Method and system for calibrating random signals in compact modeling

Abstract

Embodiments of the present disclosure relate to calibrating random signals in compact modeling. Calibrating random signals in compact modeling is provided by obtaining data of process variations in producing a resist mask; calibrating a continuous compact model of the resist mask based on the data; evaluating the continuous compact model against a data-based stochastic compact model; selecting a functional description of an edge position distribution of the stochastic compact model; mapping image parameters from the evaluation to edge distribution parameters according to the functional description; determining an edge position range for the stochastic compact model based on a scaling measure from the image parameters; calibrating a threshold value of the resist mask and updating parameters of the stochastic compact model to reduce a difference between experimental data and modeled LER values; and outputting the stochastic compact model.

Description

TECHNICAL FIELD

[0001] The present disclosure relates to calibrating stochastic signals in compact modeling in microlithography processes. BACKGROUND

[0002] Microlithography is a manufacturing process that includes making a relief image of a master pattern (called a mask or reticle) in a photosensitive material (called photoresist or resist). The goal of any lithography process is to transfer the pattern on the reticle into the photoresist film with good fidelity. The quality of the resulting pattern depends on the uniformity of the photon absorption events the photoresist pattern receives during exposure and the uniformity of the subsequent photochemical reactions in the photoresist. Because the number of photon absorption events is the first stochastic process in the chain of stochastic processes in a lithography process, photon statistics (e.g., shot noise) become more and more important as the source photon count output decreases.

[0003] When using deep ultraviolet excimer laser sources with high photon output, the effects of photon shot noise can be largely ignored. Patterns that are prone to random failures during mask correction, optimization, and verification can be of particular interest. The term "random failure" is used herein to describe any non-conformance that has a statistically or probabilistically measurable chance of occurrence but cannot be accurately predicted. However, next generation extreme ultraviolet lithography (EUVL) systems struggle with both low photon output and low light source throughput. For these reasons, photon shot noise plays a greater role in image process development for EUVL than in older technology processes.

[0004] Currently, two different approaches are taken to manage the local edge placement uniformity (LEPU) error or local critical dimension uniformity (LCDU) error associated with photon shot noise. The classical approach is to assume that the linear edge roughness (LER) around a nominal gauge that is typically measured directly on a wafer is constant and use that information as part of various optimization steps when creating features. As an alternative to using constant LER, the fabricator can assume that the LER is proportional to some quantity that is measured or calculated. The normalized image log-slope (NILS) of repeating features (e.g., lines and spaces, contact holes, repeating scribe lines, etc.) is used as such a measure to identify problem areas. Printing smaller two-dimensional features using lithography technology means that there is no method established for calculating edge placement errors and designing or optimizing features to compensate for these random variations. SUMMARY

[0005] In one aspect, the present disclosure provides a method comprising: obtaining data of process variations in producing a resist mask; calibrating a continuous compact model of the resist mask based on the data; evaluating the continuous compact model against a stochastic compact model based on the data; selecting a functional description of an edge position distribution for the stochastic compact model; mapping image parameters from the evaluation to edge distribution parameters according to the functional description; determining an edge position range for the stochastic compact model based on scaling measurements from the image parameters; calibrating a threshold of the resist mask and updating parameters of the stochastic compact model to reduce a difference between the data and modeled linear edge roughness (LER) values; and outputting the stochastic compact model.

[0006] In one aspect, the present disclosure provides a method comprising: generating a continuous compact model for a resist mask pattern calibrated to production data; evaluating the production data according to the continuous compact model; selecting a functional description to represent an edge position range of the resist mask pattern in a stochastic compact model of the resist mask pattern; mapping the evaluated production data to the selected functional description; and generating the stochastic compact model that probabilistically models the edge position range of the resist mask pattern based on the mapped production data.

[0007] In one aspect, the present disclosure provides a system comprising a processor; and a memory including instructions that, when executed by the processor, are configured to perform operations comprising: obtaining data of process variations in producing a resist mask; calibrating a continuous compact model of the resist mask based on the data; evaluating the continuous compact model against a stochastic compact model based on the data; selecting a functional description of an edge position distribution for the stochastic compact model; mapping image parameters from the evaluation to edge distribution parameters according to the functional description; calculating an edge position range for the stochastic compact model based on scaling measurements from the image parameters; calibrating a threshold of the resist mask and updating parameters of the stochastic compact model to reduce a difference between the data and modeled linear edge roughness (LER) values; and outputting the stochastic compact model. BRIEF DESCRIPTION OF DRAWINGS

[0008] The present disclosure will be more fully understood from the following detailed description taken in connection with the accompanying drawings, in which:

[0009] Figure 1 A general compact model flow of a stochastic model following an optical model and a resist model is depicted in accordance with embodiments of the present disclosure.

[0010] Figure 2An improvement of band value prediction using newer models is depicted according to embodiments of the present disclosure.

[0011] Figure 3 A Gaussian distribution for emulating a function F is depicted according to embodiments of the present disclosure.

[0012] Figure 4 How different confidence bands can be calibrated on a model at once is depicted according to embodiments of the present disclosure.

[0013] Figure 5 Edge profiles of a first lithography pattern and a second lithography pattern are respectively illustrated according to embodiments of the present disclosure.

[0014] Figure 6 is a flowchart of a method for calibrating a stochastic compact model to experimental data according to embodiments of the present disclosure.

[0015] Figure 7A and Figure 7B Data types recommended for fitting a stochastic compact model to two different pattern types on a mask are illustrated according to embodiments of the present disclosure.

[0016] Figures 8A-8D A noise floor for roughness measurement is illustrated according to embodiments of the present disclosure.

[0017] Figure 9A and Figure 9B QQ plots of two different distributions are illustrated according to embodiments of the present disclosure.

[0018] Figure 10 Three potential mappings to provide a modeling device with the ability to match experimental data to non-Gaussian distributions using selected mean, standard deviation, variance, and skewness are illustrated according to embodiments of the present disclosure.

[0019] Figure 11 is a flowchart of an application method of a stochastic compact model according to embodiments of the present disclosure.

[0020] Figure 12 A flowchart of various processes used during the design and fabrication of integrated circuits is depicted according to some examples of the present disclosure.

[0021] Figure 13 A representation of an example computer system in which examples of the present disclosure can operate is depicted. DETAILED DESCRIPTION

[0022] Aspects described herein relate to calibrating stochastic signals when modeling compactly in microlithography. At the single molecule level, all chemical processes are stochastic in nature. When the number of molecules, ions, and photons involved in an imaging process becomes very large, the stochastic nature of the reactions can be described by a continuous process. Reducing the photon count in EUVL (extreme ultraviolet lithography) processes and reducing the number of photoresist molecules involved in these processes increases the stochastic variation of all processes that occur after photon absorption. The increased variance causes printed edges to deviate from the average edge with some probability. As a result, some structures can be lost or unusable at random.

[0023] The present disclosure is based on a method that uses pre-computed continuous signals and measured or rigorously simulated LEPU or LCDU values to calibrate a confidence band around a profile from the continuous signal. This process adds a small amount of overhead to existing simulation flows. Tests show that the accuracy in predicting the 3 sigma confidence interval of a flow is relatively better (e.g., four times better) compared to considering NILS alone. The calibration approach can be simplified to include only the spatial image of a wafer. Using signals that include material interactions with the spatial image can improve the accuracy of the method. The inhibitor concentration field after exposure and post-apply bake is an example of a signal that improves accuracy.

