Methods and associated apparatus for fitting measured data to models and modeling performance parameter distributions

Abstract

A method of fitting measurement data to a model is disclosed. The method includes: obtaining measurement data associated with a performance parameter of at least a portion of a substrate; and fitting the measurement data to the model by minimizing a complexity index of fitting parameters applied to the model while not allowing a deviation between the measurement data and the fitted model to exceed a threshold.

Description

[0001] CROSS-REFERENCE TO RELATED APPLICATIONS

[0002] This application claims the priority of European application No. 19203752.1 filed on October 17, 2019 and European application No. 20193618.4 filed on August 31, 2020, the entire contents of which are incorporated herein by reference. Technical Field

[0003] The present invention relates to methods and apparatus for applying a pattern to a substrate in a lithographic process. Background Art

[0004] A lithographic apparatus is a machine that applies a desired pattern to a substrate (usually to a target portion of the substrate). A lithographic apparatus can be used, for example, to manufacture an integrated circuit (IC). In this case, a pattern forming device, alternatively referred to as a mask or reticle, can be used to produce a circuit pattern to be formed on a separate layer of the IC. The pattern can be transferred to a target portion (e.g., a portion including a die, a die, or several dies) on a substrate (e.g., a silicon wafer). The transfer of the pattern is usually performed by imaging onto a layer of radiation-sensitive material (resist) provided on the substrate. In general, a single substrate will contain a network of adjacent target portions that are patterned continuously. Known lithographic apparatus include: a so-called stepper, in which each target portion is irradiated by exposing the entire pattern to the target portion at one time; and a so-called scanner, in which each target portion is irradiated by scanning the substrate synchronously parallel or antiparallel to a given direction (the "scanning" direction) while scanning the pattern by a radiation beam. It is also possible to transfer the pattern from the pattern forming device to the substrate by imprinting the pattern onto the substrate.

[0005] In order to monitor the lithography process, parameters of the patterned substrate are measured. The parameters may include, for example, the overlay error between successive layers formed in or on the patterned substrate and the critical line width (CD) of the developed photoresist. The measurement may be performed on a production substrate and / or on a dedicated measurement target. There are various techniques for measuring microstructures formed during the lithography process, including the use of scanning electron microscopes and various specialized tools. A fast and non-invasive form of specialized inspection tool is a scatterometer, in which a radiation beam is directed onto a target on the surface of a substrate and the properties of the scattered or reflected beam are measured. Two main types of scatterometers are known. Spectral scatterometers direct a broadband radiation beam onto a substrate and measure the spectrum (intensity as a function of wavelength) of the radiation scattered into a specific narrow angular range. Angular resolved scatterometers use a monochromatic radiation beam and measure the intensity of the scattered radiation as a function of angle.

[0006] Examples of known scatterometers include angle-resolved scatterometers of the type described in US2006033921A1 and US2010201963A1. The target used by such scatterometers is a relatively large (e.g., 40 μm×40 μm) grating, and the measurement beam produces a light spot that is smaller than the grating (i.e., the grating is underfilled). In addition to measurement of feature shape by reconstruction, such devices can also be used to measure diffraction-based overlay, as described in published patent application US2006066855A1. Diffraction-based overlay measurement using dark field imaging of diffraction orders enables overlay measurement of smaller targets. Examples of dark field imaging measurements can be found in international patent applications WO2009 / 078708 and WO2009 / 106279, the entire contents of which are hereby incorporated by reference into this document. Further developments of this technology have been described in published patent publications US20110027704A, US20110043791A, US2011102753A1, US20120044470A, US20120123581A, US20130258310A, US20130271740A and WO2013178422A1. These targets can be smaller than the illumination spot and can be surrounded by product structures on the wafer. Multiple gratings can be measured in one image using a composite grating target. The contents of all these applications are also incorporated herein by reference.

[0007] When performing a lithographic process such as applying a pattern to a substrate or measuring such a pattern, process control methods are used to monitor and control the process. Such process control techniques are typically performed to obtain corrections to the control of the lithographic process. It would be desirable to improve such process control methods. Summary of the invention

[0008] In a first aspect of the present invention, a method for fitting measurement data to a model is provided, the method comprising: obtaining measurement data associated with performance parameters of at least a portion of a substrate; and fitting the measurement data to the model by minimizing a complexity index of fitting parameters applied to the model while not allowing the deviation between the measurement data and the fitted model to exceed a threshold.

[0009] In a second aspect of the present invention, a method for modeling a performance parameter distribution is provided, the method comprising: obtaining measurement data associated with a performance parameter of at least a portion of a substrate; and modeling the performance parameter distribution based on the measurement data by optimizing the model, wherein the optimization minimizes a cost function representing the complexity of the modeled performance parameter distribution subject to the following constraint: substantially all points included in the measurement data are within a threshold from the modeled performance parameter distribution.

[0010] In other aspects of the invention, there is provided: a computer program comprising program instructions operable to perform a method as described in the first aspect when run on a suitable device; a processing device comprising a processor and a memory having such a computer program; and a lithographic device having such a processing device.

[0011] Other aspects, features and advantages of the present invention, as well as the structure and operation of various embodiments of the present invention are described in detail below with reference to the accompanying drawings. It should be noted that the present invention is not limited to the specific embodiments described herein. Such embodiments are presented in the present invention only for illustrative purposes. Based on the teachings contained herein, additional embodiments will be apparent to those skilled in the relevant art. BRIEF DESCRIPTION OF THE DRAWINGS

[0012] Embodiments of the present invention will now be described by way of example with reference to the accompanying drawings, in which:

[0013] Figure 1 Depicts a lithographic apparatus along with other equipment forming a production facility for semiconductor devices;

[0014] Figure 2 shows exemplary sources of processing parameters;

[0015] Figure 3 schematically illustrates a current method of determining a correction for controlling a lithographic apparatus;

[0016] Figure 4 is an overlay graph conceptually illustrating support vector machine regression optimization;

[0017] Figure 5 (a) and Figure 5 (b) Cumulative yield curves of percentage yield relative to overlay error in the x-direction and y-direction, respectively;

[0018] Figure 6 is a conceptual diagram describing the “model assumptions” that map between input space and feature space and fit from feature space to output space; and

[0019] Figure 7 is a graph of the output space OS (values ​​for the parameter of interest) versus the input space IS (wafer locations) for actual distinctive signatures and KB SVM estimations obtained according to an embodiment of the present invention. DETAILED DESCRIPTION

[0020] Before describing embodiments of the present invention in detail, it is helpful to present an exemplary environment in which embodiments of the present invention may be implemented.

[0021] Figure 1 At 200, a lithographic apparatus LA is shown as part of an industrial production facility that implements a relatively large number of lithographic manufacturing processes. In this example, the manufacturing process is applicable to the manufacture of semiconductor products (integrated circuits) on substrates such as semiconductor wafers. The skilled person will appreciate that a variety of products can be manufactured by processing different types of substrates in variations of this process. The production of semiconductor products is used only as an example of great commercial significance today.

[0022] In a lithographic apparatus (or simply "lithography tool" 200), a measurement station MEA is shown at 202 and an exposure station EXP is shown at 204. A control unit LACU is shown at 206. In this example, each substrate visits a measurement station and an exposure station in order to have a pattern applied. In an optical lithographic apparatus, for example, a pattern transfer unit or projection system is used to transfer a product pattern from a pattern forming device MA to a substrate using conditioned radiation and a projection system. This is achieved by forming a pattern image in a layer of radiation-sensitive resist material.

[0023] The term "projection system" as used herein should be broadly interpreted to cover any type of projection system suitable for the exposure radiation used or for other factors such as the use of immersion liquid or the use of vacuum, including refractive, reflective, reflective-refractive, magnetic, electromagnetic and electrostatic optical systems, or any combination thereof. The pattern forming device MA can be a mask or reticle that imparts a pattern to a radiation beam transmitted or reflected by the pattern forming device. Well-known operating modes include stepping mode and scanning mode. It is well known that the projection system can cooperate with the support and positioning system for the substrate and the pattern forming device in a variety of ways to apply the desired pattern to many target portions on the entire substrate. A programmable pattern forming device can be used instead of a mask with a fixed pattern. For example, the radiation can include electromagnetic radiation in the deep ultraviolet (DUV) band or the extreme ultraviolet (EUV) band. The present invention is also applicable to other types of lithography processes, such as imprint lithography and direct write lithography, for example, by electron beam.

