Method for modeling measurement data on a substrate region and associated device

Abstract

A method and associated apparatus for modeling measurement data on a substrate region are disclosed. The method comprises: obtaining substrate measurement data describing spatial variations in substrate parameters; and fitting a model to the substrate measurement data to obtain a fitted model; wherein the fitting step comprises regularizing the fitting using a plurality of regularization parameters, the plurality of regularization parameters having been individually optimized to penalize the spatial components of the substrate measurement data according to the influence of the spatial components of the substrate measurement data on a parameter of interest.

Description

Cross-references to related applications

[0001] This application claims priority to European application 23166036.6, filed on March 31, 2023, which is incorporated herein by reference in its entirety. Technical Field

[0002] This disclosure relates to the processing of substrates used in the production of, for example, semiconductor devices. Background Technology

[0003] A lithography apparatus is a machine configured to apply a desired pattern onto a substrate. Lithography apparatus can be used, for example, in the manufacture of integrated circuits (ICs). A lithography apparatus can project a pattern (also often referred to as a “design layout” or “design”) from a patterning apparatus (e.g., a mask) onto a radiation-sensitive material (resist) layer disposed on a substrate (e.g., a wafer).

[0004] To project a pattern onto a substrate, a photolithography apparatus can use radiation. The wavelength of this radiation determines the minimum size of the feature that can be formed on the substrate. Typical wavelengths currently in use are approximately 365 nm (i-line), approximately 248 nm, approximately 193 nm, and approximately 13 nm. Compared to photolithography apparatuses using radiation with a wavelength, for example, 193 nm, photolithography apparatuses using extreme ultraviolet (EUV) radiation with wavelengths in the range of 4 nm to 20 nm (e.g., 6.7 nm or 13.5 nm) can be used to form smaller features on the substrate.

[0005] Low-k1 lithography can be used to process features with dimensions smaller than the classical resolution limit of a lithography apparatus. In such a process, the resolution can be expressed as CD = k1 × λ / NA, where λ is the wavelength of the radiation used, NA is the numerical aperture of the projection optics in the lithography apparatus, CD is the “critical size” (typically the size of the smallest feature to be printed, but in this case, half a pitch), and k1 is an empirical resolution factor. Generally, the smaller k1 is, the more difficult it becomes to reproduce patterns on the substrate that resemble the shape and size planned by the circuit designer to achieve specific electrical functionality and performance. To overcome these difficulties, complex fine-tuning steps can be applied to the lithography projection apparatus and / or design layout. These steps include, for example, but not limited to, optimization of the numerical aperture (NA), custom illumination schemes, use of one or more phase-shifting patterning devices, optimization of the design layout such as optical proximity correction (OPC) in the design layout, or other methods generally defined as “resolution enhancement techniques” (RET). Additionally or alternatively, one or more tightly controlled loops for controlling the stability of the lithography apparatus can be used to improve pattern reproduction at low k1.

[0006] The effectiveness of control over a lithography apparatus can depend on the characteristics of the individual substrates. For example, a first substrate (or any other process step in the manufacturing process, collectively referred to herein as a manufacturing process step) that is processed by a first processing tool before being processed by the lithography apparatus can benefit from (slightly) different control parameters than a second substrate that is processed by a second processing tool before being processed by the lithography apparatus.

[0007] Accurately placing patterns on a substrate is a major challenge in reducing the size of circuit components and other products that can be produced through photolithography. Specifically, the challenge of accurately measuring the features already laid on the substrate is a critical step, which lies in being able to align successive layers of features in the stack with sufficient accuracy to produce functional devices with high yield. Generally speaking, in today's submicron semiconductor devices, so-called overlap should be achieved within tens of nanometers, and in the most critical layers, it should be achieved within a few nanometers.

[0008] Therefore, modern lithography equipment involves numerous measurement or 'mapping' operations prior to the actual exposure or other patterning of the substrate at target locations. So-called advanced alignment models have been developed and continue to be developed to more accurately model and correct nonlinear deformations of the wafer 'grid' caused by processing steps and / or the lithography equipment itself. However, not all deformations can be corrected during exposure, and it remains important to track and eliminate as many of the causes of such deformations as possible.

[0009] These deformations of the wafer grid are represented by measurement data associated with the marking positions. Measurement data is obtained from measurements of the wafer. One example of such measurement is the alignment measurement of alignment marks performed before exposure using an alignment system in a lithography apparatus. Another example of such measurement is the overlap measurement of overlapping targets performed after exposure using a metrology system.

[0010] Regularization can be employed to improve fit and prevent overfitting of the model (e.g., alignment model) to the measurement data (e.g., alignment data). Improvements to current regularization techniques are desired. Summary of the Invention

[0011] In a first aspect of the invention, a method is provided for modeling substrate measurement data associated with a substrate in a photolithography process, the method comprising: obtaining substrate measurement data describing spatial variations of substrate parameters; and fitting a model to the substrate measurement data to obtain a fitted model, wherein the fitting step includes regularizing the fit using a plurality of regularization parameters, the plurality of regularization parameters being individually optimized to penalize the spatial components of the substrate measurement data based on the influence of the spatial components of the substrate measurement data on parameters of interest.

[0012] In another aspect of the invention, a computer program is provided, the computer program including program instructions operable to perform the method according to the first aspect when run on a suitable device, and an associated processing device and a lithography device are provided. Attached Figure Description

[0013] Embodiments of the invention will now be described by way of example only with reference to the accompanying schematic diagrams, in which:

[0014] Figure 1 A schematic overview diagram of a photolithography apparatus is depicted;

[0015] Figure 2 A schematic overview diagram of the photolithography unit is depicted;

[0016] Figure 3 schematically shown Figure 1 and Figure 2 The use of lithography equipment and lithography units, together with one or more other facilities forming a manufacturing facility for, for example, semiconductor devices, wherein the facility implements a control strategy according to embodiments of the present invention; and

[0017] Figure 4 This is a flowchart describing a method according to an embodiment of the present invention. Detailed Implementation

[0018] Figure 1 A lithography apparatus LA is schematically depicted. The lithography apparatus LA includes: an irradiation system (also referred to as an irradiator) IL configured to modulate a radiation beam B (e.g., UV radiation, DUV radiation, or EUV radiation); a support (e.g., a mask stage) T configured to support a patterning apparatus (e.g., a mask) MA and connected to a first positioner PM configured to accurately position the patterning apparatus MA according to certain parameters; one or more substrate supports (e.g., wafer stages) WTa and WTb configured to hold a substrate (e.g., a resist-coated wafer) W and connected to a second positioner PW configured to accurately position the substrate supports according to certain parameters; and a projection system (e.g., a refractive projection lens system) PS configured to project a pattern imparted by the radiation beam B by the patterning apparatus MA onto a target portion C (e.g., comprising one or more dies) of the substrate W.

