Method and related apparatus for modeling measurement data of a substrate area

Abstract

A method is disclosed for modeling measurement data for a substrate to an area of ​​the substrate in a lithography process, the method comprising obtaining measurement data for the substrate and performing a joint fitting to the measurement data at least a first inter-field model describing distortion of the substrate and a field distortion model describing distortion within the exposure field, wherein the at least first inter-field model comprises a radial basis function model or an elastic energy minimizing spline model, or the method further comprises fitting a radial basis function model or an elastic energy minimizing spline model to distortion residuals of the joint fits of different inter-field models and field distortion models.

Description

[Technical Field]

[0001] Cross-reference of related applications

[0001] This application claims priority to European application 20207862.2 filed November 16, 2020, which is incorporated herein by reference in its entirety.

[0002]

[0002] This disclosure relates, for example, to the processing of substrates for the manufacture of semiconductor devices. [Background technology]

[0003]

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

[0004]

[0004] To project a pattern onto a substrate, a lithography 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. For example, to form a smaller feature on a substrate than a lithography apparatus using radiation with a wavelength of approximately 193 nm can be used, a lithography apparatus using extreme ultraviolet (EUV) radiation with a wavelength in the range of 4 nm to 20 nm, for example, 6.7 nm or 13.5 nm, can be used.

[0005]

[0005] Low k1 lithography is sometimes used to process features having dimensions smaller than the classical resolution limit of a lithography apparatus. In such a process, the resolution formula may 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 of the lithography apparatus, CD is the "critical dimension" (generally the smallest feature size to be printed, but in this case half-pitch), and k1 is the empirical resolution factor. Generally, the smaller k1, the more difficult it becomes to reproduce on the substrate a pattern that resembles the shape and dimensions planned by the circuit designer to achieve a particular electrical functionality and performance. To overcome such difficulties, advanced fine-tuning steps may be applied to the lithography projection apparatus and / or design layout. These include, but are not limited to, optimizing the numerical aperture (NA), customizing the illumination scheme, using one or more phase-shift patterning devices, optimizing 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 strict control loops may be used to control the stability of the lithography apparatus in order to improve pattern reproduction at low k1.

[0006]

[0006] The effect of controlling the lithography apparatus may depend on the characteristics of individual substrates. For example, a first substrate processed by a first processing tool prior to processing by the lithography apparatus (or any other process step of the manufacturing process, which is generally referred to herein as a manufacturing process step) may benefit from (slightly) different control parameters than a second substrate processed by a second processing tool prior to processing by the lithography apparatus.

[0007]

[0007] Precise placement of patterns on a substrate is a major challenge in reducing the size of circuit components and other products that can be produced by lithography. In particular, the challenge of accurately measuring features already placed on a substrate is a crucial step in being able to overlap consecutive layers of features with sufficient accuracy to manufacture working devices with a high yield. So-called overlays generally need to be achieved within tens of nanometers in today's submicron semiconductor devices, and down to a few nanometers in the most critical layers.

[0008]

[0008] As a result, modern lithography equipment involves extensive measurement or "mapping" operations before the step of actually exposing or patterning the substrate at the target location. So-called advanced alignment models continue to be developed to more accurately model and correct the nonlinear distortion of the wafer "grid" caused by the processing steps and / or the lithography equipment itself. However, not all distortion can be corrected during exposure, and it remains important to track and eliminate as many of the causes of such distortion as possible.

[0009]

[0009] These distortions in the wafer grid are represented by measurement data associated with the mark positions. The measurement data is obtained from wafer measurements. An example of such measurement is the alignment measurement of alignment marks performed using an alignment system in the lithography apparatus before exposure.

[0010]

[0010] It is desirable to improve the modeling of these distortions. [Overview of the Initiative]

[0011]

[0011] In a first aspect of the present invention, a method for modeling measurement data regarding a substrate in a lithography process, the method comprising: obtaining measurement data regarding the substrate; performing a combined fitting that adapts at least a first inter-field model describing the distortion of the substrate and a field distortion model describing the distortion within an exposure field to the measurement data, wherein the at least first inter-field model includes a radial basis function model that describes the distortion of the substrate with respect to a radial basis function or an elastic energy minimization spline model that describes the distortion of the substrate with respect to a basis function that minimizes a certain functional of the model; or the method further comprises adapting a radial basis function model that describes the distortion of the substrate with respect to a radial basis function or an elastic energy minimization spline model that describes the distortion of the substrate with respect to a basis function that minimizes a certain functional of the model to the distortion residuals of a combined fit of different inter-field models and field distortion models.

[0012]

[0012] In a second aspect of the present invention, a method for modeling measurement data regarding a substrate in a lithography process, the method comprising: obtaining measurement data regarding the substrate; and performing a fitting that adapts a field distortion model describing the distortion within an exposure field to the measurement data by minimizing a cost function that is a regularization term depending on the parameters of the field distortion model and that includes a regularization term related to the bending energy of the field distortion model.

