a first center value under a first manufacturing condition of the sample and a second center value under a second manufacturing condition of the sample; assigning a second constant scatter value; Calculating a third median value under a third manufacturing condition of the sample and a fourth median value under a fourth manufacturing condition of the sample; determining an optimized center value and an optimized spread value; generating a process window based on the optimized center value and the optimized spread value; 2. The method of clause 1, further comprising: 6. The method of clause 5, wherein determining the optimized center value is by selecting the smallest root mean square error of the first center value, the second center value, the third center value and the fourth center value. 7. The method of clause 5, wherein the fabrication conditions are lithographic patterning conditions. 8. The method according to clause 7, wherein the fabrication conditions are focus conditions, dose conditions or combined focus and dose conditions. 9. The method of clause 5, wherein the optimized center value is fitted as a function of focus only. 10. The method of clause 5, wherein the optimized dispersion value is assumed to be constant for all production conditions. 11. Assigning a first scatter value according to a first set of manufacturing conditions and calculating a first set of center values; assigning a second scatter value according to the first set of manufacturing conditions and calculating a second set of center values; Calculating an optimized dispersion value according to a first set of manufacturing conditions; assigning a first scatter value according to a second set of manufacturing conditions and calculating a first set of center values; assigning a second scatter value according to a second set of manufacturing conditions and calculating a second set of median values; Calculating an optimized dispersion value according to a second set of manufacturing conditions; fitting the optimized scatter values ​​according to the first set of manufacturing conditions and the second set of manufacturing conditions to calculate an optimized center value; generating a process window based on the optimized spread values ​​and the optimized center values ​​according to the first and second sets of fabrication conditions; 2. The method of clause 1, further comprising: 12. The method of claim 11, wherein calculating the optimized dispersion values ​​according to the first set of production conditions is by minimizing the 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 conditions includes a first constant focus condition and at least three different dose conditions. 14. The method of clause 11, wherein the second set of fabrication conditions includes a second constant focus condition and at least three different dose conditions. 15. The method of clause 11, wherein the optimized dispersion values ​​from the first set of production conditions and the second set of production conditions are fitted as a function of focus. 16. The method of clause 11, wherein the optimized center value is fitted as a function of focus. 17. A method for parameterizing non-fault probability based on a sigmoid function, comprising: assigning a first constant scatter value; Calculating a first center value under a first manufacturing condition of the sample and a second center value under a second manufacturing condition of the sample; assigning a second constant scatter value; Calculating a third median value under a third manufacturing condition of the sample and a fourth median value under a fourth manufacturing condition of the sample; determining an optimized center value and an optimized spread value; generating a process window based on the optimized center value and the optimized spread value; A method comprising: 18. The method of clause 17, wherein determining the optimized center value is by selecting the smallest root mean square error of the first center value, the second center value, the third center value, and the fourth center value. 19. The method of clause 17, wherein the fabrication conditions are lithographic patterning conditions. 20. The method of clause 19, wherein the fabrication conditions are focus conditions, dose conditions, or combined focus and dose conditions. 21. The method of clause 17, wherein the optimized center value is fitted as a function of focus only. 22. The method of clause 17, wherein the optimized dispersion value is assumed to be constant for all production conditions. 23. A method for parameterizing non-fault probability based on a sigmoid function, comprising: assigning a first scatter value according to a first set of manufacturing conditions and calculating a first set of center values; assigning a second scatter value according to the first set of manufacturing conditions and calculating a second set of center values; Calculating an optimized dispersion value according to a first set of manufacturing conditions; assigning a first scatter value according to a second set of manufacturing conditions and calculating a first set of center values; assigning a second scatter value according to a second set of manufacturing conditions and calculating a second set of median values; Calculating an optimized dispersion value according to a second set of manufacturing conditions; fitting the optimized scatter values ​​according to the first set of manufacturing conditions and the second set of manufacturing conditions to calculate an optimized center value; generating a process window based on the optimized spread values ​​and the optimized center values ​​according to the first and second sets of fabrication conditions; A method comprising: 24. The method of clause 23, wherein calculating the optimized dispersion values ​​according to the first set of production conditions is by minimizing the root mean square error of the first set of center values ​​and the second set of center values. 25. The method of clause 23, wherein the first set of fabrication conditions includes a first constant focus condition and at least three different dose conditions. 26. The method of clause 23, wherein the second set of fabrication conditions includes a second constant focus condition and at least three different dose conditions. 27. The method of clause 23, wherein the optimized dispersion values ​​from the first and second sets of production conditions are fitted as a function of focus. 28. The method of clause 23, wherein the optimized center value is fitted as a function of focus. 29. A non-transitory computer-readable medium comprising a set of instructions executable by one or more processors of a computing device to cause the computing device to perform a method for modeling defect probabilities, the method comprising: obtaining critical dimension measurements and defect data from the sample; determining a lower defect threshold limit and an upper defect threshold limit based on a sigmoid function, the lower defect threshold limit and the upper defect threshold limit including a center value and a scatter value; 1. A non-transitory computer-readable medium comprising: 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 corresponds to smaller defects. 32. The non-transitory computer-readable medium of clause 29, wherein the defect threshold upper limit corresponds to larger defects. 