[0019] During operation, the irradiation system IL receives a radiation beam from a radiation source SO (e.g., via a beam delivery system BD). The irradiation system IL may include various types of optical components for guiding, shaping, and / or controlling the radiation, such as refractive optical components, reflective optical components, magnetic optical components, electromagnetic optical components, electrostatic optical components, and / or other types of optical components, or any combination thereof. The irradiator IL can be used to adjust the radiation beam B to have a desired spatial and angular intensity distribution in a cross-section of the radiation beam B located at the plane of the pattern forming apparatus MA.

[0020] Depending on the exposure radiation being used and / or other factors such as the use of immersion liquids or vacuum, the term "projection system" PS as used herein should be interpreted broadly to encompass all types of projection systems, including refractive optical systems, reflective optical systems, reflective-refractive optical systems, distorting optical systems, magnetic optical systems, electromagnetic optical systems, and / or electrostatic optical systems, or any combination thereof. Any use of the term "projection lens" herein may be considered synonymous with the more general term "projection system" PS.

[0021] A lithography apparatus LA can be of the type in which at least a portion of the substrate can be covered by a liquid (e.g., water) having a relatively high refractive index to fill the space between the projection system PS and the substrate W; this is also known as immersion lithography. Further information on immersion techniques is given in U.S. Patent No. 6,952,253, which is incorporated herein by reference.

[0022] The lithography apparatus LA in this example is a so-called dual-platform type, which has two substrate stages WTa and WTb and two stations—an exposure station and a measurement station—between which the substrate stages can move. While one substrate on one stage is being exposed at the exposure station EXP, another substrate can be loaded onto the other substrate stage, for example, at the measurement station MEA, or at another location (not shown), or can be processed at the measurement station MEA. The substrate stage with the substrate can be located at the measurement station MEA, allowing various preparation steps to be performed. Preparation steps may include mapping the surface height of the substrate using a level sensor LS and / or measuring the position of alignment marks on the substrate using an alignment sensor AS. The alignment marks are nominally arranged in a regular grid pattern. However, due to inaccuracies in the creation of the marks and also due to substrate deformation that occurs throughout its processing, the marks may deviate from the ideal grid. Therefore, in addition to measuring the position and orientation of the substrate, the alignment sensor can actually measure the position of many marks over the entire substrate area in detail if the apparatus LA is to print product features in the correct positions with high accuracy. Therefore, the measurement of alignment marks can be time-consuming, and providing two substrate stages significantly increases the throughput of the apparatus. If the position sensor IF cannot measure the position of the substrate stage when it is at the measurement station and the exposure station, a second position sensor can be provided to enable tracking of the substrate stage position at both stations. Embodiments of the invention can be applied to devices having only one substrate stage or more than two substrate stages.

[0023] In addition to having one or more substrate supports, the lithography apparatus LA may also include a measurement platform (not shown). The measurement platform is arranged to hold sensors and / or cleaning devices. The sensors may be arranged to measure the properties of the projection system PS or the properties of the radiation beam B. The measurement platform may hold multiple sensors. The cleaning devices may be arranged to clean part of the lithography apparatus, for example, part of the projection system PS or part of a system providing an immersion liquid. The measurement platform may move below the projection system PS as the substrate support WT moves away from the projection system PS.

[0024] A radiation beam B is incident on a patterning apparatus (e.g., a mask MA), which is held on a support structure (e.g., a mask stage) MT, and the radiation beam B is patterned by the patterning apparatus MA. After passing through the patterning apparatus MA, the radiation beam B passes through a projection system PS, which focuses the beam onto the target portion C of the substrate W. The substrate stage WTa / WTb can be accurately moved, for example, to position different target portions C within the path of the radiation beam B, by means of a second positioner PW and a position sensor IF (e.g., an interferometric device, a linear encoder, or a capacitive sensor). Similarly, for example, after mechanical retrieval from a mask library or during scanning, a first positioner PM and another position sensor (the other position sensor is located in…) Figure 1 (Not explicitly depicted) can be used to accurately position the pattern forming apparatus MA relative to the path of the radiation beam B. Generally, the moving support structure MT can be implemented by means of a long-stroke module (coarse positioning) and a short-stroke module (fine positioning), which form part of the first positioner PM. Similarly, the long-stroke module and the short-stroke module can be used to move the substrate stage WTa / WTb, which form part of the second positioner PW. In the case of a stepper (as opposed to a scanner), the support structure MT can be connected only to the short-stroke actuator, or it can be fixed. The pattern forming apparatus MA and the substrate W can be aligned using pattern forming apparatus alignment marks M1, M2 and substrate alignment marks P1, P2. Although the substrate alignment marks, as illustrated, occupy a dedicated target portion, they can be located in the space between the target portions (these target portions are referred to as scribing alignment marks). Similarly, in cases where more than one die is provided on the pattern forming apparatus MA, the pattern forming apparatus alignment marks can be located between the dies.

[0025] The equipment also includes a Lithography Equipment Control Unit (LACU), which controls all movement and measurement of the various actuators and sensors (such as those described) within the lithography equipment. The LACU also includes signal and data processing capabilities to perform desired calculations related to the operation of the equipment. In practice, the LACU will be implemented as a system of multiple sub-units, each performing real-time data acquisition, processing, and control of subsystems or components within the equipment. For example, one processing subsystem could be dedicated to the servo control of the substrate positioner (PW). A separate unit could even handle coarse and fine adjustment actuators or different axes. Another unit could be dedicated to the readout of the position sensor (IF). Overall control of the equipment can be controlled by a central processing unit that communicates with these subsystem processing units, the operator, and other equipment involved in the lithography process.

[0026] like Figure 2 As shown, a lithography apparatus LA can form part of a lithography unit LC, sometimes referred to as a lithography unit or (lithography) cluster. This lithography unit or (lithography) cluster may also include equipment for performing pre-exposure and post-exposure processes on a substrate W. Typically, this equipment includes one or more spin coaters SC for depositing a resist layer, one or more developers DE for developing the exposed resist, one or more chillers CH for regulating the temperature of the substrate W (e.g., for regulating the solvent in the resist layer), and one or more baking plates BK. A substrate processor or robot RO picks up the substrate from input / output ports I / O1, I / O2, moves the substrate between different processing devices, and then transfers the substrate W to the loading stage LB of the lithography apparatus LA. These devices in the lithography unit, often collectively referred to as the track or coating / developing system, are typically under the control of a track control unit or coating / developing system control unit TCU, which is itself controlled by a management control system SCS, which also controls the lithography apparatus LA, for example, via a lithography control unit LACU.