[0013]

[0013] In a further aspect of the present invention, a computer program comprising program instructions operable to perform the method of the first aspect when operating on a suitable device, as well as a related processing device and a lithography device are provided.

Brief Description of the Drawings

[0014]

[0014] Embodiments of the present invention will be described below by way of example only with reference to the accompanying schematic diagrams.

[0015] [Figure 1]

[0015] A schematic appearance of the lithography apparatus is shown. [Figure 2]

[0016] This shows a schematic appearance of a lithography cell. [Figure 3]

[0017] Figures 1 and 2 schematically illustrate the use of the lithography apparatus and lithography cell in conjunction with one or more other devices forming a manufacturing facility for semiconductor devices, for example, to implement a control strategy according to one embodiment of the present invention. [Modes for carrying out the invention]

[0016]

[0018] Figure 1 schematically shows a lithography apparatus LA. The lithography apparatus LA includes an illumination system (also called an illuminator) IL configured to adjust a radiation beam B (e.g., UV radiation, DUV radiation, or EUV radiation); a support (e.g., a mask table) T connected to a first positioner PM, which is constructed to support a patterning device (e.g., a mask) MA and configured to precisely position the patterning device MA according to specific parameters; one or more substrate supports (e.g., wafer tables) WTa and WTb connected to a second positioner PW, which is constructed to hold a substrate (e.g., a resist-coated wafer) W and configured to precisely position the substrate according to specific parameters; and a projection system (e.g., a refractive projection lens system) PS configured to project the pattern applied to the radiation beam B by the patterning device MA onto a target portion C of the substrate W (e.g., including one or more dies).

[0017]

[0019] During operation, the illumination system IL receives a radiated beam from the radiation source SO, for example, via the beam delivery system BD. The illumination system IL may include various types of optical components, such as refractive, reflective, magnetic, electromagnetic, electrostatic, and / or other types of optical components, or any combination thereof, for inducing, shaping, and / or controlling the radiation. The radiated beam B may be tuned using the illuminator IL so that a desired spatial and angular intensity distribution is obtained across the cross-section in the plane of the patterning device MA.

[0018]

[0020] As used herein, the term “projection system” PS should be interpreted broadly to encompass a variety of projection systems, including refractive optical systems, reflective optical systems, reflective-refractory optical systems, anamorphic optical systems, magneto-optical systems, electromagnetic optical systems, and / or electrostatic optical systems, or any combination thereof, as appropriate to the exposure radiation used and / or other factors such as the use of immersion liquid or vacuum. Where the term “projection lens” is used herein, it can be considered synonymous with the more general term “projection system” PS.

[0019]

[0021] The lithography apparatus LA may be of a type that can cover at least a portion of the substrate with a liquid having a relatively high refractive index, such as water, to fill the space between the projection system PS and the substrate W. This is also called immersion lithography. Further information relating to immersion techniques is given in U.S. Patent No. 6,952,253, which is included by reference in this application.

[0020]

[0022] The lithography apparatus LA in this example is a so-called dual-stage type, having two substrate tables WTa and WTb, and two stations (an exposure station and a measurement station) that can move the substrate tables between stations. While one substrate on one substrate table is being exposed at the exposure station EXP, another substrate may be loaded onto the other substrate table or processed at the measurement station MEA, for example, at the measurement station MEA or another location (not shown). The substrate table containing the substrates may be positioned at the measurement station MEA so that various preparation steps can 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. Due to inaccuracies in creating the marks and the deformation of the substrate that occurs throughout the process, the set of marks may have undergone more complex transformations than translation and rotation. As a result, if the apparatus LA wants to print product features in the correct location with high precision, in addition to measuring the position and orientation of the substrate, the alignment sensor may actually measure the position of many marks in detail across the entire substrate area. Therefore, measuring alignment marks can be time-consuming, and providing two substrate tables can significantly improve the throughput of the apparatus. If the position sensor IF cannot measure the position of the substrate table while it is in the measurement station and the exposure station, a second position sensor may be provided to enable the position of the substrate table to be tracked at both stations. Embodiments of the present invention may be applicable to apparatuses with only one substrate table or with three or more substrate tables.

[0021]

[0023] In addition to having one or more substrate supports, the lithography apparatus LA may include a measurement stage (not shown). The measurement stage is positioned to hold sensors and / or cleaning devices. Sensors may be positioned to measure the characteristics of the projection system PS or the characteristics of the radiation beam B. The measurement stage may hold multiple sensors. Cleaning devices may be positioned to clean parts of the lithography apparatus, such as parts of the projection system PS or parts of the system that provides the immersion fluid. If the substrate support WT is separated from the projection system PS, the measurement stage may move below the projection system PS.