33. Assigning a first constant dispersion value; Calculating a first center value under a first manufacturing condition of the sample and a second center value under a second manufacturing condition of the sample; assigning a second constant scatter value; Calculating a third median value under a third manufacturing condition of the sample and a fourth median value under a fourth manufacturing condition of the sample; determining an optimized center value and an optimized spread value; generating a process window based on the optimized center value and the optimized spread value; 29. The non-transitory computer-readable medium of claim 29, further comprising: 34. The non-transitory computer-readable medium of clause 33, wherein determining the optimized center value is by selecting a minimum root mean square error of the first center value, the second center value, the third center value, and the fourth center value. 35. The non-transitory computer-readable medium of clause 33, wherein the fabrication conditions are lithographic patterning conditions. 36. The non-transitory computer-readable medium of clause 35, wherein the fabrication conditions are focus conditions, dose conditions, or combined focus and dose conditions. 37. The non-transitory computer-readable medium of clause 33, wherein the optimized center value is fitted as a function of focus only. 38. The non-transitory computer-readable medium of clause 33, wherein the optimized dispersion value is assumed to be constant for all production conditions. 39. Assigning a first scatter value according to a first set of manufacturing conditions and calculating a first set of center values; assigning a second scatter value according to the first set of manufacturing conditions and calculating a second set of center values; Calculating an optimized dispersion value according to a first set of manufacturing conditions; assigning a first scatter value according to a second set of manufacturing conditions and calculating a first set of center values; assigning a second scatter value according to a second set of manufacturing conditions and calculating a second set of median values; Calculating an optimized dispersion value according to a second set of manufacturing conditions; fitting the optimized scatter values ​​according to the first set of manufacturing conditions and the second set of manufacturing conditions to calculate an optimized center value; generating a process window based on the optimized spread values ​​and the optimized center values ​​according to the first and second sets of fabrication conditions; 29. The non-transitory computer-readable medium of claim 29, further comprising: 40. The non-transitory computer-readable medium of clause 39, wherein calculating the optimized dispersion values ​​for the first set of production conditions is by minimizing the 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 conditions includes a first constant focus condition and at least three different dose conditions. 42. The non-transitory computer-readable medium of clause 39, wherein the second set of fabrication conditions includes a second constant focus condition and at least three different dose conditions. 43. The non-transitory computer-readable medium of clause 39, wherein the optimized dispersion values ​​from the first set of fabrication conditions and the second set of fabrication conditions are fitted as a function of focus. 44. The non-transitory computer-readable medium of clause 39, wherein the optimized center value is fitted as a function of focus. 45. A non-transitory computer-readable medium comprising a set of instructions executable by one or more processors of a computing device to cause the computing device to perform a method for parameterizing non-fault probabilities based on a sigmoid function, the method comprising: assigning a first constant scatter value; Calculating a first center value under a first manufacturing condition of the sample and a second center value under a second manufacturing condition of the sample; assigning a second constant scatter value; Calculating a third median value under a third manufacturing condition of the sample and a fourth median value under a fourth manufacturing condition of the sample; determining an optimized center value and an optimized spread value; generating a process window based on the optimized center value and the optimized spread value; 1. A non-transitory computer-readable medium comprising: 46. ​​The non-transitory computer-readable medium of clause 45, wherein determining the optimized center value is by selecting a minimum root mean square error of the first center value, the second center value, the third center value, and the fourth center value. 47. The non-transitory computer-readable medium of clause 45, wherein the fabrication conditions are lithographic patterning conditions. 48. The non-transitory computer-readable medium of clause 47, wherein the fabrication conditions are focus conditions, dose conditions, or combined focus and dose conditions. 49. The non-transitory computer-readable medium of clause 45, wherein the optimized center value is fitted as a function of focus only. 50. The non-transitory computer-readable medium of clause 45, wherein the optimized dispersion value is assumed to be constant for all production conditions. 51. A non-transitory computer-readable medium comprising a set of instructions executable by one or more processors of a computing device to cause the computing device to perform a method for parameterizing non-fault probabilities based on a sigmoid function, the method comprising: assigning a first scatter value according to a first set of manufacturing conditions and calculating a first set of center values; assigning a second scatter value according to the first set of manufacturing conditions and calculating a second set of center values; Calculating an optimized dispersion value according to a first set of manufacturing conditions; assigning a first scatter value according to a second set of manufacturing conditions and calculating a first set of center values; assigning a second scatter value according to a second set of manufacturing conditions and calculating a second set of median values; Calculating an optimized dispersion value according to a second set of manufacturing conditions; fitting the optimized scatter values ​​according to the first set of manufacturing conditions and the second set of manufacturing conditions to calculate an optimized center value; generating a process window based on the optimized spread values ​​and the optimized center values ​​according to the first and second sets of fabrication conditions; 1. A non-transitory computer-readable medium comprising: 52. The non-transitory computer-readable medium of clause 51, wherein calculating the optimized dispersion values ​​according to the first set of production conditions is by minimizing the 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 clause 51, wherein the first set of fabrication conditions includes a first constant focus condition and at least three different dose conditions. 54. The non-transitory computer-readable medium of clause 51, wherein the second set of fabrication conditions includes a second constant focus condition and at least three different dose conditions. 55. The non-transitory computer-readable medium of clause 51, wherein the optimized dispersion values ​​from the first set of production conditions and the second set of production conditions are fitted as a function of focus. 56. The non-transitory computer-readable medium of clause 51, wherein the optimized center value is fitted as a function of focus.