[0027] To ensure correct and consistent exposure of the substrate W exposed by the lithography apparatus LA, it is desirable to inspect the substrate to measure properties of the patterned structure, such as overlap error between subsequent layers, line thickness, critical dimension (CD), etc. For this purpose, one or more inspection tools (not shown) may be included in the lithography unit LC. If an error is detected, adjustments can be made, for example, to the exposure of subsequent substrates or other processing steps to be performed on the substrate W, especially if the inspection was performed before other substrates W in the same batch or batch are still to be exposed or processed.

[0028] An inspection device (also referred to as a metrology device or measurement tool) is used to determine one or more properties of a substrate W, particularly how one or more properties vary between different substrates W or how one or more properties associated with different layers of the same substrate W vary between layers. The inspection device can be configured to identify defects on the substrate W and can be, for example, part of a lithography unit (LC), integrated into a lithography apparatus (LA), or even a stand-alone device. The inspection device can measure one or more properties of: a latent image (an image in a resist layer after exposure), or a semi-latent image (an image in a resist layer after a post-exposure baking step), or a developed resist image (where the exposed or unexposed portions of the resist have been removed), or even an etched image (after a pattern transfer step such as etching).

[0029] Figure 3A lithography apparatus LA and a lithography unit LC are shown in the context of an industrial manufacturing facility for, for example, semiconductor products. Within the lithography 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. As already described, the lithography tool 200 forms part of a "lithography unit" or "lithography cluster," which also includes a coating apparatus SC 208 for applying photoresist and / or one or more other coatings to a substrate W for patterning via the apparatus 200. At the output side of the apparatus 200, a baking apparatus BK 210 and a developing apparatus DE 212 are provided for developing the exposed pattern into a physical or solid resist pattern. (The remaining text is omitted for clarity.) Figure 3 Other components shown.

[0030] Once the pattern has been applied and developed, the patterned substrate 220 is transferred to other processing equipment, such as those shown as 222, 224, and 226. In a typical manufacturing facility, a wide range of processing steps are performed using various equipment. For illustrative purposes, equipment 222 in this embodiment is an etching station, and equipment 224 performs a post-etch annealing step. Other physical and / or chemical processing steps are applied in other equipment such as 226. Many types of operations (such as material deposition, modification of surface material properties (oxidation, doping, ion implantation, etc.), chemical mechanical polishing (CMP), etc.) may be required to fabricate actual devices. In practice, apparatus 226 can represent a series of different processing steps performed in one or more devices.

[0031] The semiconductor manufacturing process described herein, which includes a series of patterning process steps, is merely one example of an industrial process in which the techniques disclosed herein can be applied. The semiconductor manufacturing process includes a series of patterning steps. Each patterning process step includes a patterning operation (e.g., photolithographic patterning operation) and several other chemical and / or physical operations.

[0032] Manufacturing semiconductor devices involves repeated processing to build a device structure with appropriate materials and patterns layer by layer on a substrate. For example, modern device manufacturing processes may include 40 or 50 individual patterning steps. Thus, the substrate 230 arriving at the photolithography cluster may be a newly prepared substrate, or it may be a substrate previously processed in that cluster 232 or entirely in another device. Similarly, depending on the required processing, the substrate leaving device 226 may be returned for subsequent patterning operations in the same photolithography cluster (e.g., substrate 232), the substrate leaving device 226 may be designated for patterning operations in a different cluster (e.g., substrate 234), or the substrate leaving device 226 may be a finished product to be sent for dicing and packaging (e.g., substrate 234).

[0033] Each layer of a product structure typically comprises a different set of process steps, and the equipment used at each layer can be entirely different in type. Furthermore, even when the processing steps to be performed by the equipment are nominally identical, in large facilities, there may be several supposedly identical machines operating in parallel to perform processing on different substrates. Minor differences in setup or malfunction between these machines can mean that the setup or malfunction affects the different substrates in different ways. Even relatively common steps for each layer (such as etching (equipment 222)) can be implemented by several etching devices that are nominally identical but operate in parallel to maximize throughput. Parallel processing can also be performed in different chambers within a larger facility. Moreover, in practice, depending on the details and specific requirements of the material to be etched (such as, for example, anisotropic etching), different layers often involve different etching processes, such as chemical etching, plasma etching, etc.

[0034] As previously mentioned, preceding and / or subsequent processes can be performed in other lithography equipment, and even in different types of lithography equipment. For example, in device fabrication, one or more layers with very high requirements, such as resolution and / or overlap, can be performed in more advanced lithography tools compared to one or more other layers with lower requirements. Therefore, one or more layers can be exposed in an immersion lithography tool while one or more other layers are exposed in a "drying" tool. One or more layers can be exposed in a tool operating at DUV wavelengths while one or more other layers are exposed using EUV wavelength radiation.

[0035] Figure 3A metrology equipment (MET) 240 is also shown, which is provided for measuring parameters of the product at a desired stage in the manufacturing process. A common example of a metrology station in modern lithography manufacturing facilities is a scatterometer, such as an angle-resolved scatterometer or a spectral scatterometer, which can be used to measure one or more properties of the developed substrate at 220 prior to etching in apparatus 222. Using metrology equipment 240, performance parameter data PDAT 252 can be determined. Based on this performance parameter data PDAT 252, it can also be determined whether performance parameters such as overlap or critical dimension (CD) 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 one or more of the substrates 220 through the lithography cluster. Moreover, by fine-tuning over time, the measurement results from metrology equipment 240 can be used to maintain accurate execution of patterning operations in the lithography cluster, thereby reducing or minimizing the risk of manufacturing defective products and requiring rework. Of course, measurement device 240 and / or one or more other measurement devices (not shown) can be used to measure one or more properties of the processed substrates 232, 234 and / or the substrate 230 being input.

[0036] Typically, the patterning process in photolithography (LA) equipment is one of the most critical steps in the process, involving the precise dimensional determination and placement of structures on the substrate W. To help ensure this high accuracy, techniques such as... Figure 3 The schematic diagram illustrates a control environment that combines three systems. One of these systems is a lithography tool 200, which is (virtually) connected to a metrology device 240 (the second system) and a computer system CL 250 (the third system). The aim of this environment is to optimize or improve the collaboration between these three systems to enhance the overall so-called “process window” and provide one or more tight control loops to help ensure that the patterning performed by the lithography device LA remains within the process window. The process window defines a range of values ​​for multiple process parameters (e.g., two or more selected from dose, focus, overlap, etc.) within which a particular manufacturing process produces a defined result (e.g., a functional semiconductor device)—this range is typically the range within which the values ​​of process parameters during the lithography or patterning process can vary while producing (e.g., a range specified according to the acceptable range of CD (such as ±10% of the nominal CD)) an appropriate structure.