[0024] For processes that are repeatable and performed using the same equipment and methods, as with lithography processes, the pure stochastic variation should be predictable for a stochastic edge placement error distribution. The exact formation location of an edge cannot be known, but when a large number of experiments are performed, a confidence band of possible edge locations can be assigned. Existing contouring methods only provide a good estimate of the implied error band given by the LER. The present disclosure calculates a confidence band around the placement of a corrected edge profile. The confidence band identifies the number of times an edge location occurs within a given range of positions from the average position. For example, when an edge location occurs between position X and position Y and a confidence level of Z% is specified, a confidence band is provided from position X to position Y (relative to a nominal or average position) to indicate that the edge location occurs Z% of the time. There is a non-zero chance that the edge location occurs outside the range specified by position X and position Y (e.g., 100-Z), and different values of X, Y, and Z can be set to describe different confidence intervals (e.g., a range of X1-Y1 nm occurs Z1% of the time and a range of X2-Y2 nm occurs Z2% of the time, where X1-Y1 ≠ X2-Y2 and Z1 ≠ Z2).

[0025] The confidence band information can be used to identify random non-robust portions of the layout or optical proximity correction / resolution enhancement technology / inverse lithography technology (OPC / RET / ILT / ) corrections. Further, OPC or source mask optimization (SMO) can be performed to improve the robustness of the pattern to improve any weak spots that have been identified as prone to failure or not meeting specified tolerances. Additionally, the robustness of critical patterns can be improved to increase yield. The confidence level of the error band can be improved via gathering more data via calibrated rigorous simulators or large scale collection and analysis of printed features using metrology tools such as critical dimension scanning electron microscopes (CD-SEMs). By designing appropriate loss functions, one can use the obtained information to perform defect window centering, stable OPC for random defects, yield estimation, etc. The obtained information can be used to understand the LCDU variations to improve inverse lithography technology (ILT).

[0026] Figure 1 is a high level view of a flow 100 for creating a stochastic model 170 according to one embodiment of the present disclosure. As shown, during a first calibration at 110, resist process parameters 120 are used with an optical model 130 to generate a resist model 140. The resist process parameters 120 can include placement position of the resist mask, thickness of the resist mask, material composition of the resist mask, intensity and frequency of light applied to the resist mask, duration of time to allow the resist mask to sit before exposure, exposure duration, etc. Additionally, the resist process parameters 120 can include a post-exposure bake (PEB) operation where the wafer is heated to a specified temperature for a specified time after the initial light exposure operation to affect diffusion of dopants in the wafer.

[0027] During the stochastic calibration, at 150, stochastic parameters 160 are used with the resist model 140 to generate a stochastic model 170. The stochastic model 170 identifies a confidence band for the mean identifying the position of the line edge, allowing the designer to identify the stochastic variations of the overall process.

[0028] In some embodiments, the optical model 130 and the resist model 140 determine optical signal strengths and optical signal thresholds associated with the optical signals. In some embodiments, the stochastic model 170 inherits the determined optical signal strengths and the determined optical signal thresholds associated with the optical signals from the optical model 130 or the resist model 140. In some embodiments, the stochastic model 170 determines optical signal strengths and optical signal thresholds independent of the optical model 130 and the resist model 140.

[0029] In some embodiments, an optical signal profile is extracted using an optical signal threshold. The extraction of the optical signal profile can be performed randomly. Generally, a contour line of a function for two variables, such as optical signal intensity I(x) and a point x on the surface where the optical signal arrives, is a curve along which the function has a constant value, to connect points of equal value. In some embodiments, the optical signal profile is a planar cross-section of a two-dimensional graph of the function I(x) that is parallel to the x-plane.

[0030] In some embodiments, a signal field associated with the optical signal is created. Thereafter, an optical signal threshold for pattern transfer associated with the optical signal is determined. In some embodiments, the optical signal threshold is a constant. In some embodiments, the optical signal threshold is defined by a number of photons expected to arrive at a point (e.g., a cut-off dose) divided by a dose associated with a region. For example, if the optical signal threshold is 0.2 and the dose associated with a certain region is 20 photons per square nanometer, then the probability that a particular region that has a cross-section of one square nanometer receives a dose that is less than or greater than the cut-off dose of 4 photons (e.g., 0.2 * 20 = 4) indicates uncertainty in forming an edge on the particular region. In some embodiments, the edge does not have an infinite slope and curvature at the optical signal threshold location, which can create finite blur once the related effects of neighboring voxels (i.e., three-dimensional pixels) are considered. As one example, for positive development, it is assumed that optical signals above the optical signal threshold are cleared, while optical signals below the optical signal threshold remain untouched. .2). In some embodiments, the edge does not have an infinite slope and curvature at the optical signal threshold location, which can create finite blur once the related effects of neighboring voxels (i.e., three-dimensional pixels) are considered. As one example, for positive development, it is assumed that optical signals above the optical signal threshold are cleared, while optical signals below the optical signal threshold remain untouched.

[0031] Figure 2 Improvements to band value prediction using stochastic models are depicted in accordance with one embodiment of the disclosure. When using the stochastic compact model described herein, stochastic base error lines 210 and stochastic head error lines 220 identify differences from the confidence band at the base and head of the line, respectively, in contrast to continuous base error lines 230 and continuous head error lines 240, as would be the case using a traditional continuous compact model. The placement of the confidence band depends on the derivative of the optical signal at the optical signal threshold. Additionally, in some embodiments, the placement of the confidence band depends on the values within a relevant length of the confidence band.

[0032] In some embodiments, the optical signal arriving at the wafer has a photon count that is distributed according to a Poisson distribution or other distribution curve. A Poisson distribution is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space if these events occur with a known constant mean rate and independently of the time since the last event. In some embodiments, a distribution curve is used to specify a number of edge formations in a distance or region. For example, if a discrete random variable x has a Poisson distribution with parameter λ > 0, then for k = 0, 1, 2,..., the probability mass function of x is given by:

[0033] Equation 1

[0034] where e is Euler's number (e = 2.71828...), k! is the factorial of k, and the positive real number λ is equal to the expected value of x.

[0035] In some embodiments, a continuous signal field of a nominal metric specifying occurrence of an event is determined. The continuous signal field can belong to at least one of: a mean, a median, a mode, or any metric of the number of occurrences of the event. In some embodiments, a distribution curve is used to extract a confidence band in which an edge is formed. The extraction of the confidence band is performed based on a mean number of photons arriving at the location. Additionally, in some embodiments, the mean number of photons arriving at the location is given by the dose applied and modulated by a pattern optical signal variation.

[0036] In some embodiments, a continuous optical signal is a position function, which is an output of a calibrated continuous model. A constant optical signal threshold (Th) can be obtained from the continuous model. In some embodiments, a method for predicting formation of a defect at a location by an optical signal in a lithographic process uses calibration of placement of a gauge end. Additionally or alternatively, in some embodiments, a method for predicting formation of a defect at a location by an optical signal in a lithographic process uses critical dimension (CD) variability. CD indicates a process in which a dimension (e.g., width) of a feature printed in resist is measured at a particular height above a substrate. In such embodiments, an indirect metric of variability (e.g., LER or CD uniformity (CDU)) is used in place of the optical signal. CDU is used to ensure good performance and reliable functionality of a sample. In some embodiments, CDU includes total wafer CDU, mask CDU, resist process, scanner and lens fingerprint, wafer topography, etc. Further, in some embodiments, a metric of a defect count or pixNOK value is used for the optical signal.

[0037] In some embodiments, the optical signal threshold is calibrated via a shift. In such embodiments, the shift (e.g., Ad and Ab) is a constant. The shift to the optical signal threshold for forming or clearing a feature can result from a decrease or increase in the signal.