[0022]

[0024] The radiant beam B is incident on a patterning device (e.g., a mask) MA held on a support structure (e.g., a mask table) MT, and a pattern is formed by the patterning device. After crossing the patterning device MA, the radiant beam B passes through a projection system PS, which focuses the beam onto a target portion C on the substrate W. With the help of a second positioner PW and position sensors IF (e.g., an interferometer device, a linear encoder, a 2D encoder, or a capacitive sensor), the substrate tables WTa / WTb can be precisely moved to position, for example, various target portions C along the path of the radiant beam B. Similarly, the patterning device MA can be precisely positioned relative to the path of the radiant beam B using a first positioner PM and another position sensor (not shown in Figure 1), such as after mechanical removal from the mask library or during scanning. In general, the movement of the support structure MT can be achieved with the help of long-stroke modules (coarse positioning) and short-stroke modules (fine positioning) that form part of the first positioner PM. Similarly, movement of the substrate tables WTa / WTb can be achieved using long-stroke and short-stroke modules that form part of the second positioner PW. In the case of a stepper (as opposed to a scanner), the support structure MT may be connected to or fixed only to the short-stroke actuator. The patterning device MA and the substrate W can be aligned using patterning device alignment marks M1, M2 and substrate alignment marks P1, P2. The illustrated substrate alignment marks occupy dedicated target areas, but the substrate alignment marks can also be placed in the space between the target areas (these are known as scribe line alignment marks). Similarly, if multiple dies are provided on the patterning device MA, the patterning device alignment marks may be placed between the dies.

[0023]

[0025] The apparatus further comprises a lithography apparatus control unit (LACU) that controls all the movement and measurement of various actuators and sensors of the lithography apparatus (as described). The control unit LACU also has signal processing and data processing capabilities to implement desired calculations related to the operation of the apparatus. In practice, the control unit LACU will be implemented as a system of numerous subunits, each handling real-time data acquisition, processing, and control of subsystems or components within the apparatus. For example, one processing subsystem may be specialized in the servo control of a substrate positioner PW. A separate unit may handle coarse and fine actuators, or even different axes. Another unit may be specialized in the readout of a position sensor IF. The overall control of the apparatus may be controlled by a central processing unit that communicates with these subsystem processing units, operators, and other apparatus involved in the lithography manufacturing process.

[0024]

[0026] As shown in Figure 2, the lithography apparatus LA may constitute part of a lithographic cell LC, sometimes referred to as a lithocell or (litho)cluster, and the lithographic cell often includes equipment for performing pre-exposure and post-exposure processes on the substrate W. Conventionally, these devices include one or more spin coaters SC for depositing a resist layer, one or more developers DE for developing the exposed resist, and one or more cooling plates CH and one or more bake plates BK for adjusting the temperature of the substrate W, for example, to prepare the solvent in the resist layer. A substrate handler or robot RO picks up the substrate W from input / output ports I / O1 and I / O2, moves it between different processing units, and delivers the substrate W to the loading bay LB of the lithography apparatus LA. The devices within the lithocell, often collectively referred to as a track, are generally under the control of a track control unit TCU, which itself may be controlled by a monitoring and control system SCS, which can also control the lithography apparatus LA, for example, via a lithography control unit LACU.

[0025]

[0027] For accurate and stable exposure of the substrate W by the lithography apparatus LA, it is desirable to inspect the substrate and measure the characteristics of the patterned structure, such as overlay error between subsequent layers, line width, and critical dimension (CD). For this purpose, the lithocell LC may include one or more inspection tools (not shown). If an error is detected, adjustments may be made to the exposure of subsequent substrates or to other processing steps performed on the substrate W, especially if the inspection is performed before other substrates of the same batch or lot are exposed or processed.

[0026]

[0028] A metronome, also called a metronome, is used to determine one or more properties of a substrate W, particularly how one or more properties of different substrates W differ, or how one or more properties related to different layers of the same substrate W differ from layer to layer. The metronome may be constructed to identify defects on the substrate W and may be, for example, part of a lithocell LC, integrated into a lithography system LA, or a standalone device. The metronome may measure one or more properties of a latent image (an image of the resist layer after exposure), or a semi-latent image (an image of the resist layer after a post-exposure bake step), or a developed resist image (after the exposed or unexposed areas of the resist have been removed), or an etched image (after a pattern transfer step such as etching).

[0027]

[0029] Figure 3 shows a lithography apparatus LA and lithocell LC in relation to industrial production equipment for semiconductor products, for example. Within the lithography apparatus (or abbreviated as "lithotool" 200), the measurement station MEA is shown as 202, and the exposure station EXP is shown as 204. The control unit LACU is shown as 206. As already described, the lithotool 200 constitutes part of a "lithocell" or "lithocluster" which also includes a coating apparatus SC208 for applying a photosensitive resist and / or one or more other coatings to the substrate W for patterning by the apparatus 200. On the output side of the apparatus 200 are a baking apparatus BK210 and a developing apparatus DE212 for developing the exposed pattern into a physical resist pattern. Other components shown in Figure 3 are omitted for clarity.