[0024] The lithographic apparatus control unit LACU controls all movements and measurements of the various actuators and sensors to receive the substrate W and mask MA and perform the patterning operation. The LACU also includes signal processing and data processing capabilities for performing the desired calculations associated with the operation of the apparatus. In practice, the control unit LACU will be implemented as a system of many sub-units, each handling real-time data acquisition, processing and control of a subsystem or component within the apparatus.

[0025] Before the pattern is applied to the substrate at the exposure station EXP, the substrate is processed at the measurement station MEA so that various preparatory steps can be performed. The preparatory steps may include: using a level sensor to map the surface height of the substrate, and using an alignment sensor to measure the position of the alignment marks on the substrate. The alignment marks are nominally arranged in a regular grid pattern. However, due to inaccuracies in the generation of the marks and due to deformations of the substrate during its entire processing, the marks deviate from the ideal grid. Therefore, if the device is to print product features at the correct position with very high accuracy, in addition to measuring the position and orientation of the substrate, the alignment sensor must actually measure the positions of many marks over the entire substrate area in detail. The device may be a so-called dual-platform type with two substrate stages, each of which has a positioning system controlled by a control unit LACU. While a substrate on a substrate stage is being exposed at the exposure station EXP, another substrate can be loaded onto another substrate stage at the measurement station MEA so that various preparatory steps can be performed. Therefore, the measurement of the alignment marks is very time-consuming, and the provision of two substrate stages can significantly increase the throughput of the device. If the position sensor IF is not capable of measuring the position of the substrate table when the substrate table is at the measurement station and at the exposure station, a second position sensor may be provided so that the position of the substrate table can be tracked at both stations. The lithographic apparatus LA may, for example, be of a so-called dual stage type having two substrate tables and two stations - an exposure station and a measurement station - between which the substrate table can be exchanged.

[0026] Within a production facility, the apparatus 200 forms part of a "lithography cell" or "lithography cluster" which also includes a coating apparatus 208 for applying photoresist and other coatings to a substrate W for patterning by the apparatus 200. At the output side of the apparatus 200, a baking apparatus 210 and a developing apparatus 212 are provided for developing the exposed pattern into a physical resist pattern. Between all these apparatuses, a substrate handling system is responsible for supporting the substrate and transferring the substrate from one apparatus to the next. These apparatuses, which are generally collectively referred to as tracks, are controlled by a track control unit, which itself is controlled by a management control system SCS, which also controls the lithography apparatus via a lithography apparatus control unit LACU. Thus, different apparatuses can be operated to maximize throughput and processing efficiency. The management control system SCS receives recipe information R which provides in great detail the definition of the steps to be performed to produce each patterned substrate.

[0027] Once the pattern has been applied and developed in the lithography unit, the patterned substrate 220 is transferred to other processing equipment such as those described at 222, 224, and 226. A wide range of processing steps are implemented by various equipment in a typical manufacturing facility. For the purpose of example, the equipment 222 in this embodiment is an etching station, and the equipment 224 performs a post-etching annealing step. In other equipment 226, etc., other physical and / or chemical processing steps are applied. Many types of operations may be required to make a real device, such as the deposition of materials, modification of surface material properties (oxidation, doping, ion implantation, etc.), chemical mechanical polishing (CMP), etc. In fact, the equipment 226 can represent a series of different processing steps performed in one or more equipment. As another example, equipment and processing steps for implementing self-aligned multiple patterning can be provided to produce multiple smaller features based on a precursor pattern laid by a lithography device.

[0028] As is well known, the manufacture of semiconductor devices involves many repetitions of such processing to build device structures layer by layer on a substrate using appropriate materials and patterns. Thus, the substrate 230 that arrives at the lithography cluster may be a newly prepared substrate, or the substrate 230 may be a substrate that has been completely processed previously in this cluster or in another device. Similarly, depending on the processing required, the substrate 232 on the device 226 that leaves may be returned for subsequent patterning operations in the same lithography cluster, the substrate 232 may be designated for patterning operations in a different cluster, or the substrate 232 may be a finished product to be sent for cutting and packaging.

[0029] Each layer of the product structure requires a different set of process steps, and the equipment 226 used at each layer can be completely different in type. In addition, even if the processing steps to be applied by the equipment 226 are nominally the same in a large facility, there may be several assumed identical machines working in parallel to perform steps 226 on different substrates. Smaller settings or failure differences between these machines can mean that these machines affect different substrates in different ways. Even relatively common steps for each layer, such as etching (equipment 222), can be implemented by several etching equipment that are nominally the same but work in parallel to maximize production. In addition, in practice, different layers require different etching processes, such as chemical etching, plasma etching, depending on the details of the material to be etched and special requirements such as, for example, anisotropic etching.

[0030] Previous and / or subsequent processes may be performed in other lithographic apparatuses as just mentioned, and may even be performed in different types of lithographic apparatuses. For example, some layers that are very demanding in terms of parameters such as resolution and overlay during device fabrication may be performed in more advanced lithographic tools than other layers that are less demanding. Thus, some layers may be exposed in immersion lithographic tools, while other layers are exposed in "dry" tools. Some layers may be exposed in tools operating at DUV wavelengths, while other layers are exposed using EUV wavelength radiation.

[0031] In order to correctly and consistently expose the substrate exposed by the lithography equipment, it is necessary to check the exposed substrate to measure properties such as overlay errors between subsequent layers, line thickness, critical dimensions (CD), etc. Therefore, the manufacturing facility in which the lithography unit LC is located also includes a measurement system, which receives some or all of the substrates W that have been processed in the lithography unit. The measurement results are provided directly or indirectly to the management control system SCS. In particular, if errors are detected, the exposure of subsequent substrates can be adjusted when the measurement can be performed quickly and quickly enough so that other substrates of the same batch are still to be exposed. In addition, the exposed substrate may be stripped and reworked to improve the yield, or discarded, thereby avoiding further processing of the substrate known to be defective. In the case where only some target portions of the substrate are defective, further exposure can be performed only on those target portions that are good.

[0032] Figure 1 Also shown is a metrology device 240, which is provided for measuring parameters of the product at a desired platform during the manufacturing process. A common example of a metrology station in a modern lithography production facility is a scatterometer (e.g., a dark field scatterometer, an angular analysis scatterometer, or a spectral scatterometer), and it can be used to measure the properties of the substrate developed at 220 before etching in the device 222. With the use of the metrology device 240, it can be determined that important performance parameters such as overlay or critical dimension (CD) do not meet the specified accuracy requirements in the developed resist. Prior to the etching step, there is an opportunity to strip the developed resist and reprocess the substrate 220 via the lithography cluster. By managing minor adjustments over time by the control system SCS and / or control unit LACU 206, the measurement results 242 from the device 240 can be used to maintain accurate performance of the patterning operation in the lithography cluster, thereby minimizing the risk of making unqualified products and requiring rework.

[0033] Additionally, metrology equipment 240 and / or other metrology equipment (not shown) may be employed to measure properties of processed substrates 232, 234 and incoming substrate 230. Metrology equipment may be used on processed substrates to determine important parameters such as overlay or CD.

[0034] A variety of techniques can be used to improve the accuracy of pattern reproduction onto a substrate. Accurate reproduction of a pattern onto a substrate is not the only concern in IC production. Another concern is yield, which typically measures how many functional devices a device manufacturer or device manufacturing process can produce per substrate. A variety of methods can be used to improve yield. One such method attempts to make the production of a device (e.g., using a lithography device such as a scanner to image a portion of a design layout onto a substrate) more tolerant to perturbations of at least one processing parameter during processing of the substrate (e.g., during the use of a lithography device to image a portion of a design layout onto a substrate). The concept of an overlap process window (OPW) is a useful tool for this method. The production of a device (e.g., an IC) can include other steps, such as substrate measurement before, after, or during imaging; loading or unloading a substrate; loading or unloading a patterning device; positioning a die under a projection optical device before exposure; stepping from one die to another, etc. In addition, various patterns on a patterning device can have different process windows (i.e., the space of processing parameters under which a pattern will be produced within specifications). Examples of pattern specifications associated with potential systematic defects include checking for necking, line pullback, line thinning, CD, edge placement, overlap, resist top loss, resist undercut and / or bridging. The process windows of all or some (usually patterns within a specific area) of the patterns on the pattern forming device can be obtained by merging (e.g., coinciding) the process windows of each individual pattern. The process windows of these patterns are thus referred to as overlapping process windows. The boundaries of the OPW may contain the boundaries of the process windows of some of the individual patterns. In other words, these individual patterns limit the OPW. These individual patterns may be referred to as "hot spots" or "process window limiting patterns (PWLP)", which may be used interchangeably herein. When controlling the lithography process, it is possible and usually economical to focus on hot spots. When there are no defects in the hot spots, it is likely that all patterns are defect-free. When the value of the processing parameter is outside the OPW, the value of the processing parameter is closer to the OPW, or when the value of the processing parameter is within the OPW, the value of the processing parameter is further away from the boundary of the OPW, imaging becomes more tolerant to disturbances.