[0038] In some embodiments, predicting formation of a defect at a location during a lithographic process by an optical signal comprises: determining a threshold for formation of the defect. In one embodiment, predicting formation of a defect at a location during a lithographic process by an optical signal further comprises: determining a random optical signal profile based on the determined threshold. To determine the random optical signal profile, the optical signal threshold is determined. In one embodiment, the random optical signal profile is determined using offset optical signal thresholds (e.g., Th-Ad) and (Th-Ab) based on successive optical signals , for example, . In some embodiments, the optical signal profile is calibrated accordingly using the measured error band placement of each gauge. In some embodiments, the optical signal profile is calibrated accordingly using the rigorously calculated error band placement of each gauge. The calibrated profile (e.g., and ) can be used as the random optical signal profile.

[0039] In some embodiments, the random optical signal profile is determined based on a contribution kernel. To determine the random optical signal based on a contribution kernel K, the contribution kernel is calibrated. In some embodiments, at larger distances, the contribution kernel approaches or equals zero. In some embodiments, the calibrated contribution kernel volume is normalized for computation. Additionally, in some embodiments, the values of the calibrated contribution kernel are non-negative real numbers for all spatial domains defining the calibrated contribution kernel.

[0040] As one example, the contribution kernel comprises a properly normalized constant. As another example, the contribution kernel comprises a disk. As yet another example, the contribution kernel comprises a Gaussian, Lorentzian, Cauchy, or Voigt sphere or a linear combination thereof.

[0041] In some embodiments, the random optics are determined based on Equation 2.

[0042] Equation 2

[0043] where denotes a convolution operator that produces a random optical signal. The produced random optical signal expresses a way in which the contribution kernel modifies the shape of the optical signal.

[0044] In some embodiments, the random optical signal is determined based on a cumulative probability of a random variable. In such embodiments, F(X{0 i}) is used as the cumulative probability of a random variable X parameterized by a parameter set {0 i}. In some embodiments, is used as the parameter estimate set ( ). In some embodiments, the random optical signal is determined based on Equation 3. Figure 3A simulation function F is depicted to determine a Gaussian distribution of a random optical signal.

[0045] Equation 3

[0046] In some embodiments, a suitable probability distribution and parameters associated with the probability distribution of the random variable X are determined using calibrated data obtained from a wafer or simulation. Also, in some embodiments, a suitable probability distribution and parameters associated with the probability distribution of the random variable X are determined by numerical and analytical computational methods. In some embodiments, a signal band is calibrated. In some embodiments, portions of a signal band are calibrated.

[0047] In some embodiments, a defect metric is determined for each gauge. To determine the defect metric, the determined random optical signal is used. The defect metric is determined based on Equation 4.

[0048] Equation 4

[0049] In Equation 3, the defect range starts and ends at pre-determined locations and is integrated over the defect domain region. In such embodiments, the definition of the defect range specifies a probability, distance, or percentage change in edge placement or CD variation. In some embodiments, the determined D gauge value is further tuned via calibration data. Alternatively, in some embodiments, the determined D gauge value is used for validation against simulated data or measured data. The defect metric is used for repeating patterns, such as lines and spaces, contact holes, scribe lines, etc., or generalized two-dimensional fields and regions of interest.

[0050] Figure 3 How a continuous signal value is used to determine an edge clear probability is illustrated. In some embodiments, a continuous model determines an optical signal threshold 310, which in turn determines an edge position. In some embodiments, for a positive tone development (PTD) resist, when the optical signal intensity at a determined position is higher than the optical signal threshold 310, the position is assigned as a space where no resist is applied. Then, an optical signal profile is drawn around the position.

[0051] In some embodiments, the optical signal field generated for a continuous optical signal gives the median value of acquiring such an optical signal. Additionally, in some embodiments, the probability of an event occurring at a given distance from the median position is given by a random signal 320 with a normal or Gaussian distribution. The photoresist edge is predicted to form in the region to the right of a vertical line 340 that lies at the intersection of the point 330, i.e., the point where the random signal 320 intersects the optical signal threshold 310.

[0052] The probability of forming an edge at point 330 (i.e., the point at which the stochastic signal 320 intersects the resist model defined by the vertical line 340) is determined by the area of region 360 of the Gaussian distribution 350, which is shown to have a mean of μ and a standard deviation of σ. As can be seen, region 360 is shown to be outside of ±1σ (i.e., more than one standard deviation from the mean).

[0053] In Figure 3 some embodiments, a value of a variance of the Gaussian distribution is determined based on at least one of: a dose at the location (D), a resist light absorption factor (abs), a secondary electron yield (SEY), an acid concentration (C A ), an acid yield (AY), a quencher concentration (C Q ), a PEB enhancement factor (K PEB ), and a constant due to dark loss. In some embodiments, the value of the variance is determined based on Equation 5.

[0054] Equation 5

[0055] Additionally or alternatively, in some embodiments, parameters in Equation 5 are grouped using two parameters a and b (which can be calibrated) to derive Equation 6. In some embodiments, a has a value that is positive and greater than zero. In some embodiments, b can have a value that is positive or negative. In some embodiments, the value of b is limited.

[0056] Equation 6

[0057] In some embodiments, the stochastic optical signal is determined based on Equation 7.

[0058] Equation 7

[0059] where Y is a random variable and is determined by Equation 8,

[0060] Equation 8

[0061] where are normal distribution parameters corresponding to the mean and standard deviation.

[0062] In some embodiments, an optical signal threshold defined for the optical signal is used to determine the nominal signal. In some embodiments, the region is an open frame region. In such embodiments, the optical signal intensity is determined by multiplying the optical signal intensity by a scalar factor. In some embodiments, the effective number of photons is increased from the relevant occurring in the vicinity of the voxel by a local voxel (i.e., a three-dimensional pixel). In such embodiments, the calibration factor k pfor determining the optical signal intensity. In some embodiments, the calibration parameter is the same for understanding the optical signal threshold and the signal intensity. In some embodiments, a fraction of the photons reaching position x is absorbed. Thus, in some embodiments, a correction factor k is determined to account for the fraction of the optical signal that is absorbed. c to account for the fraction of the optical signal that is absorbed.

[0063] Afterwards, the probability distribution is determined based on equations 9 and 10.

[0064] Equation 9

[0065] Equation 10

[0066] As can be seen from equation 10, according to one aspect of the present disclosure, the probability distribution function of forming a defect at point x is determined according to the defined threshold Th, the intensity of the optical signal at point x, the parameters a and b defined above, and the error function (erf). In some embodiments, for the other side of the distribution, P(x) is determined by equation 11.

[0067] Equation 11

[0068] As an example, to calibrate to P(X) = 0.95, the optical signal threshold is set to 0.45. Figure 4 It is depicted how different confidence bands are calibrated in parallel on the stochastic model 170. Figure 4 Residuals 410-450 from this approach are also plotted in FIG. 4, where residuals 410-450 of multiple bands (e.g., of confidence bands with increasing confidence percentages) are compared to a rigorous model calculation 460.

[0069] In some embodiments, the information used to generate the continuous profile is provided by the optical signal and the optical signal threshold. Further, in some embodiments, the optical signal values are known for the entire optical signal field. Additionally, in some embodiments, the data is obtained from a rigorous simulator such as s-litho, or directly from wafer measurements. In such embodiments, the stochastic boundary for profile generation is determined based on equation 12.

[0070]

[0071] In some embodiments, whether a location is developed or not depends on the optical signal set around the location. For example, if the probability of the signal generates an optical signal that exceeds the optical signal threshold, the transformation is applied to the optical signal. In some embodiments, represents the cumulative probability, and is therefore monotonic. The probability is then determined by equation 13.