[0050]

[0061] It will be understood that the embodiments of the present disclosure are not limited to the exact construction described above and illustrated in the accompanying drawings, and that various modifications and changes can be made therein without departing from the scope thereof. While the present disclosure has been described in connection with various embodiments, other embodiments of the present 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

1. 1. A non-transitory computer-readable medium comprising a set of instructions executable by one or more processors of a computing device to cause the computing device to perform a method for modeling defect probabilities, the method comprising: obtaining critical dimension measurements and defect data from the sample; determining a defect threshold lower limit and a defect threshold upper limit based on a sigmoid function, wherein the defect threshold lower limit and the defect threshold upper limit include a center value and a scatter value; 1. A non-transitory computer-readable medium comprising:

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 corresponds to smaller defects.

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

5. assigning a first constant spread value; Calculating a first median value under a first manufacturing condition of the sample and a second median value under a second manufacturing condition of the sample; assigning a second constant spread value; Calculating a third median value under a third manufacturing condition of the sample and a fourth median value under a fourth manufacturing condition of the sample; determining an optimized center value and an optimized spread value; generating a process window based on the optimized center value and the optimized spread value; The non-transitory computer-readable medium of claim 1 , further comprising:

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

7. The non-transitory computer-readable medium of claim 5 , wherein the fabrication conditions are lithographic patterning conditions.

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

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

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

11. assigning a first scatter value according to a first set of manufacturing 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 median values; Calculating an optimized dispersion value according to the first set of production conditions; assigning a first scatter value according to a second set of manufacturing conditions and calculating a first set of center values; assigning a second scatter value according to a second set of fabrication conditions and calculating a second set of median values; Calculating an optimized dispersion value according to the second set of production conditions; fitting the optimized scatter values ​​with the first set of manufacturing conditions and the second set of manufacturing conditions to calculate an optimized center value; generating a process window based on the optimized spread values ​​and the optimized center value according to the first and second sets of fabrication conditions; The non-transitory computer-readable medium of claim 1 , further comprising:

12. 12. The non-transitory computer-readable medium of claim 11, wherein calculating the optimized scatter values ​​according to the first set of production 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 conditions includes a first constant focus condition and at least three different dose conditions.

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

15. 12. The non-transitory computer-readable medium of claim 11, wherein the optimized dispersion values ​​from the first set of fabrication conditions and the second set of fabrication conditions are fitted as a function of focus.

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