[0035] Figure 2Exemplary sources of processing parameters 250 are shown. One source may be data 210 of a processing apparatus, such as parameters of a source of a lithographic apparatus, projection optics, substrate stage, etc., parameters of a track, etc. Another source may be data 220 from various substrate metrology tools, such as substrate height maps, focus maps, critical dimension uniformity (CDU) maps, etc. The data 220 may be obtained before the applicable substrate undergoes a step (e.g., development) that prevents rework of the substrate. Another source may be data 230 from one or more patterning device metrology tools, patterning device CDU maps, patterning device (e.g., mask) film stack parameter variations, etc. Yet another source may be data 240 from an operator of a processing apparatus.

[0036] The control of the lithography process is typically based on feedback or feedforward measurements, which are then modeled using, for example, an inter-field (distinguishing mark across the substrate) or intra-field (distinguishing mark across the field) model. Within the die, there may be separate functional areas such as storage areas, logic areas, contact areas, and the like. Each different functional area or different functional area types may have a different process window, each process window having a different process window center. For example, different functional area types may have different heights and therefore different optimal focus settings. In addition, different functional area types may have different structural complexities and therefore have different focus tolerances (focus process windows) around each optimal focus. However, due to control grid resolution limitations, the same focus (or dose or position, etc.) setting will typically be used to form each of these different functional areas.

[0037] Lithographic control is typically performed using off-line calculations of one or more set point corrections to one or more specific control degrees of freedom, based on, for example, measurements of previously formed structures. Set point corrections may include corrections to specific process parameters, and may include corrections to the settings of specific degrees of freedom to compensate for any drift or error so that the measured process parameter remains within specification (e.g., within an allowed variation (e.g., OPW or process window) relative to an optimal set point or optimal value). For example, an important process parameter is focal length, and focal length errors may manifest themselves in imperfect structures formed on a substrate. In a typical focus control loop, a focus feedback method may be used. This method may include a metrology step that may measure the focus setting used on an already formed structure, for example by using a diffraction based focus (DBF) technique, wherein a target is formed having a focus-dependent asymmetry so that the focus setting may then be determined by measurement of the asymmetry on the target. The measured focus setting can then be used to determine corrections to the lithography process offline; for example, a position correction to one or both of the reticle stage or substrate stage to correct for focus offset (defocus). This offline position correction can then be transmitted to the scanner as a set-point best focus correction for direct actuation by the scanner. The measurement results can be obtained over several batches by averaging the best focus correction (over the batch) applied to each substrate in one or more subsequent batches. A similar control loop is used in the other two dimensions (substrate plane) to control and minimize overlay error.

[0038] Figure 3 Describe this method. Figure 3Product information 305, such as product layout, irradiation pattern, product microtopography, etc., and metrology data 310 (e.g., defocus data or overlay data from previously generated substrate measurements) are shown fed to an offline processing device 315 executing an optimization algorithm 320. The output of the optimization algorithm 320 is, for example, one or more set point corrections / offsets 325 for actuators used to control the reticle stage and / or substrate table positioning within a scanner 335 (in any direction, i.e., in x, y, and / or z directions, where x and y are substrate plane directions and z is perpendicular to x and y); the set point corrections 325 are calculated to compensate for any offsets / errors (e.g., defocus, dose, or overlay offsets / errors) included in the metrology data 310. The control algorithm 340 (e.g., a leveling algorithm) uses the substrate-specific metrology data 350 to calculate control set points 345. For example, a leveling exposure trajectory (e.g., determining a relative movement or acceleration profile for positioning the substrate table relative to the reticle stage during the lithography process) may be calculated using the leveling data (e.g., wafer height map) and output as a position setpoint 345 for the scanner actuator. Also for each substrate, the scanner 335 applies the setpoint correction 325 directly to the calculated setpoint 345. In other control configurations, optimization may be performed within the scanner to provide optimized corrections on a per wafer basis (wafer-to-wafer control).

[0039] The optimization algorithm (e.g., when executed in an offline processing device and / or within a scanner) can be based on several different merit functions, one for each control mechanism. Thus, in the above example, the leveling (or focus) merit function is used for focus control (scanner z-direction control), which is different from the overlap (scanner x / y-direction control) merit function, the lens aberration correction merit function, etc. In other embodiments, the control can be optimized jointly for one or more of these control mechanisms.

[0040] Regardless of the control mechanism and control aspect of the optimization, existing optimization methods generally rely on performing optimization based on least squares (e.g., root mean square (RMS)) regression or similar such regression. Although some measurements suffer more noise and uncorrectable errors than other measurements, such methods make all measurements have equal importance. More importantly, existing methods can try to correct the tube core with smaller overlap errors, and therefore will potentially produce at the expense of making the tube core that is otherwise slightly qualified not conform to the specification anyway. When all measurement results have the same weight, the estimator tries to find a compromise between all measurement results to reduce errors everywhere. This means that even if the point that is easy to qualify is pressed down, this may make other tube cores not conform to the specification. Such methods are sensitive to noisy data and lack measurement points. In addition, such methods can estimate too high values ​​for distinguishing marks, which may waste actuator possibilities (actuation range) in optimization later without additional benefits. Note that the larger the estimated distinguishing mark parameter, the higher the risk of reaching the limit of the actuator capability in the optimization.

[0041] Such RMS-type regression methods have a tendency to overfit or underfit, and there is no direct control over the degree of fit. In the case of overfitting, the calculated distinguishing marks exceed the actual values, which can be very problematic. Normalized model uncertainty (nMU) together with projection ratios can be used to predict and prevent overfitting by reducing the complexity of the model; however, these methods limit the choice of models. For example, it is well known that a 3rd order model cannot be fitted to only two data points, etc. However, this can be made possible by adding other constraints or cost functions to the fitting problem. This practice, known as regularization in machine learning, can help fit models with lower out-of-sample errors in a probabilistic sense.

[0042] To address these issues, it is proposed to use an improved version of the Support Vector Machine (SVM) regression technique in the estimation part of the optimization instead of using least squares fitting. Compared with the existing least squares method, this optimization technique will use a different cost function and a different set of constraints.

[0043] Therefore, this article discloses a method for controlling a lithography device configured to provide a product structure to a substrate during a lithography process, the method comprising: obtaining measurement data associated with the substrate; and optimizing a control evaluation function for the lithography device based on the measurement data, the optimization comprising performing support vector machine regression on the control evaluation function.

[0044] The objectives of this approach include determining the distinguishing mark so that:

[0045] · The distinctive mark is robust to noisy data.

[0046] Differentiated tags can easily handle less or sparse measurement data. This can reduce measurement load and increase throughput.

[0047] • The distinctive mark is as small as possible (but not smaller than possible) so that the actuator range is not wasted. This can free up budget for other corrections.

[0048] No overfitting is possible: In order to keep the out-of-sample error as close as possible to the in-sample error, machine learning techniques (including SVM) try to achieve a model with the smallest possible variance for sampling. This is achieved via margin maximization and regularization. This technique will statistically have smaller errors at non-measured locations. In contrast, least squares methods only minimize the error for in-sample errors (measured points).

[0049] The estimated differential identification model describes the measured data well enough.