[0028]

[0030] Once the pattern is applied and developed, the patterned substrate 220 is transferred to other processing equipment, indicated by 222, 224, 226, etc. Various processing steps are performed by various pieces of equipment in a typical manufacturing setup. For example, in this embodiment, equipment 222 is an etching station, and equipment 224 performs the post-etching annealing step. Further physical and / or chemical processing steps are applied to further equipment 226, etc. Various types of operations, such as material deposition, modification of surface material properties (oxidation, doping, ion implantation, etc.), and chemical mechanical polishing (CMP), may be required to fabricate an actual device. Equipment 226 may practically represent a series of various processing steps performed by one or more pieces of equipment.

[0029]

[0031] The semiconductor manufacturing process described above, which includes a series of patterning process steps, is merely one example of an industrial process to which the technology disclosed herein may be applied. The semiconductor manufacturing process includes a series of patterning steps. Each patterning process step includes a patterning operation, such as a lithography patterning operation, and several other chemical and / or physical operations.

[0030]

[0032] The manufacturing of semiconductor devices requires the repeated use of appropriate materials and patterns for each layer of the substrate to construct the device structure. Modern device manufacturing processes may include, for example, 40 or 50 individual patterning steps. Therefore, the substrate 230 arriving at the litho cluster may be a newly prepared substrate, or it may have already been processed in this cluster 232 or in a completely different apparatus. Similarly, depending on the required processing, the substrate leaving apparatus 226 may be returned to subsequent patterning operations in the same litho cluster (e.g., substrate 232), sent to patterning operations in a different cluster (e.g., substrate 234), or sent as a finished product for dicing and packaging (e.g., substrate 234).

[0031]

[0033] Each layer of a product structure generally requires a different set of process steps, and the equipment used for each layer may be of completely different types. Furthermore, even when the processing steps applied by the equipment are nominally the same, in large-scale facilities, there may be several seemingly identical machines operating in parallel to process various substrates. Small differences in configuration or defects between these machines can mean that these small differences will affect various substrates in different ways. Even steps that are relatively common to each layer, such as etching (equipment 222), may be carried out by several etching machines operating in parallel to maximize throughput, even if they are nominally identical. Parallel processing may also be performed in various chambers within a larger facility. Moreover, in practice, different layers often require different etching processes, such as chemical etching or plasma etching, depending on the details of the material being etched and any special requirements, such as anisotropic etching.

[0032]

[0034] As described above, the preceding and / or following processes may be carried out with other lithography equipment, and may even be carried out with various types of lithography equipment. For example, one or more layers with extremely stringent requirements in terms of resolution and / or overlay may be carried out with more advanced lithography tools than one or more other layers with less stringent requirements in the device manufacturing process. Thus, one or more layers may be exposed with an immersion-type lithography tool, while one or more other layers may be exposed with a "dry" tool. One or more layers may be exposed with a tool operating at DUV wavelengths, while one or more other layers may be exposed using EUV wavelength radiation.

[0033]

[0035] Figure 3 also shows a metronome (MET) 240, which is provided for measuring product parameters at a desired stage of the manufacturing process. A common example of a metronome station in a modern lithography manufacturing facility is a scattermeter, such as an angle-resolved scattermeter or a spectroscopic scattermeter, which may be applied to measure one or more properties of the substrate developed in 220 before etching in apparatus 222. The metronome 240 may be used to determine performance parameter data PDAT 252. From this performance parameter data PDAT 252, it may be further determined if performance parameters such as overlay or critical dimension (CD) do not meet certain precision requirements in the developed resist. Before the etching step, there is an opportunity to remove the developed resist and reprocess one or more of the substrates 220 through the litho cluster. Furthermore, the metronome results from the metronome 240 may be used to maintain the precise performance of the patterning operation in the litho cluster by making small adjustments over time, thereby reducing or minimizing the risk of the product being out of specification and requiring reprocessing. Naturally, a metronome 240 and / or one or more other metronome devices (not shown) may be used to measure one or more characteristics of the processed substrates 232, 234 and / or the incoming substrate 230.