[0072]

[0073] wherein represents a contribution kernel. In some embodiments, the contribution kernel has been reduced to a vanishing value very far from the center. In some embodiments, locations near x have a higher contribution kernel compared to distance values. In some embodiments, the contribution kernel is represented by a Gaussian, a hyper / hyper- Gaussian, a Lorentz, a Voigt, etc. representation. Alternatively, in some embodiments, the contribution kernel is represented numerically. Equation 14 can determine a random band.

[0074]

[0075] In some embodiments, the method is repeated for a random band that infers the other side of the optical signal with a different optical signal threshold. If the contribution kernel is well known and the formation mechanism of the optical signal profile for all optical signal values is the same, then both random bands use the same contribution kernel, otherwise it can be necessary to determine a contribution kernel specific to the random band.

[0076] Figure 5 Measured edge profiles 510a and 510b for a first lithography pattern and a second lithography pattern, respectively, are illustrated in accordance with embodiments of the present disclosure. The first edge profile 510a is shown with respect to a strong pattern (also referred to as a positive pattern), where the edge (nominal) is formed at the edge of the applied photoresist pattern. The second edge profile 510b is shown with respect to a weak pattern (also referred to as a negative pattern), where the edge (nominal) is formed where no photoresist was applied (i.e., a negative image is formed with respect to the applied pattern). All edges in any lithography pattern (whether strong / weak or positive / negative) can appear in a different relative position than where the edge was placed on the mask. A random process variation causes each edge to be governed by a probability distribution. When the variance of this distribution is small, this manifests as LER, but is less likely to cause random failures (e.g., in the case of a nonconformance that occurs within a statistically or probabilistically measurable chance, but cannot be precisely predicted). When the variance is larger or when edges are closer together, random failures due to edge variation are more likely to occur. The probability distribution can appear to be a normal distribution, but this is not the case.

[0077] Monte Carlo simulations can provide a statistically accurate simulation for small areas (on the order of single pm 2 This type of simulation is resource intensive and is not practical for a full chip, which can be as large as 858 mm 2(mm2). This type of edge placement variation can be predicted at full chip level for EUVL processes, which is essential for the development of the process. The prediction model allows OPC techniques to compensate and design for the random effects in the chip pattern.

[0078] The present disclosure provides embodiments that simplify the modeling process by fitting a stochastic compact model directly to experimental data. The present disclosure also provides embodiments that reduce the time involved in fitting a stochastic compact model to experimental data.

[0079] As used herein, the CD of a feature refers to the width of the feature at a given location, and is measured by a scanning electron microscope (SEM). LER is a common problem in modern lithography processes, so the mean CD is often of most interest to the lithographer. LER is also measured by CD-SEM, and is calculated as the standard deviation of the edge placement. This number is often scaled by three standard deviations and referred to as the "3σ LER". Alternatively, the standard deviation of the CD can be used for scaling. In this case, the measurement is referred to as line width roughness (LWR) if the pattern is one-dimensional, or CDU if the pattern is two-dimensional. The present disclosure can sometimes use LER, LWR, CDU, or just "roughness" interchangeably. Unless otherwise noted, where one of these terms is applicable, the others are applicable as well.

[0080] As used herein, a compact model refers to a model that is fast to evaluate and applicable at full chip level for correction (via, e.g., optical proximity correction, inverse lithography techniques) or verification (e.g., lithography rule checks). A compact model takes as input a set of graphs representing a mask pattern. The output of the model is a signal that can be evaluated at any point in the plane bounded by the input graphs. The compact model also includes a threshold that can be used to find a contour from the model signal. The goal of the compact model is to predict the continuous mean edge placement of photoresist for the contour found from the model signal. In the present disclosure, a traditional compact model can be referred to as a continuous model, a continuous compact model, a persistent model, a persistent compact model, etc.

[0081] A stochastic compact model can be applied at full chip level, but in contrast to a continuous compact model, the goal of the stochastic compact model is to predict a probabilistic edge placement. In other words, a continuous model identifies a mean edge placement, while a stochastic model identifies a range of positions (e.g., edge placement range) in which the expected edge placement is to be formed within a given confidence interval. The present disclosure relates to fitting a stochastic compact model using experimental roughness data.

[0082] Figure 6This is a flowchart of a method 600 for calibrating a randomized compact model to experimental data according to embodiments of the present disclosure. Method 600 begins at 610, in which a modeling device acquires experimental data to calibrate a randomized model. In various embodiments, the modeling device may be a dedicated computer or a general-purpose computer, such as… Figure 13 The computer system 1300 discussed. The experimental data in this disclosure may include two or more distinct datasets collected from mask production runs, which identify process variations during mask production.

[0083] The first set includes data used to calibrate a continuous (traditional) compact model. This data may include (but is not limited to) CD measurements and contours extracted from SEM photomicrographs. The data in the first set may include a range of modified process conditions. For example, in addition to nominal conditions, the first dataset may also include data on defocus locations and exposure doses.

[0084] The second dataset, derived from the experimental dataset, is specific to calibrating stochastic compact models. These data include statistical measures of process variations, which may include (but are not limited to) linewidth roughness (LWR), linear edge roughness (LER), and critical dimensional uniformity (CDU). In addition to nominal conditions, these data include process conditions. Possible variations include, but are not limited to, defocusing, dosage, and photoresist thickness. These variations may include process conditions that are close to or significantly different from the nominal conditions. Therefore, collecting process conditions that replicate the full range of LER variability can enhance the fitting of process and stochastic models.

[0085] For the two different pattern types on the mask, Figure 7A and Figure 7B The document shows examples of data types recommended for fitting stochastic compact models. In this case, CD data 710a and 710b are used to calibrate continuous compact models, while linear edge roughness (LER) data 720a and 720b are used to calibrate stochastic compact models.

[0086] Measurements were performed on various mask geometries across two experimental datasets. Each measurement site on the mask is referred to as a gauge in this paper. The various mask geometries were randomly selected, including user-defined subsets of mask geometries, all mask geometries accessible to the modeling device, and so on.

[0087] The high-energy electron beam of the SEM can alter the photoresist pattern being measured. To account for this issue, calibration data for one or both of the continuous model and the stochastic model can optionally be collected using different SEM settings (e.g., number of scans, electron beam energy, etc.). The differences between various measurements of the same pattern under different SEM settings can give information about the impact of the SEM on the measurements. For example, measuring the same pattern twice using two different electron beam energies can give different CD values. The differences between these measurements are due to the SEM and are unique to each pattern. This information can be used to correct the CD measurements, LWR measurements, LER measurements, and CDU measurements before the subsequent operations of the method 600.

[0088] At 620, the modeling device calibrates a continuous compact model of the resist mask based on the experimental data. In one embodiment, the continuous compact model represents a continuous and smooth function to describe the average edge position observed on different measurement locations in the production data across various specified locations on the wafer design. The gauge used to calibrate the continuous model can be, but is not necessarily, the same gauge used for stochastic calibration. For example, a model that is fit directly to SEM profiles can be calibrated in a different manner than a model that is fit to discrete CD measurements. Ultimately, either model is compatible with the present disclosure, and the calibration process of the continuous model does not affect the systems and methods described herein.

[0089] After collecting the data and calibrating the continuous compact model, the method 600 proceeds to 630, where the modeling device evaluates the continuous model based on the patterns corresponding to the gauges to be used to calibrate the stochastic compact model (per block 660). In this case, the patterns refer to the digital representation of the mask design. This simulation is used to extract image parameters from the stochastic calibration gauges. The calibration and extraction can include, but are not limited to, edge position, maximum signal along the gauge, slope at the mask edge, etc. Any quantity that can be derived from the compact model signal can be collected during calibration and used for de-biasing, edge placement, simulation, and fabrication.