[0050] The SVM regression method works by essentially sacrificing / compromising where the overlap value is small (e.g., within a threshold ∈), and using the degrees of freedom to correct a die with a larger error (e.g., a die that would otherwise be almost a qualified die). More specifically, the SVM regression method attempts to find a function f(x) that has the largest ∈ deviation from the known values ​​for all training data (e.g., the training data), while being as flat (non-complex) as possible. In other words, if the error is less than ∈, the error is accepted and ignored. Deviations greater than ∈ are not allowed in basic SVM regression, however, in practical environments, the resulting optimization problem will generally be infeasible. To address this issue, the slack variable ξ i , Can be used to adjust for outliers.

[0051] Figure 4 Conceptually explain SVM regression. Figure 4is an overlap graph (e.g., a graph of overlap components (e.g., dx or dy) versus wafer position coordinates) in which each point on the graph represents an overlap error value. Note that this is a 2D graph for ease of representation only, and in actual overlap modeling, both the dx and dy overlap components will be modeled as functions of x and y. The parameter ∈ defines the acceptable margin or overlap error and can be selected by the user. The white points within the dashed line HP (the white points refer to the range of the hyperplane defined by the margin ∈), that is, those points with magnitudes less than ∈, do not incur any cost. In other words, these values ​​are essentially ignored when performing SVM regression; these values ​​are considered to represent a sufficiently good overlap and therefore do not require any correction. The gray points are the points closest to the hyperplane; these points are called support vector points. The support vector points are the basis functions of the SVM that determine the SVM regression (solid line). The black dots are outliers or error support vectors. Slack variables are used to process these points so that the distance of these points from the dashed line is minimized (e.g., the first norm). Thus, the model SVM produced by SVM regression depends only on a subset of the training data, because the cost function used to build the model ignores any training data that is close to the model prediction (within a threshold ∈). For comparison, the least squares fit LS to the same data points is also shown (dotted line), which shows signs of overfitting (overcomplexity).

[0052] A highly simplified mathematical description of the difference between least squares regression and SVM regression will now be described. Although the example uses overlap as a direct use case, the method is by no means specific to estimating the distinctive identity of overlap. The SVM regression technique disclosed herein is equally suitable for any parameter such as focal length, critical dimension (CD), alignment, edge placement error, etc. and / or any optimized distinctive identity estimation included in the lithography process control.

[0053] For both the least squares and SVM regression cases, the models can be expressed as:

[0054] Ax=b

[0055] Where A is the so-called "design matrix" that is generated by evaluating the overlap (or other parameter) model on a measured grid; the term x is the so-called "model parameter" and is a vector comprising the distinguishing identity parameters: for example the "k-parameters" or the parameters of a typical six-parameter model (x / y translation parameters: Tx, Ty, symmetric / asymmetric magnification parameters: Ms, Ma, symmetric / asymmetric rotation parameters: Rs, Ra) or any other suitable model for modeling distinguishing identities; and the term b is a vector comprising all measured overlap values ​​in both the x-direction and the y-direction (i.e., measurement data). The goal of least squares regression optimization is to find the model parameters x that minimize Ax-b; i.e., the least squares method minimizes the 2-norm of the error in the equation Ax=b:

[0056]

[0057] where ||.|| is the 2-norm operator. Note that italic "x" will always be used to refer to model parameter terms, in contrast to non-italic "x" which refers to spatial coordinates.

[0058] In contrast, in the SVM regression technique, the optimization aims to minimize the "complexity" of the discriminative identification parameters subject to the constraint that all measurements are "fully explained" by the model.

[0059] The complexity of distinguishing identification parameters can be defined as the 2-norm of the vector holding parameter values ​​​​except for any zero-order parameters (e.g., translation parameters Tx and Ty in the overlap model). In order to better understand the concept of complexity in this context, the following concepts from machine learning should be understood:

[0060] Generalization: Assume that a model is to be fitted to a data set. The first portion of the data (e.g., the first half) is used to train (fit) your model and the second portion of the data (e.g., the second half) is used to validate the model once it has been trained. The first portion of the data is often referred to as the in-sample data and the second portion of the data is often referred to as the out-of-sample data. The ratio between the in-sample error and the out-of-sample error is a measure of the generalization of the model; that is, a measure of how successful the model is in representing out-of-sample data that was not used (not considered) in the fitting process.

[0061] VC dimension: The Vapnik-Chervonenkis (VC) dimension is a measure of the complexity of the model. In neural networks, the VC dimension is usually measured using a dichotomy. In general: the lower the VC dimension, the more generally applicable or generalized the fit. For example: a second-order model comprising a total of three parameters on one-dimensional data can be better generalized than a third-order model with a total of four parameters fitted on the same data (in this case, the number of parameters is equal to the VC dimension). It should be understood that although it is usually specified that the number of parameters should not exceed the number of measurement results, this is usually incorrect. In fact, the number of VC dimensions (non-parametric) should be less than the number of measurement results. The number of parameters is not necessarily equal to the VC dimension. For example, a 1000 parameter model can be fit using data comprising 10 measurement results; however, the complexity of the fit as defined by the VC dimension should not be higher than 10.

[0062] It is still possible to fit a complete infinite-dimensional model to a given data set; a common practice for fitting nonlinear models such as f(A, x) = b is to use kernel functions. With such techniques, the VC dimension can be kept low while the model itself has an infinite number of parameters, which means that the out-of-sample error can be kept low.

[0063] The out-of-sample error can be kept close to the in-sample error using regularization techniques. Regularization is a technique that discourages the learning (or fitting) of complex or flexible models (i.e., regularization favors simpler models), thereby keeping the VC dimension low and avoiding the risk of overfitting.

[0064] The VC dimension of the model can be minimized based on the optimization of the 2-norm of the parameter values ​​except the zero-order term (i.e., the bias). Taking the overlap as an example, this means minimizing all parameter values ​​except the linear translation parameters (Tx and Ty). Later, the reason why the VC dimension is reduced due to this optimization will become apparent, making the VC dimension low enough to be generally applicable or generalized even for overlapping models with a very large number of parameters.

[0065] To keep the equation simple, for this example, assume that the overlapping data model can be written as:

[0066] Ax+t=b

[0067] where t represents the zeroth order (translation term). Then, the optimization problem for lower complexity becomes the minimization of the 1-norm or 2-norm of the model parameters; for example:

[0068]

[0069] Subject to the criterion that all measurements are fully explained by the model. Note that ||x|| is only one example of a complexity metric to be minimized in the methods described herein. In other embodiments, a weighted norm may be minimized, for example:

[0070]

[0071] Where Q is any positive definite rectangular matrix size of x. Q can contain information about the cost of using a certain model parameter. For example, if it is not desirable to use a first parameter p1, but to use a second parameter p2 to compensate for this (as much as possible), then the Q elements associated with parameter p1 can be given a higher weight relative to the Q elements associated with parameter p2, making it less likely that the estimator will use parameter p1 as parameter p2. Q can also be used to assign relative costs of use to parameter pairs or sets of parameters using the off-diagonal elements of the Q matrix.

[0072] This criterion means that for each measurement result j:

[0073]

[0074] where |.| denotes an absolute value. This constraint states that all measured overlap values ​​are fully explained by a model with better accuracy than ∈.

[0075] However, outliers and residuals are almost inevitable. Therefore, such outliers should be regulated but penalized at the same time. This can be done by providing slack variables, whereby the optimization problem can be written as:

[0076]

[0077] Limited by:

[0078]

[0079]

[0080]

[0081] Among them, ξ and ξ *are the upper and lower slack variables that take into account outliers, and C is the outlier penalty coefficient, also called the "complexity coefficient". The constant C (>0) determines the trade-off between the flatness (complexity) of the fit and the degree to which deviations greater than ∈ are tolerated by penalizing outliers. The higher the complexity coefficient, the greater the degrees of freedom used to select a complex model in order to better represent the in-sample data. At one extreme, regardless of the overlapping model used to generate the A matrix, if C=0, the solution will simply be just a zero-order translation. At the other extreme, C equal to infinity would mean that regardless of the complexity, the maximum error always remains less than a certain value; for example, similar to L ∞ Norm (absolute maximum) optimization (L ∞ <∈).

[0082] The optimization should determine the complexity coefficient C, the margin ∈ and the slack variable ξ so that all measured data are represented by the model within an accuracy less than the (e.g., user-defined) margin ∈; otherwise, when this is not possible, their error (ξ) should be kept to a minimum, with the constraint that the solution does not become too complex as a result.