[0034]

[0036] Typically, the patterning process in a lithography apparatus LA is one of the most critical steps in the process, requiring high precision in dimensionalizing and positioning structures on a substrate W. To ensure this high precision, three systems may be incorporated into a control environment schematically shown in Figure 3. One of these systems is the lithotool 200, which is (virtually) connected to the metrologic apparatus 240 (second system) and the computer system CL250 (third system). The requirements of such an environment are to optimize or improve the coordination between these three systems to strengthen the entire so-called "process window" and to provide one or more rigorous control loops to help the patterning performed by the lithography apparatus LA stay within the process window. A process window defines a range of values ​​for several process parameters (e.g., two or more selected from dose, focus, overlay, etc.) within which a particular manufacturing process produces a specified result (e.g., a functional semiconductor device). Typically, within this range, the values ​​of process parameters in a lithography or patterning process can vary while producing a suitable structure (e.g., specified within a CD tolerance (e.g., ±10% of nominal CD)).

[0035]

[0037] The computer system CL may use (part of) the design layout to be patterned to predict which of one or more resolution enhancement techniques should be used, and perform computer lithography simulations and calculations to determine the patterning device layout and lithography apparatus settings that achieve maximization of the entire process window of the patterning process (indicated by the bidirectional arrows on the first dial SC1 in Figure 3). Typically, the resolution enhancement techniques are configured to fit the patterning capabilities of the lithography apparatus LA. The computer system CL may also be used to predict whether defects may exist, for example, for suboptimal processing, by detecting where in the process window the lithography apparatus LA is currently operating (for example, using input from the metronography tool MET) (indicated by the arrow pointing to "0" on the second dial SC2 in Figure 3).

[0036]

[0038] The metrology tool MET may provide input to the computer system CL that enables accurate simulation and prediction, for example, by providing feedback to the lithography apparatus LA to identify potential drifts in the calibration status of the lithography apparatus LA (indicated by multiple arrows on the third scale plate SC3 in Figure 3).

[0037]

[0039] The computer system 250 can control the process based on a combination of (i) “pre-processing metrology data” associated with the substrate before it is processed in a given processing step (e.g., a lithography step) (e.g., scanner metrology data LADAT254 and external pre-processing metrology ExDAT260) and (ii) performance data, i.e., “post-processing data” PDAT252, associated with the substrate after it has been processed.

[0038]

[0040] A first set of preprocessing metrometry data LADAT254 (referred to herein as scanner metrometry data, as it is data generated by the lithography apparatus LA200 or a scanner) may include alignment data conventionally obtained by the lithography apparatus LA200 using an alignment sensor AS at the measurement station 202. Alternatively, or in addition to alignment data, the scanner metrometry data LADAT254 may include height data obtained using a level sensor LS, and / or a "wafer quality" signal from an alignment sensor AS, etc. Thus, the scanner metrometry data LADAT254 may include data relating to the alignment grid for the substrate and the deformation (flatness) of the substrate. For example, the scanner metrometry data LADAT254 may be generated prior to exposure by the measurement station MEA202 of a twin-stage lithography apparatus LA200 (typically equipped with an alignment sensor and a leveling sensor), enabling simultaneous measurement and exposure operations. Such twin-stage lithography apparatuses are well known.

[0039]

[0041] The use of external pre-exposure metrometry tools (e.g., standalone) like the ExM270 is increasing for pre-exposure measurements of lithography systems. Such external pre-exposure metrometry tools like the ExM270 are distinct from the MEA202 measurement station of the LA200 twin-stage lithography system. Any pre-exposure measurements performed within the track are considered external measurements. To maintain sufficient exposure throughput, the scanner metrometry data LADAT (e.g., alignment grid and substrate deformation grid) measured by the MEA202 measurement station is based on a sparse set of desired measurement results. This generally means that such a measurement station cannot collect enough measurement data for higher-order corrections, specifically corrections beyond the third order. In addition, the use of opaque hard masks can make it difficult to accurately measure wafer grid alignment.

[0040]

[0042] The external pre-exposure metrology tool ExM270 enables much higher-density measurements to be performed on each substrate prior to exposure. These ExM270 pre-exposure metrology tools measure and / or predict wafer grid deformation at the same or faster throughput than scanners, and at a much higher measurement density than could be achieved using alignment and level sensors, even when these are contained within a separate measurement station MEA202. Pre-exposure metrology tools include, for example, substrate shape inspection tools and / or standalone alignment stations.

[0041]

[0043] Figure 3 shows separate storage units 252, 254, and 260 for performance data PDAT, scanner metrology data LADAT, and pre-exposure data ExDAT, respectively. However, these different types of data may be stored in a single shared storage unit or distributed across multiple storage units, allowing specific data items to be retrieved as needed.

[0042]

[0044] Alignment models are used to illustrate alignment measurements on and / or on the wafer. The primary purpose of an alignment model is to provide a mechanism for interpolating and / or extrapolating available measurement data across the entire wafer so that an exposure grid can be created on each exposure field. Measurement data can become sparse because it is not at all practical to measure as many measurement areas as desired in terms of overlay accuracy, i.e., the time and consequently throughput overhead is too high. The secondary purpose of an alignment model is to provide noise suppression. This can be achieved by using fewer model parameters than the measured values ​​or by using regularization.