[0090] When evaluating the continuous model, the modeling device calibrates the continuous model of pairs of patterns (e.g., vertical line images with relatively high print probability) and measured gauges (e.g., critical dimension measurements) to compute a continuous signal field for new patterns (e.g., vertical line images with relatively low print probability) and measured gauges (e.g., critical dimension and line edge roughness measurements), including stochastic measurements of edge position. The stochastic measurements indicate the probability of an edge position occurring at a given physical location, and are used by the modeling device to identify and map functions that describe these stochastic characteristics.

[0091] In various embodiments, although a subset of the entire kernel set can be used to perform calibration, a compact model can be composed of several overlapping convolution kernels. For example, a typical compact model contains a kernel corresponding to the optics of the lithographic process and other kernels corresponding to the photoresist. Signals derived using only the optical kernel are considered a subset of the continuous compact model. Thus, many different signals can be computed from the compact model to calculate the image parameters.

[0092] At 640, the modeling device de-biases the noise from the LER measurements. All experimental measurements will include some degree of noise. The largest source of noise comes from the SEM, but this is not the only source. SEM noise is tolerable when fitting a traditional compact model, but can severely impact the ability to fit a stochastic compact model.

[0093] A common practice in lithography is to analyze process variations in the frequency domain, as shown in Figures 8A-8D These curves can be generated by transforming the edge profile into the frequency domain via a Fourier transform, and then computing the power spectral density (PSD). The PSD obtained from a single line can be noisy, but can be improved by averaging the PSDs of many nominally identical features, as shown in Figures 8A-8C The areas of the regions 810a-c are shown in. The first region 810a corresponds to the PSD obtained from a single line, the second region 810b corresponds to the PSD obtained from one hundred lines, and the third region 810c corresponds to the PSD obtained from one thousand lines. As will be seen, the area of the regions 810a-c is indicative of the noise level in the measurement, which decreases as the number of lines measured increases.

[0094] The area under the PSD is related to the variance of the process. Thus, this area is not directly equal to the LER or LWR measured by the CD-SEM, but is related to it. Nonetheless, there is a point in the high frequency regime where the PSD flattens out. This point corresponds to the noise in the process, and is referred to as the noise floor. This is illustrated in Figure 8D , where several ideal PSDs 820, 830, 840 are plotted for different roughness values (e.g., the highest roughness value in ideal PSD 840, the middle roughness value in ideal PSD 830, the lowest roughness value in ideal PSD 820), as well as the noise floor 850.

[0095] In one embodiment of the disclosure, the modeling device removes the noise from the measurements prior to model calibration. One possible way to eliminate the noise is to assume that the noise is constant between measurements, and thus represents a constant area under the PSD. Then, the unbiased measurements can be found according to Equation 15 below:

[0096] σ unbiased = √(σ 2 biased – σ2 Noise ) Equation 15

[0097] The present disclosure does not require the use of a specific de-biasing procedure, but can remove process noise from the measurements through various procedures. In various embodiments, de-biasing can be omitted entirely.

[0098] At 650, the modeling device selects the edge distribution. Many different probability distributions can arise in different cases. Each source of random variation in the lithography process can have a separate and unique distribution. The statistical central limit theorem states that the distribution of the sum of independent random processes tends toward a normal distribution. However, this does not mean that the edge distribution should be assumed to be normally distributed. Random events in the lithography process can not be statistically independent, and the central limit theorem only predicts a tendency of the final distribution.

[0099] In practice, edge distributions such as that shown in Figure 5 may be close to normal distributions, but slightly skewed. Distributions can be compared to normal distributions by comparing quantiles of the distributions to quantiles of a normal distribution. This is referred to as a QQ plot, and is shown for two different distributions in Figure 9A and Figure 9B .

[0100] In Figure 9A , the normal distribution of the measurements 910 is compared to a normal distribution, so the QQ plot is perfectly linear. In Figure 9B , the same analysis is performed on the skew-normal distribution of the measurements 920. In this case, the skewness of the distribution causes some quantiles to occur earlier or later than those that would occur in a normal distribution.

[0101] Thus, the present disclosure does not require the assumption of normality of the distributions. However, in some embodiments, the type of edge probability distribution is selected a priori. Possible types include, but are not limited to, normal, skew-normal, and Poisson. In one embodiment, each edge is allowed to have a separate probability distribution. In other embodiments, the two edges of each gage can have the same distribution, but each gage has a different distribution. In yet alternative embodiments, each gage shares the same distribution.

[0102] At 660, the modeling device maps the image parameters (e.g., from 630) to the edge distribution parameters (selected at 650) according to a functional description of the distribution of edge positions. The functional description models the variation or perturbation of the mean position value (represented in a continuous compact model), and can be represented as two functions (one for positive variation, one for negative variation) or a non-symmetric function to account for different behaviors that cause the edge position range to be higher or lower than the mean value. Figure 10Three possible mappings 1010a, 1010b, 1010c are illustrated, which provide the modeling device with the ability to match experimental data from the input space 1020 to non-Gaussian distributions in the respective output spaces 1030a-c using selected mean, standard deviation, variance, and skewness, which correspond to different edge profile parameters according to the selected functional description.

[0103] At 670, the modeling device scales the position measurements and probabilistically computes a new edge position range based on the selected and mapped edge profile. In various embodiments, the modeling device scales the position measurements from 1σ to a desired amount of deviation (e.g., xσ) to determine where the edge can extend to within a desired probability range. In other words, the collected data is extrapolated out based on the confidence band selected for the stochastic compact model. The new edge position range indicates the outer bounds of the predicted edge position within the desired amount of deviation from the mean.

[0104] At 680, the modeling device sets (e.g., calibrates) the new threshold and accordingly sets (e.g., calibrates / optimizes) the stochastic model parameters. In other words, at 680, the modeling device applies the stochastic compact model to the resist mask design to allow the developer to adjust the resist mask to produce a chip with a confidence of manufacturing yield. The threshold indicates a position in the profile of the optical signal that the parameters of the stochastic model are adjusted to ensure that the optical signal is able to pass through the chip in the specified manner. The method 600 can perform 680 for several iterations (e.g., in an optimization loop), adjusting and updating the parameters and / or threshold of the stochastic compact model until the output signal from the stochastic compact model meets the design requirements of the new threshold. Once the stochastic model is complete, the modeling device can output the stochastic model to the fabricator to produce a chip or wafer using the resist mask modeled by the stochastic model.

[0105] Figure 11 is a flowchart of a method 1100 for applying a stochastic compact model according to embodiments of the present disclosure. Figure 11 A re-interpretation of the methods discussed with respect to Figure 6 The concepts described with respect to the method 600 can be used in conjunction with the concepts described with respect to the method 1100, and vice versa.

[0106] Method 1100 begins at 1110, where the modeling device generates a continuous compact model of a mask pattern calibrated to production data. For several measurement sites in a wafer design, the continuous compact model identifies a mean of those locations where an edge of the applied mask pattern has been observed (e.g., by SEM) over a range of production runs. For a given mask design using various dopant concentrations, exposure times / intensities, and PEB times / heat, these production data can include several different locations of the edge that are averaged to identify a mean location of the edge used in the continuous compact model. In various embodiments, the continuous compact model can be based on profiles measured via SEM or discrete measurements of CD at particular locations by SEM.

[0107] For example, the modeling device can measure the locations of edge positions (averaged) at X production runs at Y different measurement sites in a wafer design to identify a corresponding Y different mean for the edge position at each corresponding measurement site.