[0083] In order to convert this optimization problem into a quadratic programming optimization, the method of Lagrange multipliers can be used. This method converts the constrained problem into the following form, so that the derivative test of the unconstrained problem can still be applied. At any stationary point of the function that also satisfies the equality constraints, the gradient of the function at that point can be expressed as a linear combination of the gradients of the constraints at that point, where the Lagrange multipliers act as coefficients. The relationship between the gradient of the function and the gradient of the constraints leads to a reformulation of the initial problem, which is called the Lagrangian function. Therefore, the Lagrangian multipliers α, α*, η, η* can be defined, and the Lagrangian function L is written as:

[0084]

[0085] The Lagrangian function L can be easily transformed into a simple quadratic program in the adjoint formulation, where the inner product of the data forms the cost function and C forms the inequality constraint:

[0086]

[0087] Limited by:

[0088]

[0089]

[0090] The initial model parameters x are a linear combination of the design matrix and the obtained optimal Lagrange multipliers:

[0091]

[0092] After solving the optimization problem, it becomes apparent that α (*) Most of (i.e., α j and ) is zero. (*) Values ​​include non-zero values. Non-zero α (*) The number of values ​​is the VC dimension of this problem. Therefore, the entire model parameters can be written as a linear combination of only a few measurement points:

[0093] Even if the overlapping model is of very high order (e.g., 100 parameters), if only a few (e.g., 6) α (*) If the value is non-zero, the complexity (VC dimension) of the model is 6, and the model is as generally applicable or generalized as a six-parameter ("6par") model. However, both the in-sample error and the out-of-sample error are as low as the 100-parameter model.

[0094] Corresponding to non-zero α (*) And each of the data values ​​(columns of matrix A) that also help distinguish the parameters x are called support vectors, because they are the vectors that support the hyperplane in the high-dimensional space (hence the name support vector machine). In the specific example in the previous paragraph, there are 6 support vectors, where each of these support vectors is 100-dimensional and together support the 100-dimensional hyperplane. It should be understood that what is optimized is not the error, nor the parameters, but α (*) The deviation (or translation parameter for the overlapping case) is determined after optimization (eg, using Karush-Kuhn-Tucker (KKT) conditions) and is not necessarily equal to the mean of the data.

[0095] In summary, the use of SVM regression to fit parameter signatures (e.g., overlap) as part of lithography process optimization is proposed. Due to its 2D nature, SVM regression in its currently known form cannot be directly applied to signature data, and SVM in its general form can only handle one-dimensional data. Therefore, an improved version of the SVM technique that can be applied to 2D signature data is described herein.

[0096] Figure 5 An example of the results of SVM modeling with a target margin ε of 0.45 nm is shown compared to modeling using a least squares fitting (LSQ) method. Figure 5 (a) and Figure 5 (b) Each shows a cumulative graph of the in-sample error (ie, the modeled error at the measurement point). The y-axis shows the overlap value OV below or equal to dx , OV dy (corresponding to Figure 5 (a) and Figure 5 (b) The cumulative number of measurement points with in-sample error values ​​for the target margin ∈ (in the form of a percentage). Because the SVM ignores measurement points within the target margin ∈, SVM modeling typically results in fewer measurement points with in-sample errors below the target margin ∈ than modeling using the LSQ method. However, SVM modeling typically results in multiple measurement points with in-sample errors that are above the target margin (corresponding to the vertical portion at ∈ for each curve graph). Therefore, since the SVM sacrifices at low error points to gain at high error points, SVM modeling is expected to result in better modeling (i.e., more measurement points have modeling errors less than or equal to the target margin) than modeling using the LSQ method. Therefore, the SVM can improve yield by focusing all correction possibilities on larger errors rather than wasting correction possibilities on smaller errors.

[0097] Often in overlap modeling (or modeling of another parameter of interest), and in the case of the foregoing specific example, one needs to assume a distinguishing identity model prior to fitting; e.g., Zernike, regular polynomial, or any other model. However, by definition, it is not possible to know / guarantee that there is no model mismatch. This means that a "hypothetical" overlap model must not necessarily be used to accurately model the potential overlap.

[0098] Having a fixed predefined signature model requires a certain sampling layout that fits the assumption. For example, it is not possible to update a signature for a first type of model (e.g., a correction per exposure (CPE) signature that determines a correction per field) using, for example, an overlap measurement that is only applicable to a second type of model. For a fixed predefined "hypothetical" model, the model granularity is categorical. For example, model categories may include per-field model, average field model, scan up and down (SUSD) dependent model, per wafer, per chuck, or per batch model. However, a model cannot be partially one of these categories; for example, a model may not be "slightly per field", "slightly per wafer", etc. Such an inflexible approach is not ideal. The real overlap will be the result of machine overlap and process signatures, which do not necessarily follow the model definition. For example, reticle heating induced changes occur partially between fields (inter-field component); however, reticle heating induced changes may also occur partially throughout the average field (intra-field component). Although chuck 1 may be slightly different from chuck 2, the lens contribution for these two chucks may be the same, etc. These chuck contributions from different chucks may be modeled using models with different granularities.However, using a kernel function, the kernel function may model reticle heating and / or these different chuck contributions without defining a granularity of differentiation.

[0099] The essence of the embodiment described below is to use the kernel function to define the category of the model in an abstract way, rather than directly specifying the model to be fitted. After this, the optimized kernel function can be formed by the model category defined by the kernel function, and fit to the formed kernel function at the same time.

[0100] In order to understand the idea behind this concept, it is important to examine the estimation / modeling task in detail. The basic concept of modeling overlap / focal length / cd (or other parameters of interest) is:

[0101] • Assume that the measured overlay / focal length / cd values ​​can be described by a set of (eg polynomial) functions.

[0102] • The coefficients of these (eg polynomial) functions are calculated by minimizing the error indicator.

[0103] For example, it can be assumed that a particular model distinguishing signature can be described using a conventional polynomial. It can be assumed that each field or wafer or batch has a different distinguishing signature. Each of these statements is an assumption. Based on the assumption, the weights or "distinguishing signature parameters" assumed in the model are calculated, for example, by minimizing the collective overlap error (e.g., the second norm) at the measurement locations. In this approach, the model complexity and the number of distinguishing signature parameters that can be assumed are limited by the number (and validity) of measurement points. Mathematically, this is true for a least squares solution, but not necessarily for an SVM.

[0104] In this embodiment, it is proposed to replace both the aforementioned assumptions and the calculation steps with a new optimization problem, which is mathematically equivalent to assuming an "infinite parameter" (or at least very high-dimensional) model. Very high-dimensional models may include, for example: more than 500 dimensions, more than 1000 dimensions, more than 5000 dimensions, more than 50000 dimensions, more than five million dimensions, or infinite dimensions.

[0105] There are many advantages to this, including:

[0106] • Model mismatch can be avoided or at least reduced. No model selection is required and no manual input is required (thus, failure modes are removed). In practice, the parameters of the knowledge and context of interest are accumulated in a so-called kernel function.

[0107] • Some process / scanner knowledge can be used to give meaning to content context abstraction, and thus estimate very complex and accurate distinctive signatures from sparse data.

[0108] • Time can be given meaning in the context of the content, enabling predictions for future batches rather than temporal filtering. Note that temporal filtering reduces noise at the expense of increasing phase lag or some delay that degrades performance.

[0109] • The distinctive signature is robust to noisy data (due to the dense dead bands of ε).

[0110] The method can more easily handle less and non-uniform measurement data. This can reduce the measurement load and increase the throughput of the wafer fab.

[0111] Model the distinctive feature as small as possible to use the actuator range more efficiently. For example, in the case where two mathematical descriptions can describe the same distinctive feature, the smallest one can be chosen so as not to waste actuation capacity. This can free up budget for other corrections.

[0112] No overfitting and no underfitting: In order to keep the out-of-sample error as close as possible to the in-sample error, machine learning techniques (including SVM) try to obtain the model with the smallest possible variance for sampling. This is achieved via margin maximization and regularization. Such techniques can have statistically smaller errors at non-measured locations.

[0113] The estimated discriminant model describes the measured data well enough. Discriminants that cannot be captured by any other model are easily captured using this technique.