[0037] The computer system CL can use a portion of the design layout to be patterned to predict which one or more resolution enhancement techniques will be used and perform computational lithography simulations and calculations to determine which patterning apparatus layout and lithography equipment setup will achieve the maximum overall process window (within) the patterning process. Figure 3(Depicted by double arrows in the first dial SC1). Typically, resolution enhancement techniques are arranged to match the patterning possibilities of the lithography apparatus LA. The computer system CL can also be used (e.g., using input from the metrology tool MET) to detect where the lithography apparatus LA is currently operating within the process window to predict whether defects may exist due to, for example, suboptimal processing (in...). Figure 3 The arrow pointing to "0" is depicted in the second dial SC2.

[0038] The measurement tool MET can provide input to the computer system CL for accurate simulation and prediction, and can provide feedback to the lithography equipment LA to identify possible drifts in, for example, the calibration status of the lithography equipment LA. Figure 3 (The text is depicted by multiple arrows on the third dial SC3).

[0039] The computer system 250 can control the process based on a combination of the following: (i) “preprocessing measurement data” (e.g., including scanner measurement data LADAT 254 and external preprocessing measurement data ExDAT 260), which is associated with the substrate before it is processed in a given processing step (e.g., a photolithography step); and (ii) performance data or “postprocessing data” PDAT 252, which is associated with the substrate after it has been processed.

[0040] The first set of preprocessed measurement data LADAT 254 (referred to herein as scanner measurement data because it is generated by the lithography apparatus LA 200 or a scanner) may include alignment data routinely obtained by the lithography apparatus LA 200 using the alignment sensor AS in the measurement station 202. Alternatively, or in addition to alignment data, the scanner measurement data LADAT 254 may include height data obtained using the level sensor LS and / or a “wafer quality” signal from the alignment sensor AS, etc. Thus, the scanner measurement data LADAT 254 may include the alignment grid of the substrate and data related to substrate deformation (flatness). For example, the scanner measurement data LADAT 254 may be generated by the measurement station MEA 202 of the dual-platform lithography apparatus LA 200 before exposure (e.g., because the measurement station MEA 202 typically includes an alignment sensor and a leveling sensor), thereby enabling simultaneous measurement and exposure operations. Such dual-platform lithography apparatuses are well known.

[0041] A separate alignment tool or external pre-exposure meter, ExM 270, can also (optionally) be used for measurements prior to exposure on the lithography equipment. This external pre-exposure meter, ExM 270, is different from the measurement station MEA 202 of the dual-platform lithography equipment LA 200. Any of the measurement or alignment data described herein can be measured by this external pre-exposure meter, which serves as an alternative to or supplement to the alignment tool within the scanner or lithography equipment.

[0042] Although Figure 3 Separate storage devices 252, 254, and 260 are shown for each of the performance data PDAT, scanner measurement data LADAT, and external pre-exposure data ExDAT, but it will be appreciated that these different types of data may be stored in a common storage unit or distributed across a greater number of storage units from which specific data items may be retrieved when needed.

[0043] To represent alignment measurements on the wafer and / or field, an alignment model is used. The primary purpose of the alignment model is to provide a mechanism for interpolating and / or extrapolating available measurement data across the entire wafer, allowing the creation of an exposure grid on each die. The measurement data will be sparse because, from a modeling perspective, measuring as many measurement areas as desired is impractical: the time and therefore yield overhead would be too high. The secondary purpose of the alignment model is to provide noise suppression. This can be achieved by using fewer model parameters than the measurements themselves.

[0044] While standard models (e.g., linear models) may use fewer than ten parameters, advanced alignment models typically use more than 15 or even 30 parameters. Examples of advanced models are the High-Order Wafer Alignment (HOWA) model, Zone Alignment (ZA) models, and alignment models based on Radial Basis Functions (RBF). HOWA is a publicly available technique based on second-, third-, and higher-order polynomial functions. RBF modeling is described in US 2012218533 A1, which is incorporated herein by reference, and essentially involves interpolation methods using thin-plate splines based on radial basis functions, for example. Different versions and extensions of these advanced models are conceivable. Advanced models generate complex descriptions of the wafer grid, which are corrected during exposure of the target layer. Recent versions of RBF and HOWA provide particularly complex descriptions based on dozens of parameters. This means that a large number of measurements are required to obtain a wafer grid with sufficient detail.

[0045] US2012218533A1 describes an RBF modeling method comprising the steps of: generating radial basis functions using marked measurement locations; and calculating model parameters of the substrate within the device using the generated radial basis functions as basis functions across the substrate. RBF It is a real-valued function whose value depends only on the distance from the origin, or alternatively on the distance x from some other point called the center c, such that:

[0046] The functional approximation using RBF can be established in the following form: Among them, the approximate function It is represented as the sum of N radial basis functions (RBFs), each of which is associated with a distinct center c and is determined by appropriate coefficients w. i Weighted, and ||·|| is the notation used for the standard Euclidean vector norm. This can satisfy the interpolation condition Y(x) i ) = y i The method of calculating weight w using the least squares method. i A linear system with weighted coefficients might look like this: in, and It is the distance between two points (e.g., the distance between two markers). Note that there are as many weighting coefficients (i.e., degrees of freedom) as there are interpolation conditions. The resulting system of equations is nonsingular (invertible) under very mild conditions and therefore has a unique solution. For many radial basis functions (RBFs), the only restriction is that at least three points are not on a straight line.

[0047] Several alternatives to the RBF are possible, such as Gaussian functions, inverse basis functions, multiple quadratic basis functions, inverse quadratic basis functions, spline-degree k-basis functions, and thin-plate spline basis functions. Note that other RBFs are also possible. Two main classes of RBFs are given below: infinite smooth (whose derivative exists at every point); and spline (whose derivative may not exist at some points). Thin-plate spline (TPS) refers to a physical simulation involving the bending of a thin metal sheet. In the physical setting, the deflection is in the z-direction orthogonal to the sheet plane. To apply this idea to the problem of substrate deformation during photolithography, the lifting of the sheet can be interpreted as a displacement in the x or y coordinates within the plane. TPS has been widely used as a non-rigid transformation model in image alignment and shape matching.