[0108] Thus, in generating the continuous compact model, the modeling device can measure edge positions of a resist mask pattern at multiple measurement sites on several wafers. From these measurements taken across different production runs, the modeling device determines a mean position of the resist mask pattern at each of the multiple measurement sites and maps or fits a function to the determined means to define a continuous compact model for the edge position.

[0109] At 1120, the modeling device evaluates production data according to the continuous compact model. In one embodiment, the continuous compact model represents a continuous and smooth function that describes the mean edge positions observed across different measurement sites in production data at various specified locations on a wafer design. In various embodiments, the continuous model uses a symmetric function that attempts to describe variance from mean position values using the same functionality of positive and negative changes.

[0110] Various edge positions used to determine the mean edge position can be influenced by various parameters, which the modeling device can derive from the production data. For example, the modeling device can derive parameters for the following: a gradient of the continuous model at various points, a slope (or its direction) at a given point, an extreme (high or low) value in a data set (e.g., a global maximum / minimum) or a subset of the data set (e.g., a local maximum / minimum), a contrast (e.g., (maximum - minimum) / (maximum + minimum)) within a given region between a local maximum and a local minimum, etc. Additionally or alternatively, various parameters can include various production parameters including various dopant concentrations, exposure times / intensities, and PEB times / heat, etc.

[0111] At 1130, the modeling device selects a functional description of how to probabilistically represent edge position as a range from a mean to define a stochastic compact model for the mask pattern. The functional description models variation or perturbation of the mean position value (represented in the continuous compact model) and can be represented as two functions (one for positive variation from the mean and one for negative variation from the mean) or one asymmetric function to account for different behavior in producing ranges of edge position above or below the mean. In various embodiments, the functional description selected to model statistical variation from the mean uses one or more of the parameters (identified from 1120) as variables to describe the probabilistic position of the edge. As such, the selected function can be a univariate or multivariate function to probabilistically model edge position in the stochastic compact model.

[0112] At 1140, the modeling device maps the evaluated production data to the functional description (selected from 1130). In various embodiments, the modeling device selects the same locations in the production data (from 1120) used to evaluate the continuous model, can use different locations or use a combination of at least one of the same locations and at least some different locations to use in mapping the production data to the functional description (selected from 1130).

[0113] In some embodiments, the modeling device uses a subset of the measurement locations used in developing the continuous model (e.g., some but not all) when mapping the evaluated production data to the stochastic model. For example, when the modeling device maps production data from Y different measurement sites to the continuous model, the modeling device can use production data from fewer than Y measurement sites. The selected measurement sites can include an interpolated subset of the continuous model measurement sites (e.g., every other one of the Y measurement sites), a randomly selected subset of the measurement sites, or some other metric or selection criteria.

[0114] In various embodiments, mapping the evaluated production data can include de-biasing for noise from the LER measurements in the stochastic compact model. Noise sources include SEM and process variation from multiple different production runs, which the modeling device can de-bias using a constant noise value (e.g., an estimated constant noise) or other methods.

[0115] At 1150, the modeling device generates a stochastic compact model that probabilistically models edge position of a resist mask as a range from a mean based on the production data mapped to the functional description. In various embodiments, the mapped production data is scaled to a specified standard deviation (e.g., xσ) from the production data to define locations where the edge is within a desired confidence interval (e.g., 99.7% of the time for 3σ, 99.99% of the time for 4σ, etc.).

[0116] At 1160, the modeling device provides the stochastic compact model to a production device to fabricate a wafer using the resist mask modeled according to the stochastic compact model. In various embodiments, the production device receives the stochastic compact model from the modeling device, or can operate as a combined modeling and production device. Once received, the production device uses the stochastic compact model to determine whether the entire wafer or certain identifiable areas on the wafer can be produced via a single exposure EUV patterning. While some non-conformities are expected (e.g., micro-bridges / missing contacts or broken / merged contacts due to incorrect application of edges of the resist pattern), and can be part of a conforming wafer if the number is not too large, when the number of non-conformities is expected to exceed a failure threshold, the production device indicates that the wafer (or its identified areas) cannot be produced within a given confidence level, and alerts the designer of the expected production failure.

[0117] Various features are described in the disclosure with reference to the drawings. It should be noted that the drawings can be drawn to scale or not to scale, and that similar structural or functional elements are denoted by similar reference numbers throughout the several drawings. It should be noted that the drawings are merely intended to facilitate description of the features. They are not intended as an extensive description of the claimed subject matter or the scope thereof. In addition, the illustrated examples need not necessarily illustrate all aspects or advantages. Aspects or advantages described in connection with a particular example need not necessarily be limited to such example, and can be practiced in any of the other examples even if not so illustrated or described. Further, methods described herein can be described in a particular order, but other methods according to other examples can be implemented in various other orders (e.g., including performing various operations in different serial or parallel orders).

[0118] Further, various terms are used herein as used in the art. For example, “optimization,” “optimize,” and “optimizing” refer to a mathematical formulation of a problem for selecting some improvement (if an improvement is available) of an identified certain feature within a structure of an implemented algorithm, and does not imply an absolute or global optimum improvement of the feature (as the term is typically used in spoken language). For example, in some situations where optimization can determine a minimum, the minimum can be a local minimum, rather than a global minimum.

[0119] As described in further detail below, the methods described herein can be embodied by one or more instruction sets stored on a non-transitory computer readable medium, which can be one or more software modules. One or more processors of a computer system can be configured to read and execute the one or more instruction sets, which cause the one or more processors to perform the various operations or steps of the methods. More details are provided below. In some examples, some operations or steps of the methods can be embodied as one or more instruction sets of one or more software modules, and other operations or steps of the methods can be embodied as one or more other instruction sets of one or more other software modules. In some examples, different software modules can be distributed and stored on different non-transitory computer readable media on different computer systems for execution by respective one or more processors of the different computer systems.

[0120] Those of ordinary skill in the art will readily appreciate various data structures that can be implemented in the above-described processes. For example, a mask object class can be defined for polygons of a mask pattern and / or edges of a polygon. Likewise, a database or other storage structure can be implemented to store data for PLTs, Jacobian matrices, and / or CFGs. Different data structures and / or modified data structures can be used in different examples.

[0121] Additionally, those of ordinary skill in the art will readily appreciate various modifications to the logical and / or mathematical expressions of the examples described herein. For example, terms such as vectors, tables, and matrices are generally considered to be mathematical expressions, while related terms such as columns and rows can likewise be organization within the mathematical expressions, and can be changed to different organization. Other examples contemplate such modifications.

[0122] Figure 12 An example process set 1200 for transforming and verifying processes representing design data and instructions for integrated circuits is illustrated during design, verification, and fabrication of integrated circuits on semiconductor dies. Each of these processes can be structured and implemented as a plurality of modules or operations. The term "EDA" means electronic design automation. These processes begin at 1210, where product ideas are created using information supplied by designers. At 1212, this information is transformed to create an integrated circuit using a set of EDA processes. At 1234, when the design is complete, the design can be released, which is sending the artwork (e.g., geometric patterns) for the integrated circuit to a fabrication plant to manufacture a set of masks, which are then used to manufacture the integrated circuit. After release, at 1236, the integrated circuit is fabricated on a semiconductor die. At 1238, packaging and assembly processes are performed to produce a finished integrated circuit (also commonly referred to as a "chip" or "integrated circuit chip") at 1240.