[0114] This technique also has the same performance in the yield curve graph as that present in the normal SVM.

[0115] Mathematical description:

[0116] In SVM, even if m is less than n, the nPar model can be fitted to m number of measurements. To illustrate fitting an infinite parameter model to a finite number of measurements, an overlap example will be given. Although the example uses overlap as a direct use case, the method is by no means exclusive to overlap and can be used for other parameters of interest PoI such as focal length, CD, alignment, edge placement, etc.

[0117] As already indicated, the overlap estimation problem is usually formulated as:

[0118] Ax=b

[0119] where A is the so-called "design matrix" which is generated by evaluating the "overlap model" on the measurement grid, x is a vector containing distinguishing identification parameters such as the k parameter, and b is a vector containing all measured overlap values ​​in the x-direction and the y-direction.

[0120] The model assumptions are included in the design matrix A: each row of this matrix refers to a certain measurement location on the wafer and each column of this matrix represents a specific basis function (eg, a single term of a polynomial) assumed in the model.

[0121] Aij = jth basis function estimated at the ith measurement point

[0122] Each basis function is usually a nonlinear function of position. For example, each basis function of the 38par field model is field (x f and f ) relative to the center of the field.

[0123]

[0124] Where p and k are powers of the polynomial. Assuming a model or modeling step actually means assuming a function that maps each point on the wafer (each context parameter associated with the wafer) to another point in a higher dimensional space. For example, a 38par per field per chuck model for a wafer with 100 fields takes any 5-dimensional vector (measurement points in each field; 2 dimensions for Xf, Yf; 2 dimensions for Xw, Yw and 1 dimension for the chuck ID (ChuckID)) and then maps that arbitrary 5-dimensional vector to a 7600-dimensional space (38Par*2 chucks*100 fields=7600). This reads formally as:

[0125]

[0126] Here nPar means the number of parameters. This function affects every measurement point i. In form: is called the input space, is called the feature space, and the values ​​of the overlap (dx, dy) are called the output space.

[0127] Figure 6 The model assumptions are conceptually explained. The figure shows an implicit mapping of a layout including wafer coordinates and content context from an input space IS to a higher dimensional space or feature space FS using a distinctive identification model FP via a modeling step MOD (assumption). The feature space FS includes the columns of the design matrix A. Then, a linear fit is attempted between the feature space FS and the output space OS, including the measured or estimated overlap or other parameters of interest PoI values.

[0128] The question assumed in this article is what is needed from the design matrix A? Is a design matrix even really needed?

[0129] In least squares optimization (and many other forms of regression), it can be shown that the following conditions are generally required:

[0130]

[0131] It should be full rank, or (depending on the model) be made full rank using regularization techniques such as Tikhonov.

[0132] However, for SVM, the following conditions are required:

[0133] K=A T A (nMeas×nMeas)

[0134] It may not be full rank, and where nMeas is the number of measurements. In the context of SVM, the K matrix is ​​called the kernel function. In practice, K ij is the inner product of the i and j elements (i.e., vectors) in the feature space (associated with measurement points i and j, respectively). The inner product is a mathematical definition of the similarity of two vectors. Therefore, K ij Describes the similarity between measurement point i and measurement point j.

[0135] Different models with different numbers of parameters may output different values; however, the models will maintain a similar feel when the kernel function remains the same size and the value of the kernel function does not vary much from model to model. For example, both the first model and the second model should agree to some degree on how similar two points on a wafer are. Thus, if two points have the same value using one model, they should not have wildly different values ​​using another model.

[0136] Using the kernel function, it is not necessary to first construct the design matrix (A) in order to construct K. The K matrix can be generated by first analytically generating the kernel function k; for example:

[0137] k(X i , X j )=φ T (X i )φ(X j )

[0138] where φ is defined as a mapping function. Note that any model can be converted to a kernel function using the above equation by simply multiplying each element of the mapping function associated with the model, estimating the value at Xi, Xj, and summing these values ​​(i.e., computing the inner product of two vectors i and j in the feature space spanned by the mapping function φ). For example,

[0139] φ=[1,x,x 2 , x 3 ]

[0140]

[0141] However, the kernel function does not have to correspond to any model in order to be effective. After this, the function can be evaluated at each measurement location:

[0142] K ij = k(X i , X j )

[0143] It is exactly the same as first constructing the design matrix A and then multiplying it by itself. This method allows the creation of the kernel function matrix even in cases where it is very difficult or even impossible to create the design matrix A, for example when the kernel function describes the inner product of an infinite-dimensional space.

[0144] Mathematically, the only requirement for this kernel function to be valid is that it should be positive semi-definite in the space for which the kernel function k is defined. Therefore, there is no requirement to check whether the mapping function φ actually exists. This means that a kernel function that does not correspond to any overlapping model can be used as long as the kernel function is positive semi-definite. The kernel function can be constructed so that it corresponds to an infinite-dimensional model.

[0145] In an embodiment, the kernel function may describe a distance indicator. The distance indicator may be the inner product of two elements in the feature space. Alternatively, the distance indicator may be the sum of the absolute values ​​of the differences between the components of two elements in the feature space (e.g., k(X1, X2)=|1-1|+|X1-X2|+|X1 2 -X2 2 |+|X1 3 -X2 3 |).

[0146] To understand the principle of kernel function, the following example is given. For an exemplary measurement in two-dimensional space:

[0147] X=[x f ,y f ] T (e.g., only one field)

[0148] And the kernel function is:

[0149]

[0150] It represents the model as:

[0151]

[0152] It is all polynomials of degree at most 2.

[0153] Similarly, the kernel function

[0154]

[0155] Represents all polynomials of degree at most n.

[0156] Similarly, the Gaussian kernel function:

[0157]

[0158] represents a model with an infinite number of parameters, where σ is an arbitrary length scale. Of course, it is impossible to produce a design matrix with an infinite number of rows; however, it is still possible to produce a kernel function that represents the inner product in this particular infinite-dimensional space.

[0159] Naturally, since there is no model, it is not possible to have a discriminant parameter. However, solving the kernel-based SVM yields a (non-parametric) function that describes the overlap at any location on the wafer. This is not a linear combination of the discriminant parameters with a polynomial basis function, instead, the overlap function is:

[0160]

[0161] This problem can be solved based on an optimization problem. The input to the optimization can be:

[0162] Kernel function: k(X j , X i ) (more information about the choice of kernel function will be described); and

[0163] Measure data points (e.g., coordinates and overlap values ​​in input space)

[0164] The output of the optimization problem can be:

[0165] Translation terms tx, ty.

[0166] Support vector coefficient α p and

[0167] Support vector X p .

[0168] The number of support vectors nSPV.

[0169] The optimization problem can take the following form:

[0170]

[0171] Limited by:

[0172]

[0173]

[0174] and where ∈ is an arbitrary estimate / guess of the noise (the thickness of the band) and C is the regularization factor as already defined above.

[0175] In the same manner as the linear embodiment described earlier, the kernel-based SVM includes minimizing a complexity metric for the distinguishing identification parameters subject to the constraint of fully explaining all measurements. For the kernel-based SVM, the complexity of the distinguishing identification parameters can be conceptually the same as the complexity defined in the linear embodiment (e.g., the same as the 2-norm of the vector holding the parameter values ​​(e.g., except for Tx and Ty)); however, this complexity of the distinguishing identification parameters is not explicitly calculated.

[0176] After solving the optimization problem, we will notice that α (*) Most of α are zero. Only a small number of α (*) Will have a non-zero value. Non-zero α (*) The number of is the VC dimension of this problem. Because the entire model parameters can be written as a linear combination of fewer measurement points. After solving the optimization, you can report the function, or evaluate the function on any (dense) layout and report the overlap value.

[0177] In summary, the following table shows the algorithmic differences between SVM and Kernel-based SVM (KB SVM):

[0178]

[0179]

[0180] Choice of kernel function:

[0181] The important question is: what should the kernel function be? And how does the kernel function affect the results? The kernel function is a measure of similarity (in this case between separate measurements) based on domain knowledge. Note that this concept is about the framework of kernel-based estimation, not any specific implementation (or any specific kernel function).

[0182] The proposed concept yields a tool that can be used for different purposes; however, a judicious choice of the kernel function should preferably be made each time.