[0043]

[0045] While standard models may use fewer than 10 parameters, advanced alignment models typically use more than 15 or even more than 30 parameters. Examples of advanced models include higher-order wafer alignment (HOWA) models and alignment models based on radiative basis functions (RBF). HOWA is a published technique based on quadratic and higher-order polynomial functions. RBF modeling is described in US2012218533A1, which is incorporated herein by reference. Different versions and extensions of these advanced models may be devised. Advanced models generate complex descriptions of the wafer grid that are corrected during the exposure of the target layer. RBF and the latest versions of HOWA give particularly complex descriptions based on dozens of parameters. This suggests that many measurements are required to obtain a wafer grid with sufficient accuracy.

[0044]

[0046] Currently, polynomial-based models such as the HOWA model are primarily used for both inter-field and intra-field wafer deformation modeling. This is typically done in a cascaded manner, with intra-field modeling performed after inter-field modeling for residual wafer deformation. For example, inter-field modeling may first be performed on an inter-field layout with alignment marks at a single intra-field location across the entire wafer (i.e., the same location within each field of the wafer where the marks are measured). The results of the inter-field modeling are then applied to a second set of measurements, which typically include a common intra-field layout (intra-field layout) of multiple marks for each small subset of fields on the wafer. After this, the intra-field model is fitted to the measurements, which are then corrected by the inter-field model of the intra-field layout.

[0045]

[0047] A drawback of this cascaded modeling approach, which reduces noise propagation and / or enables more advanced models, is that measurements used for in-field modeling are not used for inter-field modeling, and vice versa. To address this, coupled layouts and coupled modeling methods are proposed for polynomial models. The coupled layout samples in-field positions in a distributed manner, i.e., various in-field positions are measured in various fields. Modeling inter-field models on such a layout can introduce crosstalk from in-field deformations to inter-field models, resulting in inaccurate wafer and field grid predictions. Therefore, coupled modeling methods are proposed for polynomial models in which inter-field polynomial (HOWA) basis functions and in-field polynomial basis functions are fitted in a single step, thereby preventing or mitigating crosstalk between inter-field and in-field polynomial modifiable deformations.

[0046]

[0048] An alternative to polynomial modeling is disclosed in US2012218533A1 (incorporated herein by reference) and is known as radial basis function (RBF) modeling. RBF modeling is an extrapolation / interpolation modeling technique that can capture local wafer deformations better than polynomial models.

[0047]

[0049] RBF modeling, as described in US2012218533A1, generates radial basis functions at several positions on the wafer, referred to as the center.

[0048]

number

[0049] The process includes the steps of using (for example, the position of alignment marks) and calculating the model parameters of the substrate in the apparatus using the generated radial basis function as a basis function across the entire substrate.

[0050]

number

[0051] This is a function whose value depends only on a certain position, for example, the distance to the origin, but in this case the position of the center is as follows.

[0052]

number

[0053] Here, the overline ￣ indicates that the variable is a column vector.

[0054]

number

[0055] This represents the Euclidean vector norm.

[0056]

[0050] Position of the RBF model

[0057]

number

[0058] The evaluation of this may be expressed as follows:

[0059]

number

[0060] Here is the approximation function

[0061]

number

[0062] These are different centers

[0063]

number

[0064] and the weight w, which is a parameter used to estimate from the measured values. j It is expressed as a weighted sum of N radial basis functions (RBFs) associated with each of them. j is residual

[0065]

number

[0066] Sometimes, the least squares method is used to calculate the sum of the squares of m i is location

[0067]

number

[0068] These are measurement results (for example, alignment measurements in one of two directions). Note that for a typical use case of a center located at any alignment mark, there are as many weights, i.e., degrees of freedom, as there are measurements. The resulting system of equations is non-singular (reversible) under very mild conditions, so a unique solution exists. For many radial basis functions (RBFs), the only constraint is that at least three points are not on a straight line.

[0069]

[0051] Many options for RBFs are possible, such as Gaussian basis 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 infinite smoothing (a derivative exists at every point) and spline (a derivative may not exist at some points).

[0070]

[0052] One specific example of RBF is thin sheet spline (TPS) modeling. TPS relates to the physical analogy of bending a thin sheet of metal. In the physical environment, deflection is in the z direction and perpendicular to the plane of the thin sheet. To apply this concept to the problem of substrate deformation in the lithography process, the lift of the sheet can be considered as a displacement in the x or y coordinate within its plane. TPS has been widely used as a non-rigid transformation model in image alignment and shape matching. TPS is popular due to several advantages. This model has no free parameters that require manual tuning, and automatic interpolation is possible. That is the fundamental solution of the two-dimensional polyharmonic operator, Considering a set of data points, the weighted combinations of thin plate splines centered on each data point give an interpolation function that precisely passes through these points while minimizing the so-called "bending energy."