[0123] The specification range of a circuit or electronic structure can range from a low-level transistor material layout to a high-level description language. High-level representations can be used to design circuits and systems using a hardware description language (HDL) such as VHDL, Verilog, SystemVerilog, SystemC, MyHDL, or OpenVera. The HDL description can be transformed into a logic-level register transfer level (RTL) description, a gate-level description, a layout-level description, or a mask-level description. Each lower level of representation description that is more detailed adds more useful detail (e.g., more detail of the modules included in the description) to the design description. The lower level of representation description that is more detailed can be generated by a computer, derived from a design library, or created by another design automation process. An example of a specification language in a lower level of representation language used to specify a more detailed description is SPICE, which is used to describe a circuit in detail using many similar components. Implementations of the description in each level of representation are used by corresponding tools (e.g., formal verification tools) of that layer. The design process can use Figure 12 the order depicted in the system. The described processes can be implemented by EDA products (or tools).

[0124] During system design, at 1214, the functionality of the integrated circuit to be manufactured is specified. The design can be optimized for desired characteristics such as power consumption, performance, area (physical and / or lines of code), and loss reduction. At this stage, the design can be partitioned into different types of modules or components.

[0125] During logic design and functional verification, at 1216, the modules or components in the circuit are specified in one or more description languages, and the specification is checked for functional accuracy. For example, the components of a circuit can be verified to generate outputs that match the specification requirements of the designed circuit or system. Functional verification can use simulators and other programs such as testbench generators, static HDL checkers, and formal verifiers. In some examples, a special component system, called a “simulator” or “prototyping system,” is used to speed up functional verification.

[0126] During synthesis and test design, at 1218, the HDL code is transformed into a netlist. In some examples, the netlist can be a graph structure where the edges of the graph structure represent the components of the circuit and where the nodes of the graph structure represent the way the components are interconnected. Both the HDL code and the netlist are layered artifacts that EDA products can use to verify that the integrated circuit performs according to the specified design when manufactured. The netlist can be optimized for the target semiconductor manufacturing technology. Additionally, the finished integrated circuit can be tested to verify that the integrated circuit meets the specification requirements.

[0127] During netlist verification, at 1220, the netlist is checked for compliance with timing constraints and correspondence with the HDL code. During design planning, at 1222, the overall floorplan of the integrated circuit is constructed and analyzed for timing and top-level routing.

[0128] During layout or physical implementation, at 1224, physical placement (positioning of circuit components such as transistors or capacitors) and routing (connections of circuit components through multiple conductors) occur, and selection of cells from a library to implement particular logic functions can be performed. As used herein, the term "cell" can designate a set of transistors, other components, and interconnections that provide a Boolean logic function (e.g., AND, OR, NOT, XOR) or a storage function (e.g., a flip-flop or latch). As used herein, a circuit "block" can refer to two or more cells. Both cells and circuit blocks can be referred to as modules or components, and can be used as both physical structures and simulations. The selected cells (based on standard cells) are designated with parameters such as size, and these parameters are made accessible in a database for use by EDA products.

[0129] During analysis and extraction, at 1226, the circuit functionality is verified at the layout level, which permits refinement of the layout design. During physical verification, at 1228, the layout design is checked to ensure that manufacturing constraints are correct, such as design rule check (DRC) constraints, electrical constraints, photolithography constraints, and that the circuit system functionality matches the HDL design specification. During resolution enhancement, at 1230, the geometry of the layout is transformed to improve the way the circuit design is manufactured.

[0130] During release, data is created for production of photolithographic masks (if appropriate, after application of photolithography enhancements). During mask data preparation, at 1232, the release data is used to produce photolithographic masks that are used to produce the finished integrated circuits.

[0131] The storage subsystem of a computer system, such as the computer system 1300 of Figure 13 The storage subsystem of a computer system, such as the computer system 1300 of

[0132] Figure 13An example computer system 1300 is illustrated, within which a set of instructions, for causing the computer system to perform any one or more of the methodologies discussed herein, can be executed. In some implementations, the computer system can be connected (e.g., networked) to other machine or computer systems within a local area network (LAN), an intranet, an extranet, and / or the Internet. The computer system can operate in the capacity of a server or a client computer system in client-server network environment, as a peer computer system in peer-to-peer (or distributed) network environment, or as a server or a client computer system in a cloud computing infrastructure or environment.

[0133] The computer system can be a personal computer (PC), a tablet PC, a set-top box (STB), a Personal Digital Assistant (PDA), a cellular telephone, a web appliance, a server, a network router, switch or bridge, or any machine capable of executing a set of instructions (sequential or other wise) that specify actions to be taken by that machine. Further, while a single computer system is illustrated, the term "computer system" shall also be taken to include any collection of computer systems that individually or jointly execute a set (or multiple sets) of instructions to perform any one or more of the methodologies discussed herein.

[0134] The example computer system 1300 includes a processing device 1302, a main memory 1304 (e.g., read-only memory (ROM), flash memory, dynamic random access memory (DRAM) such as synchronous DRAM (SDRAM), a static memory 1306 (e.g., flash memory, static random access memory (SRAM), etc.), and a data storage device 1318, which communicate with one another via a bus 1330. The main memory 1304 includes or is a non-transitory computer-readable medium. The main memory 1304 (e.g., the non-transitory computer-readable medium) can store one or more sets of instructions 1326 comprising instructions for implementing functionality of the present disclosure, including instructions for implementing the functionality of the example methods 1302. The instructions 1326 can also be stored and / or transported within any computer-readable medium for implementation by or to

[0135] Processing device 1302 represents one or more processors such as a microprocessor, a central processing unit (CPU), or the like. More particularly, the processing device 1302 can be or include a complex instruction set computing (CISC) microprocessor, reduced instruction set computing (RISC) microprocessor, very long instruction word (VLIW) microprocessor, processor implementing other instruction sets, or one or more processors implementing a combination of instruction sets. Processing device 1302 can also be one or more special-purpose processing devices such as an application specific integrated circuit (ASIC), a field programmable gate array (FPGA), a digital signal processor (DSP), network processor, or the like. Processing device 1302 can be configured to execute instructions 1326 for performing some or all of the operations, steps, methods, and processes described herein.

[0136] Computer system 1300 can also include a network interface device 1308 to communicate over the network 1320. The computer system 1300 can also include a video display unit 1310 (e.g., a liquid crystal display (LCD) or a cathode ray tube (CRT)), an alphanumeric input device 1312 (e.g., a keyboard), a cursor control device 1314 (e.g., a mouse), a graphics processing unit 1322, a signal generation device 1316 (e.g., a speaker), a graphics processing unit 1322, a video processing unit 1328, and an audio processing unit 1332.

[0137] The data storage device 1318 can include a machine-readable storage medium 1324 (e.g., a non-transitory computer-readable medium) to store one or more sets of instructions 1326 or software embodying any one or more of the methodologies or functions described herein. The instructions 1326 can also reside completely, or at least partially, within the main memory 1304 and / or within the processing device 1302 during execution thereof by the computer system 1300, the main memory 1304 and the processing device 1302 also including machine-readable storage media. The instructions 1326 can be transmitted or received over a network 1320 via the network interface device 1308.

[0138] In some implementations, the instructions 1326 include instructions to implement a function described above. Although the machine -readable storage medium 1324 is shown in an example implementation to be a single medium, the term "machine-readable storage medium" should be taken to include a single medium or multiple media (e.g., a centralized or distributed database, and / or associated caches and servers) that store the one or more sets of instructions. The term "machine-readable storage medium" shall also be taken to include any medium that is capable of storing or encoding a set of instructions for execution by the computer system and that cause the computer system and the processing device 1302 to perform any one or more of the methods described above. The term "machine-readable storage medium" shall accordingly be taken to include, but not be limited to, solid-state memories, optical media, and magnetic media.