[0183] In a first example, the kernel function may include a portion per-field, a portion global inter-field, and a portion global intra-field, all of which are polynomials of at most N order.

[0184] First, a 1-dimensional (1D) example will be given. The underlying pattern is x f 、x wA polynomial / sine / cosine function of where all fields are different but related to each other through sine / cosine relationships. The pattern is sampled / measured in random positions (e.g., a circle) and fed into a KB-SVM using a polynomial kernel function.

[0185]

[0186] Among them, at the measurement result i, x i =[x w , x f ].

[0187] The measurement layout is very random, e.g. one or more fields may have no measurements. However, KB-SVM with a simple 4th order kernel function is able to fit the data correctly even for fields where no measurements exist. Interestingly, even measurements can be ignored or discarded if it is not considered to add any additional information.

[0188] Figure 7 is a graph of the output space OS (values ​​of the parameters of interest) versus the input space IS (wafer positions on fields 1 to 6) illustrating this situation. The first curve (black line) is the actual distinguishing feature FP and the second curve (grey line) is the KB-SVM estimate using the polynomial kernel function in this example. Field 4 does not include measurement data M and therefore does not include support vectors SV. However, for all fields including field 4, the estimated KB SVM is very close to the actual distinguishing feature FP.

[0189] Applying the same principles in the 2-dimensional (2D) overlap example, a CPE (per-field correction) can be obtained based on a data set that is only suitable for global modeling using other techniques. The main advantage of this technique is that it tries to find the underlying pattern from any (incomplete) set of data that is available. More specifically, assuming a measurement layout in which some fields are densely measured and others are sparsely measured, it will be necessary to use a KB-SVM to estimate the CPE for this layout. The principle is that each field is a little different, and these differences are captured (to some extent) by the existing measurements. Then, a kernel function is constructed to capture this measure of similarity. The kernel function does not need to be exact, but should have the necessary components. For example, the following kernel function can be used:

[0190]

[0191] in,

[0192]

[0193] The first part of the kernel function essentially says: if two points are in the same field, then they are 10 times more similar than if they are not in the same field. This means: part (0.1) global within field and part (1) per field. The second part says that any within-field distinguisher can be any 5th order polynomial. The third part of the kernel function says that the between-field part of the distinguisher should be continuous (Gaussian kernel function).

[0194] The disadvantage of this technique is that it requires an expert to construct a good kernel function. Although the number of kernel functions is irrelevant, the structure of the kernel function is important.

[0195] In another example, an inter-field Gaussian kernel function is proposed. The local inter-field distinguishing feature can be such that the local inter-field distinguishing feature cannot be captured by existing distinguishing feature models, because very high-order models are required; the distinguishing feature is too local. In addition, existing per-field models give discrete, imprecise estimates. To model the distinguishing feature, the Gaussian radial kernel function can take the following form:

[0196]

[0197] Among them, X i =[x w ,y w ] is the position of the point on the wafer, and σ is a constant (greater than the distance between the two points and less than the footprint of the distinguishing mark).

[0198] Each field model gives discrete estimates of physical distinctions that should not be discrete.

[0199] Kernel-based methods require a good definition of the kernel function. This can be derived based on expert knowledge or using a data-driven approach. Another approach can include multi-kernel estimation.

[0200] In summary, the kernel-based embodiment includes constructing or selecting a kernel function to describe one or more criteria (e.g., proximity between two wafer coordinates) for evaluating a measured distinct identity. The kernel function defines a model of one or more categories (e.g., multiple model categories that may be based on a weighted combination) from which a function is generated for densifying the measured distinct identity while taking into account different granularities of the model (e.g., per unit, per die, per subfield, per field, per wafer, per batch, etc.). The SVM with the kernel function determines the function for describing the measured distinct identity.

[0201] The embodiments may be further described using the following aspects:

[0202] 1. A method for fitting measured data to a model, comprising:

[0203] obtaining measurement data related to a performance parameter of at least a portion of the substrate; and

[0204] The measured data are fitted to the model by minimizing a complexity index of fitting parameters applied to the model while not allowing the deviation between the measured data and the model being fitted to exceed a threshold.

[0205] 2. The method according to aspect 1, wherein the complexity indicator is the 1-norm or 2-norm of the model parameters, or the 1-norm or 2-norm of the weighted model parameters.

[0206] 3. A method according to aspect 1 or 2, wherein the complexity indicator also includes: one or more slack variables for adjusting any outliers included in the measurement data, the deviation between the measurement data and the fitted model being allowed to exceed a threshold of the outliers; and one or more coefficients for weighting the slack variables.

[0207] 4. A method according to aspect 3, wherein the one or more coefficients are complexity coefficients, and the one or more complexity coefficients can be selected and / or optimized to determine the degree to which the outliers are penalized for the complexity of the fit.

[0208] 5. A method according to any preceding aspect, wherein the measurement data comprises at least two-dimensional measurement data.

[0209] 6. The method according to aspect 5, wherein the fitting step comprises: determining a two-dimensional feature signature describing the spatial distribution of the performance parameter.

[0210] 7. The method according to any of the preceding aspects further comprises: defining a Lagrangian multiplier for the complexity indicator, and converting the complexity indicator into a Lagrangian function using the Lagrangian multiplier.

[0211] 8. The method according to aspect 7 comprises: converting the Lagrangian function into a quadratic programming optimization.

[0212] 9. A method according to clause 7 or 8, wherein the fitting step comprises determining model parameters as a linear combination of a design matrix and optimized values ​​for the Lagrange multipliers.

[0213] 10. A method according to any of the preceding aspects, wherein the measurement data describes one or more of: characteristics of the substrate; characteristics of a pattern forming device that defines a pattern to be applied to the substrate; the position of one or both of a substrate stage for holding the substrate and a mask stage for holding the pattern forming device; or characteristics of a pattern transfer system that transfers the pattern on the pattern forming device to the substrate.

[0214] 11. The method according to any of the preceding aspects, wherein the measurement data comprises one or more of overlay data, critical dimension data, alignment data, focus data and leveling data.

[0215] 12. A method according to any of the preceding aspects, wherein the complexity index involves controlling the lithography process to optimize one or more of the following: exposure trajectory control in a direction parallel to the substrate plane; exposure trajectory control in a direction perpendicular to the substrate plane; lens aberration correction, dose control, and laser bandwidth control of a laser source used for the lithography equipment.

[0216] 13. The method of clause 12, comprising controlling the lithography process according to the optimized control.

[0217] 14. A method according to clause 12 or 13, wherein the lithography process comprises exposing a layer on a substrate, thereby forming part of a manufacturing process for manufacturing an integrated circuit.

[0218] 15. The method of any preceding aspect, wherein the complexity index is operable to minimize one or more of overlay error, edge placement error, critical dimension error, focus error, alignment error, and leveling error.

[0219] 16. A method for modeling a performance parameter distribution, comprising:

[0220] obtaining measurement data partially related to a performance parameter of at least a portion of the substrate; and

[0221] The performance parameter distribution is modeled based on the measured data by optimization of the model, wherein the optimization minimizes a cost function representing the complexity of the modeled performance parameter distribution subject to the following constraint: substantially all points included in the measured data are within a threshold from the modeled performance parameter distribution.

[0222] 17. The method of aspect 16, wherein the measurement data includes one or more outliers, the one or more outliers are allowed to fail to satisfy the constraint condition, and the cost function further includes a penalty term to penalize the outliers that fail to satisfy the constraint condition.

[0223] 18. The method of clause 17, wherein the penalty term comprises one or more slack variables for adjusting for any outliers included in the measurement data, the constraints being relaxed for the outliers.

[0224] 19. The method of clause 18, wherein the penalty term further comprises a complexity coefficient, the complexity coefficient being selectable and / or optimized to determine the degree to which the outliers are penalized for the complexity of the fit.

[0225] 20. The method of clauses 16 to 19, further comprising defining Lagrangian multipliers for the cost function, and converting the cost function into a Lagrangian function using the Lagrangian multipliers.

[0226] 21. The method of clause 20, comprising: converting the Lagrangian function into a quadratic programming optimization.

[0227] 22. A method according to clause 20 or 21, wherein the modelling step comprises determining model parameters as a linear combination of a design matrix and optimized values ​​for the Lagrangian multipliers.