[0071]

[0053] The mathematical details of the thin plate spline are shown below. The thin plate spline is a model f(x,y) that interpolates one-dimensional data such that the functional F(f(x,y)) is minimized, and F(f(x,y)) is given by the following equation,

[0072]

number

[0073] This represents the so-called "bending energy" of the model (in a physical environment where f(x,y) describes the height of a thin metal sheet, this functional is truly proportional to the bending energy associated with bending the sheet). In its regularized form, the thin sheet spline minimizes the cost function given by the following equation:

[0074]

number

[0075] Here

[0076]

number

[0077] is the column vector of the measured values, K is the RBF model matrix, and K is the matrix element for its thin plate spline. ij It is given by the following equation.

[0078]

number

[0079] Here

[0080]

number

[0081] is a column vector containing RBF weights (fit parameters), and P is a linear polynomial inter-field model matrix.

[0082]

number

[0083] is a column vector containing six linear inter-field model parameters, and λ is the RBF regularization parameter.

[0084]

number

[0085] This is the RBF "bending energy". The cost function is constrained

[0086]

number

[0087] It may be minimized under the following conditions: when the center is at the measurement position (K C=K=K T , P C = P), which is a parameter of the cost function

[0088]

Number

[0089] and

[0090]

Number

[0091] may be done by setting the gradient in the direction to zero. By rearranging the resulting equation, the solution given by the following equation is obtained,

[0092]

Number

[0093] where I is the identity matrix. In the case of wafer alignment, the solution is calculated separately for two directions (x and y).

[0094]

[0054] Thin plate spline is also an example of an elastic energy minimization spline model next to being an RBF. The functional to be minimized is the integral of the bending energy density of the model function. Alternatively, the model can be found by minimizing the integral over different functional, e.g., different densities L depending on a model function u(x, y) generally describing the strain in the x direction, a model function v(x, y) describing the strain in the y direction, and their derivatives of any order. L(u, v, u x , v x , u y , v y , u xx , v xx , u xy , v xy ,…) Here, the subscript indicates the derivative of the function in that direction (e.g., u xy =d 2 (u / dxdy). To find the basis function that minimizes this type of functional, we can use variational calculus to derive the Euler-Lagrange equations. The spline model function is center position (x c ,y c The solution can be found by finding solutions that satisfy the Euler-Lagrangian equations everywhere except ), that is, the solution satisfies the following set of (possibly combinatorial) differential equations,

[0095]

number

[0096] Here, δ is the Dirac delta function, and w x and w y is a constant. The solution, i.e., the required spline model function, generally looks like the following equation.

[0097]

number

[0098] Here, P is the model matrix of a model that does not affect the value of the functional being minimized, and p, w, z, and q are the basis functions of the spline model, and w x,j and w y,j is the spline center position (x c,j ,y c,j These are model parameters related to (x,y) in the case of a thin plate spline model, functions w and z are zero, and functions p and q are the same, with (x,y) and (x c,j ,y c,j It depends only on the distance between the two points, i.e., it becomes a radial basis function.

[0099]

[0055] Using an RBF model or an elastic energy minimizing spline model instead of a HOWA inter-field model in cascaded modeling of the coupled layout results in crosstalk from the field strain to the RBF model. Depending on whether and to what extent regularization is used, this crosstalk may be more pronounced than in the case of polynomial inter-field modeling. Furthermore, a typical RBF model with centers at every measurement location is an interpolation model and consists of as many parameters as the measurements, making it unsuitable for a standard coupled fitting approach. Therefore, unless a compromise is made regarding the number of centers in the RBF model or elastic energy minimizing spline model, an (unconstrained) coupled fit between such a model and the field strain model results in a system of underdetermined equations.

[0100]

[0056] To overcome the limitations of cascading the RBF model or elastic energy minimizing spline model and the field strain model on a combined layout, two methods are proposed. • A combined fit of an RBF model or elastic energy minimizing spline model and a field strain model. This fit may be solved by minimizing a cost function that includes a regularization term dependent on the RBF or elastic energy minimizing spline model parameters in addition to the squared residuals of the model, and optionally includes a regularization term dependent on the field strain model parameters, to yield a clearly determined set of equations. • Cascaded fit of the residuals of a combined fit of different inter-field models and field strain models for RBF models or elastic energy minimizing spline models.

[0101]

[0057] By including a regularization term in the cost function to be minimized, the field strain model parameters are selected such that this regularization term is minimized. The regularization term may be the "bending energy" of the inter-field model. In that case, the field strain model parameters will be those that give the inter-field model the minimum "bending energy". The model parameters may be found by setting the gradient of the cost function in the direction of the model parameters to zero and solving the resulting equation.