[0139] Some portions of the preceding detailed descriptions have been presented in terms of algorithms and symbolic representations of operations on data bits within a computer memory. These algorithmic descriptions and representations are the means used by those skilled in the data processing arts to most effectively convey the substance of their work to others skilled in the art. An algorithm is here, and generally, is considered to be a self- consistent sequence of operations that leads to a desired result. The operations are those requiring physical manipulations of physical quantities. Usually, though not necessarily, these quantities take the form of electrical or magnetic signals capable of being stored, combined, compared, and otherwise manipulated. It has been convenient at times, principally for reasons of common usage, to refer to these signals as bits, values, elements, symbols, characters, terms, numbers, or the like.

[0140] It should be borne in mind, however, that all of these and similar terms are to be associated with the appropriate physical quantities and are merely convenient labels applied to these quantities. Unless specifically stated otherwise as apparent from the prior discussion, it is appreciated that throughout the description, certain terms refer to the action and processes of a computer system, or similar electronic computing device, that manipulates and transforms data represented as physical (electronic) quantities within the computer system's registers and memories into other data similarly represented as physical quantities within the computer system memories or registers or other such information storage devices.

[0141] The present disclosure also relates to an apparatus for performing the operations herein. This apparatus can be specially constructed for the intended purposes, or it can include a computer selectively activated or reconfigured by a computer program stored in the computer. Such a computer program can be stored in a computer readable storage medium, such as, but not limited to, any type of disk including floppy disks, optical disks, CD-ROMs, and magnetic-optical disks, read-only memories (ROMs), random access memories (RAMs), EPROMs, EEPROMs, magnetic or optical cards, or any type of media suitable for storing electronic instructions, each coupled to a computer system bus.

[0142] The algorithms and displays presented herein are not inherently related to any particular computer or other apparatus. Various other systems can be used with programs in accordance with the teachings herein, or it can prove convenient to construct a more specialized apparatus to perform the method. In addition, the present disclosure is not described with reference to any particular programming language. It will be appreciated that a variety of programming languages can be used to implement the teachings of the disclosure as described herein.

[0143] The present disclosure can be provided as a computer program product, or software, that can include a machine-readable medium having stored thereon instructions, which can be used to program a computer system (or other electronic devices) to perform a process according to the present disclosure. A machine-readable medium includes any mechanism for storing information in a form accessible by a machine (e.g., a computer). For example, a machine-readable (e.g., computer-readable) medium includes a machine (e.g., a computer) readable storage medium such as a read only memory (ROM), random access memory (RAM), magnetic disk storage media, optical storage media, flash memory devices, etc.

[0144] In the foregoing disclosure, implementations of the present disclosure have been described with reference to specific example implementations of the present disclosure. It is evident that various modifications can be made thereto without departing from the broader scope of the present disclosure as set forth in the following claims. In some instances, reference has been made to particular means for implementing one or more embodiments of the present disclosure. Such means are stated in terms of specific embodiments of the application. However, it is to be understood that the disclosure is not limited to the specific means disclosed herein. Rather, many modification and variations of the means described herein are possible, and all such modifications and variations are intended to be within the scope of the present disclosure. Where the disclosure refers to particular means for implementing one or more embodiments of the present disclosure, such means are stated in terms of specific embodiments of the application. However, it is to be understood that the disclosure is not limited to the specific means disclosed herein. Rather, many modification and variations of the means described herein are possible, and all such modifications and variations are intended to be within the scope of the present disclosure. Where the disclosure refers to particular elements of the present disclosure, and multiple instances of such means are described, the instances can be used either interchangeably or independently of one another.

Claims

1. A method comprising: obtaining data of process variation in production of a resist mask; calibrating a continuous compact model of the resist mask based on the data; evaluating the continuous compact model against a stochastic compact model based on the data; selecting a functional description of edge position distribution of the stochastic compact model; mapping image parameters from the evaluation to edge distribution parameters according to the functional description; determining an edge position range for the stochastic compact model based on a scaling measure from the image parameters; calibrating a threshold value of the resist mask and updating parameters of the stochastic compact model to reduce a difference between the data and modeled linear edge roughness, LER, values; and outputting the stochastic compact model.

2. The method of claim 1, further comprising: producing a resist mask for a chip according to the stochastic compact model.

3. The method of claim 1, further comprising: de-biasing noise according to LER measurements in the stochastic compact model, wherein the noise is de-biased from the LER measurements by: removing a constant noise value from the LER measurements based on a noise standard deviation.

4. The method of claim 1, wherein the edge position range indicates an outer bound of predicted edge positions within a desired deviation from a mean position in the data.

5. The method of claim 1, wherein the image parameters are converted to the scaling measure according to a specified standard deviation from the data.

6. The method of claim 1, wherein the functional description is selected to be an asymmetric function that describes deviations above a mean from the data differently than deviations below the mean.

7. The method of claim 1, wherein the modeled LER values indicate a stochastic confidence interval from a mean calculated from the data.

8. A method comprising: generating a continuous compact model for a resist mask pattern, the resist mask pattern calibrated to production data; evaluating the production data according to the continuous compact model; selecting a functional description to represent an edge position range of the resist mask pattern in a stochastic compact model of the resist mask pattern; mapping the evaluated production data to the selected functional description; and generating the stochastic compact model that probabilistically models the edge position range of the resist mask pattern based on the mapped production data.

9. The method of claim 8, further comprising: manufacturing wafers using the resist mask pattern as modeled by the stochastic compact model.

10. The method of claim 8, wherein the mapped production data is scaled to a specified standard deviation from the production data. de-biasing noise according to linear edge roughness, LER, measurements in the stochastic compact model.

11. The method of claim 8, wherein mapping the evaluated production data further comprises:

12. The method of claim 8, wherein generating the continuous compact model further comprises: ​ measuring edge positions of the resist mask pattern on a plurality of wafers patterned according to the resist mask pattern at a plurality of measurement sites across the plurality of wafers; determining a mean position of the resist mask pattern at each of the plurality of measurement sites; and mapping a function to the mean positions to predict a continuous mean edge position in the resist mask pattern.

13. The method of claim 8, wherein the function description is selected to be an asymmetric function that describes a range of positions within a given confidence interval from the mean of edge positions from the production data in a different manner for deviations above the mean than for deviations below the mean.

14. A system comprising: a processor; and a memory comprising instructions that, when executed by the processor, are configured to perform operations comprising: obtaining data of process variations in producing resist masks; calibrating a continuous compact model of the resist mask based on the data; evaluating the continuous compact model against a stochastic compact model based on the data; selecting a function description of an edge position distribution of the stochastic compact model; mapping an image parameter from the evaluation to an edge distribution parameter according to the function description; computing an edge position range for the stochastic compact model based on a scaled measurement from the image parameter; calibrating a threshold of the resist mask and updating parameters of the stochastic compact model to reduce a difference between the data and modeled linear edge roughness, LER, values; and outputting the stochastic compact model.

15. The system of claim 14, wherein the operations further comprise: producing a resist mask for a chip according to the stochastic compact model.

16. The system of claim 14, further comprising: de-biasing noise from LER measurements in the stochastic compact model, wherein the noise is de-biased from the LER measurements by: removing a constant noise value from the LER measurements based on a noise standard deviation.

17. The system of claim 14, wherein the edge position range indicates an outer boundary of a predicted edge position within a desired number of deviations from a mean position in the data.

18. The system of claim 14, wherein the mapped image parameter is converted to the scaled measurement according to a specified standard deviation from the data.

19. The system of claim 14, wherein the function description is selected to be an asymmetric function that describes deviations above the mean from the data in a different manner than describing deviations below the mean.

20. The system of claim 14, wherein the modeled LER values indicate a stochastic confidence interval from a mean calculated according to the data. ​

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