[0228] 23. A method for determining a function describing a distribution of a performance parameter, comprising:

[0229] obtaining measurement data related to a performance parameter for a sampling location on a substrate;

[0230] determining a kernel function; and

[0231] An optimization process is performed using the kernel function to determine support vectors and support values ​​that define the function.

[0232] 24. The method of clause 23, wherein the kernel function comprises a positive semi-definite matrix.

[0233] 25. A method according to clause 23 or 24, wherein determining the kernel function is based at least in part on a criterion for evaluating the measurement data.

[0234] 26. The method according to any one of aspects 23 to 25, further comprising: generating a feature space based on a mapping function.

[0235] 27. The method of clause 26, wherein the kernel function corresponds to a distance metric associated with the feature space.

[0236] 28. A method according to clause 26 or 27, wherein the dimensions of the feature space correspond to the components of the mapping function.

[0237] 29. The method of any one of aspects 26 to 28, wherein the mapping function maps the sampling positions to the feature space.

[0238] 30. A method according to any one of aspects 27 to 29, wherein the distance indicator defines the distance between elements of the feature space.

[0239] 31. A method according to any one of aspects 27 to 30, wherein the distance indicator is derived from an inner product defined for the feature space.

[0240] 32. A method according to any of aspects 23 to 31, wherein the at least one criterion comprises a measure of similarity between individual measurements of the measurement data.

[0241] 33. The method according to any one of aspects 23 to 32, comprising:

[0242] generating a kernel function; and

[0243] The kernel function is determined by evaluating the kernel function at one or more measurement locations of the measurement data.

[0244] 34. A method according to clause 33, wherein the kernel function is generated analytically.

[0245] 35. A method according to any one of aspects 23 to 34, wherein the performing the optimization process comprises: performing kernel function-based support vector machine regression using the kernel function.

[0246] 36. A method according to any one of aspects 23 to 35, wherein the kernel function-based support vector machine regression includes: using the kernel function to model the measurement data by minimizing a complexity indicator of the coefficients applied to the support vector while not allowing the deviation between the measurement data and the function to exceed a threshold.

[0247] 37. A method according to aspect 35 or 36, wherein the optimization process includes solving the kernel-based support vector machine regression to generate the function.

[0248] 38. A method according to any one of aspects 23 to 37, wherein the function comprises a non-parameterized function.

[0249] 39. A method according to any one of aspects 23 to 38, wherein the kernel function is constructed so that it corresponds to an infinite-dimensional parameter model.

[0250] 40. A method according to any one of aspects 23 to 39, wherein the kernel function is constructed such that the kernel function corresponds to a model of one or more classes.

[0251] 41. A method according to clause 40, wherein the category of the model describes the level of granularity of the model.

[0252] 42. A method according to aspect 40 or 41, wherein the kernel function is constructed such that the kernel function corresponds to models of multiple categories.

[0253] 43. A method according to any one of aspects 23 to 42, wherein the kernel function comprises a Gaussian kernel function, a polynomial kernel function and / or a discrete kernel function.

[0254] 44. A computer program comprising program instructions operable to perform the method according to any one of clauses 1 to 43 when run on a suitable device.

[0255] 45. A non-transitory computer program carrier comprising a computer program according to clause 44.

[0256] 46. ​​A processing apparatus comprising storage means including a computer program according to clause 36; and

[0257] A processor operable to perform a method according to any one of aspects 1 to 43 in response to the computer program.

[0258] 47. A lithographic apparatus configured to provide a product structure to a substrate in a lithographic process, comprising a processing apparatus according to clause 46.

[0259] 48. The lithographic apparatus according to aspect 47, further comprising:

[0260] a substrate stage for holding the substrate;

[0261] a patterning device stage for holding a patterning device; and

[0262] A pattern transfer unit is used to transfer the pattern on the pattern forming device to the substrate.

[0263] 49. The lithographic apparatus according to clause 48, comprising an actuator for at least one of the substrate table, the patterning device table and the pattern transfer unit, and operable to control the actuator based on the fitted model.

[0264] 50. A lithography unit comprising:

[0265] A lithographic apparatus according to clause 47, 48 or 49; and

[0266] A measurement system is operable to measure the measurement data.

[0267] The terms "radiation" and "beam" as used with respect to a lithographic apparatus encompass all types of electromagnetic radiation, including ultraviolet (UV) radiation (e.g., having a wavelength equal to or approximately equal to 365 nm, 355 nm, 248 nm, 193 nm, 157 nm, or 126 nm) and extreme ultraviolet (EUV) radiation (e.g., having a wavelength in the range of 5 nm to 20 nm), as well as particle beams such as ion beams or electron beams.

[0268] The term "lens" may refer, where the context permits, to any one or combination of various types of optical components, including refractive, reflective, magnetic, electromagnetic, and electrostatic optical components.

[0269] The foregoing description of specific embodiments will therefore fully disclose the general nature of the invention: without departing from the general concept of the invention, other persons can easily modify and / or adapt these specific embodiments for various applications by applying knowledge within the technical scope of the art without undue experimentation. Therefore, based on the teachings and guidance presented herein, such adaptations and modifications are intended to be within the meaning and range of equivalents of the disclosed embodiments. It should be understood that the wording or terminology herein is for the purpose of describing in an illustrative manner rather than for limiting purposes, so that the terms or wording of this specification are to be interpreted by the technician in accordance with the teachings and guidance.

[0270] The breadth and scope of the present invention should not be limited by any of the above-described exemplary embodiments, but should be defined only in accordance with the following claims and their equivalents.

Claims

1. A method for fitting measured data to a model, comprising: obtaining measurement data associated with a performance parameter of at least a portion of a substrate; as well as fitting the measured data to the model by minimizing a complexity index of fitting parameters applied to the model while not allowing the deviation between the measured data and the model being fitted to exceed a threshold, wherein the complexity indicator comprises: one or more slack variables for adjusting any outliers included in the measurement data and one or more coefficients for weighting the one or more slack variables, the deviation between the measurement data and the fitted model being allowed to exceed a threshold value of the outliers, and Therein, the one or more coefficients are complexity coefficients that can be selected and / or optimized to determine the extent to which the outliers are penalized for the complexity of the fit.

2. The method of claim 1, wherein: The complexity index also includes the 1-norm or 2-norm of the model parameters, or the 1-norm or 2-norm of the weighted model parameters.

3. The method of claim 1, wherein: The measurement data includes at least two-dimensional measurement data.

4. The method of claim 3, wherein: The step of fitting includes determining a two-dimensional distinguishing feature describing the spatial distribution of the performance parameter.

5. The method of claim 1, further comprising: Lagrangian multipliers are defined for the complexity index, the complexity index is converted to a Lagrangian function using the Lagrangian multipliers, and the Lagrangian function is converted to a quadratic programming optimization.

6. The method of claim 5, wherein: The step of fitting includes determining model parameters as a linear combination of a design matrix and optimized values ​​for the Lagrange multipliers.

7. The method of claim 1, wherein: The measurement data describes one or more of: characteristics of the substrate; characteristics of a pattern forming device that defines a pattern to be applied to the substrate; the position of one or both of a substrate stage for holding the substrate and a mask stage for holding the pattern forming device; or characteristics of a pattern transfer system that transfers the pattern on the pattern forming device to the substrate.

8. The method of claim 1, wherein: The measurement data includes one or more of overlay data, critical dimension data, alignment data, focus data, and leveling data.

9. The method of claim 1, wherein: The complexity indicator involves controlling the lithography process to optimize one or more of the following: exposure trajectory control in a direction parallel to the substrate plane, exposure trajectory control in a direction perpendicular to the substrate plane, lens aberration correction, dose control, and laser bandwidth control of a laser source used for the lithography equipment.

10. The method of claim 9, comprising: The photolithography process is controlled according to the optimized control.

11. The method of claim 9, wherein: The lithography process involves the exposure of layers on a substrate, which forms part of a manufacturing process for manufacturing integrated circuits.

12. The method of claim 1, the complexity index being operable to minimize one or more of overlay error, edge placement error, critical dimension error, focus error, alignment error, and leveling error.

13. A non-transitory computer program carrier comprising a computer program comprising program instructions operable to perform the method as claimed in claim 1 when run on a suitable device.

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