[0048] Therefore, RBF can be regularized using bending energy (BE), which minimizes the combination of the model residuals and the model's second derivative as squared integrals over an infinite plane. This approach is described, for example, in WO2022101204A1, which is incorporated herein by reference. In this BE-regularized RBF model, the data perfectly fits the least second derivative of the model function. This is equivalent to solving the problem of minimizing the fitting coefficients c: (1.1) in, It consists of both the model matrix and the BE regularization matrix, and the data is measurement data (e.g., alignment data). The model function used for this model. Including thin plate splines.

[0049] The unknown parameter in the above problem is the regularization parameter λ, which can be determined during the training phase, for example, on a training or calibration wafer from which first training data or alignment training data and corresponding second training data or overlap (or other placement errors, such as edge placement errors) training data can be measured. Alignment training data x train This can include an alignment data training matrix (size: number of wafers × alignment APD (alignment position deviation) for each marker), and overlapping training data y train This can include an overlap data training matrix (size: number of wafers × decorrected overlap APD for each overlap target). The decorrected overlap data can be overlap data that has already been decorrected for alignment correction (e.g., this could include the difference between the overlap data and the corresponding alignment correction). The first model matrix M... x (The first model parameters) may include a model matrix for aligning the grid, and a second model matrix M y (The second model parameters) can include the model matrix used for overlapping rasters. The regularization parameter λ can be optimized to minimize the (decorrected) overlap: (1.2)

[0050] However, since only a single regularization parameter is optimized, this regularization may be suboptimal. In particular, not all spatial content in the aligned and overlapping data must be penalized equally, because not all fingerprint components are equally correlated between alignment and overlap.

[0051] Wafer Alignment Model Mapping (WAMM) is a linear regression method that improves wafer alignment by training on alignment and overlap data. This method suppresses shapes (spatial components) in wafer alignment that do not improve overlap, such as shapes arising from measurement noise, marker distortion, or extrapolation errors.

[0052] In wafer alignment model mapping (WAMM), the model parameters are modified by applying a linear transformation. This linear transformation can be considered as matrix multiplication: the model parameters are arranged together as column vectors, which are multiplied by a matrix to obtain new column vectors of model parameters. The matrix involved is called the model mapping matrix. The primary purpose of the model mapping matrix is ​​as a linear filter: it aims to improve the (e.g., overlap) performance of wafer alignment by filtering out spatial components (shapes) that degrade overlap and modifying the remaining shape to further improve overlap. The concept of wafer alignment model mapping is described in WO2017060054 (which is incorporated herein by reference) and will now be briefly described.

[0053] In this method, through the model mapping matrix M wamm The wafer alignment parameters are optimized by mapping the wafer alignment parameters to overlap parameters, wherein the model mapping matrix M wamm It has a size equal to the square of the number of parameters. Model mapping matrix M wamm It can be determined by the following formula: (1.3)

[0054] Then, the model mapping matrix M can be used. wamm To scale the alignment parameters to M wamm c. These scaled parameters are used to fit the evaluation data. WAMM suffers from overfitting due to the large number of degrees of freedom in the model mapping matrix. To improve this, cross-validation is used during training. Even so, currently some number of training chips between 100 and 200 are still needed to properly train the matrix.

[0055] To address the aforementioned issues using current regularization strategies and / or WAMM, this paper discloses an improved regularization technique. In this technique, multiple (and preferably all) of the regularization matrix elements (regularization parameters) in the regularization used to optimize the fitting of the regularization model to substrate measurement data (e.g., alignment data or overlapping data) are proposed, rather than a single regularization parameter λ.

[0056] The proposed regularization reduces sensitivity to overfitting, but its correction potential as an optimal model regularization technique is less than that of WAMM. This is due to the fact that WAMM can amplify model components, while regularization can only attenuate those components by design. Furthermore, regularization can improve the interpolation and extrapolation errors of the model more effectively than WAMM. In practice, this means that regularization can achieve similar or better results with fewer training chips compared to WAMM. Note that this regularization concept can also be used for other modeling applications, for example, to model sparse overlapping layouts using optimal model regularization to achieve similar performance to modeling dense overlapping layouts.

[0057] Two specific embodiments will now be described. The first such embodiment may be referred to as Optimal Model Regularization (OMR). This method may include optimizing the full model regularization matrix K (or at least a plurality of regularization matrix elements) to minimize overlap. As previously mentioned, this can be achieved by using one or more training wafers from which first training data (e.g., alignment data x) is measured. train ) and related second training data (e.g., overlapping data y) train Optimization is performed during the training phase. This embodiment does not require any initial shape settings or correlation analysis for determining shape-specific regularization parameters.

[0058] In one particular embodiment, optimization may include: (1.4) Here, K is optimized under the constraint of being a positive symmetric matrix, such that only half of the regularized matrix elements need to be trained. This will also produce an n×n matrix, thus achieving eigenvalue decomposition. As mentioned earlier, M x It is the first model matrix or the aligned raster model matrix, and M y It can include a second model matrix or an overlapping raster model matrix.

[0059] The regularization matrix K of this model can be intuitively interpreted as a decay of a specific shape that does not improve overlap.

[0060] As a purely optional further step, the shape and its attenuation factor can be found using appropriate decomposition techniques, such as single-valued decomposition (SVD) or eigenvalue decomposition of the model regularization matrix K. For example, this can be performed for analytical purposes, such as creating aligned data, overlapping data, and overlap corrections, using orthogonal shape decomposition.

[0061] In WO2019001871A1 (in the context of WAMM), which is incorporated herein by reference, an eigenvalue decomposition for finding a specific shape is described. In short, such a decomposition may include: decomposing the model regularization matrix K to individually determine, for example, a first deformation, which in the dataset x train y train The mapping between the two deformations is scaled differently from the second deformation. The decomposition involves: computing the eigenvalue decomposition for the model regularization matrix K, and using the eigenvalues ​​in the eigenvalue matrix D representing different scaling to identify one or more eigenvectors V representing the first deformation. The decomposition can typically be represented as: (1.5) Where V can be considered as the shape decomposition matrix representing the deformation; D is a diagonal matrix containing eigenvalues; and V -1 It is a matrix containing shapes. Note that since the regularization matrix is ​​positive and symmetric, all eigenvalues ​​are positive, and the eigenvectors are orthogonal. Therefore, the inverse or reciprocal of V can also be written at the transpose of V. This can be viewed as decomposing the model regularization matrix K into shapes that are scaled by coordinate transformations and then decomposed back to the original coordinates.

[0062] For example, if the model used has a full-rank model matrix on a sparsely aligned grid, this model matrix can first be orthogonally normalized to the sparsely aligned grid. In doing so, This is transformed into an identity matrix. As a next step, the regularization matrix can be diagonalized using orthogonal eigenvalue decomposition. It can be understood that after orthogonal normalization of the model, the regularization matrix remains positive and symmetric because it is multiplied on both sides by the coordinate transformation matrix and its transpose. The remaining modeling scheme after these two fundamental transformations can be directly converted into a diagonal WAMM matrix, which includes elements equal to 1 / (1+d). ii The diagonal elements of ), where d iiThese are the eigenvalues ​​of the regularization matrix after orthogonalizing the model matrix. In the second example, if the model matrix used is not full-rank and ill-conditioned on a sparse alignment grid, then the model matrix cannot be orthogonally normalized because it is not full-rank. In this case, regularization can be used, and regularization is needed to better tune the model. A similar conclusion can be reached using Generalized Singular Value Decomposition (GSVD). As input data for GSVD, the "square roots" of the model matrix and the regularization matrix can be used. The "square roots" of the regularization matrix can be computed using, for example, Cholesky or LDL decomposition. To convert it to a WAMM, the model can be restricted to a relevant subspace such that the subspace model becomes better tuned on the alignment layout (e.g., is a full-rank model matrix). This can be done by removing the model dimension corresponding to the zero diagonal elements of the diagonal C matrix generated by GSVD (A=U*C*X' and B=V*S*X', where A is the model matrix and B is the "square root" of the regularization matrix).

[0063] The second embodiment can be referred to as Shape Improvement Based Regularization (SIBR). SIBR can determine shape-specific regularization factors based on identifying shapes in the alignment and / or overlap data that explain most of the variation, and using optimization of the overlap or using different criteria, such as the correlation of each shape between the first and second training data (e.g., obtained according to the OMR embodiment).

[0064] Therefore, the method may include: decomposing the model into specific shapes and adjusting a regularization factor on a shape-by-shape basis or on a per-shape basis. The regularization factor may be optimized for overlap or may be adjusted using other criteria; for example, the wafer-to-wafer correlation between alignment and overlap. In this way, regularization can suppress unwanted shapes that induce greater overlap during wafer alignment modeling.

[0065] For shape selection, various methods can be used, such as appropriate component analysis (e.g., Principal Component Analysis (PCA) or Independent Component Analysis (ICA)) or decomposition techniques (e.g., SVD). Taking PCA as an example, this produces orthogonal shapes that can be optimally ordered by the intensity of overlap contributions. BE regularization can be used as a starting point for regularization to be scaled / optimized (base regularization), such that the amount of BE regularization can be scaled according to the shape. The regularization optimization problem can be written as the SIBR equation: (1.6) in, ; And R is BE regularization. It can be called an SIBR matrix.

[0066] In this equation, assume matrix M x M y R has been transformed into the chosen shape basis by multiplying them with the appropriate fundamental transformation matrix, i.e., such that each column corresponds to the chosen shape.

[0067] Figure 4 This is a flowchart describing a method for finding a regularization matrix K for n model shapes. This example uses PCA-based shape selection, bending energy regularization as a starting point, and wafer-to-wafer correlation criteria. For example, training data 400 (e.g., first training data or alignment data and second training data or overlap data) is obtained by exposing and measuring training wafers. At step 410, principal component analysis is performed on the overlap dataset to find orthogonal overlapping shapes, and they are ranked by the intensity of their overlap contribution. At step 420, orthogonal shapes are fitted on the overlap dataset and the wafer alignment dataset, respectively, to obtain the alignment raster model matrix and the overlap raster model matrix. At step 430, the wafer-to-wafer correlation (e.g., correlation metric) between the overlap and alignment fitting coefficients is determined.

[0068] Step 440 includes constructing the SIBR matrix. Regularization can be applied to each shape based on the correlation metric determined in step 430, for example, according to the inverse relationship between correlation level and moderation level (i.e., such that highly correlated shapes have a low applied regularization penalty, and vice versa). Therefore, the SIBR matrix can include: (1.7) Where f(·) is the regularization penalty λ1,…,λ that maps the correlation to each shape. n A function of f. A simple example of f is f(x) = 1 - x.

[0069] Finally, at step 450, the SIBR matrix determined in the previous step is used. The model regularization matrix K is determined using the above equation (1.6).

[0070] Regardless of the implementation, once trained, the model regularization matrix K can be used to regularize the fitting of the model to substrate measurement data (e.g., alignment or overlap data). The details of this fitting can depend on the model used, but can take, for example, the following form:

[0071] Other embodiments of the invention are disclosed in the following list of numbered entries: 1. A method for modeling substrate measurement data associated with a substrate in a photolithography process, the method comprising: obtaining substrate measurement data describing spatial variations of substrate parameters; and fitting a model to the substrate measurement data to obtain a fitted model, wherein the fitting step includes regularizing the fit using a plurality of regularization parameters, the plurality of regularization parameters being individually optimized to penalize the spatial components of the substrate measurement data based on the influence of the spatial components of the substrate measurement data on parameters of interest. 2. The method according to item 1, wherein the plurality of regularization parameters are included in a regularization matrix, wherein each regularization matrix element includes a corresponding regularization parameter. 3. The method according to clause 1 or 2, wherein the plurality of regularization parameters are operable to apply a penalty level to the spatial component based on the inverse relationship of the influence of the spatial component on the parameter of interest. 4. The method according to any one of the preceding clauses, wherein the substrate measurement data includes overlap data. 5. The method according to any one of clauses 1 to 3, wherein the substrate measurement data includes alignment data. 6. The method according to any one of the preceding clauses, wherein the parameter of interest includes overlap or another placement error measure. 7. The method according to any one of the preceding clauses, wherein the individual optimization comprises: Obtain first training data associated with one or more training substrates; Obtain second training data associated with the one or more training substrates; and The plurality of regularization parameters are individually optimized to obtain the plurality of regularization parameters. 8. The method according to item 7, wherein the separate optimization step comprises: optimizing the regularization parameters using a cost function associated with the first training data and the second training data, based on first model parameters associated with the first training data, second model parameters associated with the second training data, and the plurality of regularization parameters. 9. The method according to item 7 or 8, wherein the optimization step comprises: optimizing the regularization parameter under the constraint that the regularization parameter includes a positive symmetric matrix. 10. The method according to any one of items 7 to 9 further comprises: decomposing the regularization parameter into a plurality of spatial components. 11. The method according to item 10, wherein the decomposition includes eigenvalue decomposition or singular value decomposition. 12. The method according to item 8, wherein the separate optimization step comprises: The first model parameters and the second model parameters are decomposed into the spatial components; Determine the effect of each of the spatial components on the parameter of interest; and The corresponding regularization parameter for each spatial component is determined based on the determined influence of each spatial component on the parameter of interest. 13. The method according to item 12, wherein the decomposition step comprises: applying component analysis or decomposition techniques to the second training data, and ranking the spatial components according to their contribution to the parameter of interest. 14. The method according to clause 12 or 13, wherein determining the corresponding regularization parameter for each spatial component includes applying appropriate adjustments or scaling to the base regularization. 15. The method according to clause 14, wherein the basic regularization includes bending energy regularization. 16. The method according to any one of clauses 12 to 15, wherein determining the influence on the parameter of interest comprises: determining a correlation metric describing the wafer-to-wafer correlation of the first model parameter and the second model parameter. 17. The method according to any one of clauses 11 to 16, wherein the first training data includes alignment data and the second training data includes corresponding parameter of interest data. 18. The method according to any one of the preceding clauses, comprising: performing subsequent exposure on the substrate based on the fitted model. 19. A computer program comprising program instructions operable to perform the method according to any one of claims 1 to 18 when run on a suitable device. 20. A non-transient computer program carrier, the non-transient computer program carrier comprising the computer program according to claim 19. 21. A processing apparatus comprising: a non-transient computer program carrier according to claim 20; and a processor operable to run a computer program included on the non-transient computer program carrier. 22. A lithography apparatus, the lithography apparatus comprising: an alignment sensor; a pattern forming apparatus support for supporting a pattern forming apparatus; a substrate support for supporting a substrate; and a processing apparatus according to claim 21. 23. The lithography apparatus according to item 22, wherein the alignment sensor is operable to measure a substrate to obtain measurement data of the substrate. 24. The lithography apparatus according to clause 22 or 23, wherein the processing apparatus is further operable to determine corrections for controlling the patterning apparatus and / or substrate support based on the fitted model.

[0072] While specific applications of lithography equipment in IC manufacturing are discussed in this article, it should be understood that the lithography equipment described herein can have other applications. Other possible applications include manufacturing integrated optical systems, guiding and detecting patterns for magnetic domain memories, flat panel displays, liquid crystal displays (LCDs), thin-film magnetic heads, etc. In this regard, depending on the type of product being manufactured, the processed "substrate" can be a semiconductor wafer, or it can be any other substrate.

[0073] Although embodiments of the invention may be specifically referred to in the context of lithography equipment, these embodiments can be used in other equipment. The embodiments of the invention can form part of an inspection apparatus, a measurement apparatus, or any apparatus for measuring or processing objects such as wafers (or other substrates) or masks (or other patterning apparatus). These apparatuses may generally be referred to as lithography tools. Such lithography tools can use vacuum conditions or ambient (non-vacuum) conditions.

[0074] In this document, the terms “radiation” and “beam” are used to cover all types of radiation, including ultraviolet radiation (e.g., having wavelengths of 365 nm, 248 nm, 193 nm, 157 nm, or 126 nm) and EUV (extreme ultraviolet radiation, e.g., having wavelengths in the range of about 5 nm to 100 nm).

[0075] As used herein, the terms “mask,” “mask,” or “patterning apparatus” can be broadly interpreted to refer to a general-purpose patterning apparatus that can be used to impart a patterned cross-section to an incident radiation beam, the patterned cross-section corresponding to a pattern to be generated in a target portion of a substrate. In this context, the term “optical valve” may also be used. Examples of other such patterning apparatuses, besides classical masks (transmission or reflection masks, binary masks, phase-shifting masks, hybrid masks, etc.), include programmable mirror arrays and programmable LCD arrays.

[0076] Although the use of embodiments of the invention has been specifically referenced above in the context of optical lithography, it will be appreciated that, where the context permits, the invention is not limited to optical lithography and can be used in other applications, such as imprint lithography.

[0077] As used herein, the terms “optimize” and “optimize” refer to or imply adjusting equipment (e.g., lithography equipment), processes, etc., to give the results and / or processes more desirable characteristics, such as higher accuracy in projecting a design pattern onto a substrate, a larger process window, etc. Therefore, as used herein, the terms “optimize” and “optimize” refer to or imply the process of identifying one or more values ​​of one or more parameters that, when compared to an initial set of values ​​for one or more of those parameters, provide an improvement in at least one relevant metric, such as a local optimum. “Optimum” and other related terms should be interpreted accordingly. In embodiments, optimization steps may be applied iteratively to provide further improvements in one or more metrics.

[0078] Various aspects of the present invention can be implemented in any convenient form. For example, embodiments can be implemented by one or more suitable computer programs that can be carried on a suitable carrier medium, which can be a tangible carrier medium (e.g., a disk) or an intangible carrier medium (e.g., a communication signal). Embodiments of the present invention can be implemented using suitable devices, which can specifically take the form of a programmable computer running a computer program arranged to implement the methods as described herein.

[0079] In the block diagram, the illustrated components are depicted as discrete functional blocks; however, the embodiments are not limited to systems in which the functions described herein are organized as illustrated. The functionality provided by each component may be provided by software or hardware modules organized differently from the currently depicted software or hardware modules. For example, such software or hardware may be mixed, combined, replicated, decomposed, distributed (e.g., within a data center or geographically), or otherwise differently organized. The functionality described herein may be provided by one or more processors of one or more computers executing code stored on a tangible, non-transitory, machine-readable medium. In some cases, a third-party content delivery network may store part or all of the information transmitted over the network, in which case, with regard to the information (e.g., content) being considered supplied or otherwise provided, the information may be provided by sending an instruction to retrieve the information from the content delivery network.

[0080] Unless otherwise expressly stated, it is evident from the discussion that throughout the specification, the use of terms such as “processing,” “calculation,” “accounting,” and “determining” refers to the actions or processes of a particular device such as a dedicated computer or similar dedicated electronic processing / computing apparatus.

[0081] Readers should understand that this application describes several inventions. These inventions have not been divided into multiple separate patent applications, but rather grouped into a single document, because their related subject matter makes the application process more economical. However, the different advantages and aspects of these inventions should not be conflated. In some cases, embodiments resolve all the deficiencies pointed out herein; however, it should be understood that the invention is independently useful, and some embodiments resolve only a subset of these problems or provide other unmentioned benefits that will be apparent to those skilled in the art upon review of this disclosure. Due to cost limitations, some inventions disclosed herein may not currently be claimed and may be claimed in subsequent applications (such as continuations or by amendment of these claims). Similarly, due to space limitations, the abstract or summary portion of this document should not be construed as a comprehensive enumeration of all such inventions or all aspects of such inventions.

[0082] It should be understood that the specification and drawings are not intended to limit this disclosure to the specific forms disclosed; rather, the invention is intended to cover all modifications, equivalents, and alternatives falling within the spirit and scope of the invention as defined by the appended claims.

[0083] Given this description, modifications and alternative embodiments of various aspects of the invention will be apparent to those skilled in the art. Therefore, this specification and drawings should be interpreted as illustrative only, intended to teach those skilled in the art the general manner of carrying out the invention. It should be understood that the forms of the invention shown and described herein will be considered as examples of embodiments. It will be apparent to those skilled in the art who have benefited from this specification that elements and materials may be substituted for those illustrated and described herein, components and processes may be reversed or omitted, certain features may be utilized independently, and features of embodiments or embodiments may be combined. Changes may be made to the elements described herein without departing from the spirit and scope of the invention as set forth in the appended claims. The headings used herein are for organizational purposes only and are not intended to limit the scope of this specification.

[0084] As used throughout the application, the word “may” is used in a permissive sense (i.e., meaning possible) rather than a mandatory sense (i.e., meaning must). The words “comprising,” “including,” and “including” mean, but are not limited to, other than. As used throughout the application, unless otherwise expressly stated, the singular forms “a,” “an,” and “the” include plural indicators. Thus, for example, a reference to a “one” element or a “single” element includes a combination of two or more elements, but other terms and phrases such as “one or more” are used for one or more elements. Unless otherwise stated, the term “or” is non-exclusive, i.e., it encompasses both “and” and “or.” Terms describing conditional relationships (e.g., "in response to X, Y", "at X, Y", "if X, then Y", "when X, Y") encompass causal relationships, where the antecedent is a necessary causal condition, a sufficient causal condition, or a contributing causal condition to the outcome. For example, "state X occurs when condition Y is obtained" is interchangeable with "X occurs only when Y" and "X occurs when Y and Z." This conditional relationship is not limited to results that occur immediately after the antecedent is obtained, as some results may be delayed, and in conditional statements, the antecedent is associated with its outcome; for example, the antecedent relates to the probability of the outcome occurring. Unless otherwise stated, a statement that multiple attributes or functions are mapped to multiple objects (e.g., one or more processors performing steps A, B, C, and D) encompasses all attributes or functions mapped to all of those objects, as well as a subset of attributes or functions mapped to a subset of attributes or functions (e.g., the case where each of all processors performs steps A through D, and the case where processor 1 performs step A, processor 2 performs a portion of steps B and C, and processor 3 performs a portion of steps C and D). Furthermore, unless otherwise stated, a statement that a value or action is “based on” another condition or value includes the case where the condition or value is the only factor and the case where the condition or value is one of multiple factors. Unless otherwise stated, a statement that “every” instance of a certain set has a certain property should not be interpreted as excluding the case where some members of a larger set that are otherwise identical or similar do not have that property; i.e., each does not necessarily mean every. References selected from a range include the endpoints of the range.

[0085] In the above description, any process, description, or block in the flowchart should be understood as representing a module, segment, or portion of code including one or more executable instructions for implementing a specific logical function or step in the process, and alternative implementations are included within the scope of exemplary embodiments of the present invention, wherein, depending on the functionality involved, functions may be executed in an order different from the order shown or discussed, including substantially simultaneously or in reverse order, as understood by those skilled in the art.

[0086] While specific embodiments of the invention have been described above, it will be understood that the invention can be practiced in ways different from those described. The description above is intended to be illustrative and not restrictive. Therefore, it will be apparent to those skilled in the art that modifications can be made to the described invention without departing from the scope of the claims set forth below.

Claims

1. A method for modeling substrate measurement data associated with a substrate during a photolithography process, the method comprising: Obtain substrate measurement data describing the spatial variation of substrate parameters; as well as The model is fitted to the substrate measurement data to obtain a fitted model, wherein the fitting step includes regularizing the fit using a plurality of regularization parameters, which have been individually optimized to penalize the spatial components of the substrate measurement data based on the influence of the spatial components of the substrate measurement data on the parameter of interest.

2. The method of claim 1, wherein the plurality of regularization parameters are included in a regularization matrix, wherein each regularization matrix element includes a corresponding regularization parameter.

3. The method of claim 1, wherein the plurality of regularization parameters are operable to apply a penalty level to the spatial component based on the inverse relationship of the influence of the spatial component on the parameter of interest.

4. The method of claim 1, wherein the substrate measurement data includes overlap data and / or alignment data.

5. The method of claim 1, wherein the parameter of interest is overlap or another placement error metric.

6. The method of claim 1, wherein the individual optimization comprises: Obtain first training data associated with one or more training substrates; Obtain second training data associated with the one or more training substrates; as well as The plurality of regularization parameters are individually optimized to obtain the plurality of regularization parameters.

7. The method of claim 6, wherein the separate optimization step comprises: The regularization parameters are optimized using a cost function associated with the first training data and the second training data, based on the first model parameters associated with the first training data, the second model parameters associated with the second training data, and the plurality of regularization parameters.

8. The method according to claim 6, wherein the optimization step comprises: The regularization parameters are optimized under the constraint that the regularization parameters include positive symmetric matrices.

9. The method according to claim 6, further comprising: The regularization parameter is decomposed into multiple spatial components.

10. The method of claim 9, wherein the decomposition comprises eigenvalue decomposition or singular value decomposition.

11. The method of claim 7, wherein the separate optimization step comprises: The first model parameters and the second model parameters are decomposed into the spatial components; Determine the effect of each of the spatial components on the parameter of interest; as well as The corresponding regularization parameter for each spatial component is determined based on the determined influence of each spatial component on the parameter of interest.

12. The method according to claim 11, wherein the decomposition step comprises: Component analysis or decomposition techniques are applied to the second training data, and the spatial components are ranked according to their contribution to the parameter of interest.

13. The method of claim 11, wherein determining the corresponding regularization parameter for each spatial component comprises: Apply appropriate adjustments or scaling to the basic regularization.

14. The method of claim 11, wherein determining the effect on the parameter of interest comprises: Determine a correlation metric that describes the wafer-to-wafer correlation of the first model parameters and the second model parameters.

15. A computer program comprising program instructions operable to perform the method according to any one of claims 1 to 14 when run on a suitable device.

Images