[0102]

[0058] The mathematical details of one example of such an approach are shown below. In the first example, the method involves a combined fit of the RBF model and the in-field model by minimizing a cost function that includes a regularization term with respect to the RBF parameter, which is the RBF bending energy. In this case, the cost function is given by the following equation:

[0103]

number

[0104] Here, L is the in-field model matrix,

[0105]

number

[0106] is a column vector containing the in-field model parameters. The cost function is also constrained here.

[0107]

number

[0108] It may be minimized under the following conditions: when the center is at the measurement position (K C =K=K T , P C =P), this is the parameter of the cost function.

[0109]

number

[0110] and

[0111]

number

[0112] This can be done by setting the directional gradient to zero. By rearranging the resulting equations, the following solution is obtained.

[0113]

number

[0114]

[0059] In some embodiments, improved results may be obtained by adding an i-regularization term for the field strain model. Such regularization may impose a quantity in the cost function that depends on the field strain model parameters. In some embodiments, this regularization may include the “bending energy” that the field strain model induces in the grid within the field. Alternatively, this regularization term may penalize the norm of the field strain model coefficients, the integral of the square of the field strain model evaluation across the field, the integral of any order derivative of the field strain model across the field, or different quantities that depend on the field strain model parameters. Mathematically, the cost function to be minimized here may be:

[0115]

number

[0116] Here, φ is the field strain model regularization parameter, and R is the field strain model regularization matrix (an example of the bending energy of the in-field model given below).

[0117]

number

[0118] This is the field strain model regularization term. The solution here again is the parameter of the cost function.

[0119]

number

[0120] and

[0121]

number

[0122] This can be obtained by setting the gradient in the direction to zero. By rearranging the resulting equation, the following equation is obtained.

[0123]

number

[0124]

[0060] The in-field bending energy and the corresponding in-field model regularization matrix may be determined as follows: In the case of a linear in-field model (an in-field model that can be expressed as a linearity of the model parameters), the in-field position (x f ,y f The evaluation of the model in ) can be expressed as follows:

[0125]

number

[0126] Here i is the basis function f i (x f ,y fThis is a model parameter corresponding to ). Bending energy U of an in-field model within a single field. f It can sometimes be calculated using the following formula.

[0127]

number

[0128] Here a f and b f These are the field sizes in the x and y directions, respectively. The equation can be expressed in the following matrix form:

[0129]

number

[0130] Here, the matrix elements R of the regularization matrix R ij It is given by the following equation,

[0131]

number

[0132] Here, i and j represent the row and column indices of the matrix. For polynomial field models, the model evaluation can be expressed as follows:

[0133]

number

[0134] Here n i and m i is the power of the i-th basis function in the x and y directions. In such a model, the regularization matrix elements can be approximated by the following equation.

[0135]

number

[0136] Here r w is the wafer radius, and the prefactor is included because it starts with the bending energy on a single full field and approximates the bending energy across the entire wafer.

[0137]

[0061] The advantage of this combined method for fitting the RBF or elastic energy minimizing spline model and the field strain model is that it can yield better performance than conventional RBF modeling when the above deformation includes field strain content. This is because crosstalk from the field strain to the RBF or elastic energy minimizing spline may be mitigated or prevented, and the field strain content may be modified. If field strain model regularization is not used, crosstalk from the field strain modifiable content to the RBF or elastic energy minimizing spline model may be completely prevented. However, this comes at the expense of higher noise sensitivity. According to the above embodiment of field strain model regularization, noise propagation is suppressed at the expense of some crosstalk. Thus, the model can be tuned to use-case-dependent optimal performance via the hyperparameter φ.

[0138]

[0062] The second method involves performing a coupled fit of an RBF or elastic energy minimizing spline model to the residuals of a coupled fit of different inter-field models with field strain models. Such a two-step method involves first performing coupled modeling of different inter-field models and field strain models, and then modeling an RBF or elastic energy minimizing spline for the resulting residuals.

[0139]

[0063] The combined fit in the first step may involve fitting the inter-field polynomial (HOWA) basis functions and the field strain model basis functions in a single fit. This fitting may be performed on the combined measurement layout so that more marks are used for inter-field and field strain modeling than would be used in each fit if they were fitted individually, thereby reducing noise propagation. The combined measurement layout may include measurement positions distributed on the wafer at various in-field positions for each field.

[0140]

[0064] The methods proposed above have been described in relation to RBF or elastic energy minimizing splines and (polynomial) in-field models, but the concept is not necessarily limited to such embodiments. For example, instead of an in-field model that continuously describes the entire field (e.g., a polynomial in-field model), the model may be an average field model, i.e., a model that describes the average of the measurements with a translation parameter for each in-field position. In this way, the complete in-field strain may be described at the measurement position, thereby eliminating all crosstalk from within the field to between fields. In contrast, an underdetermined in-field polynomial model always leaves an in-field uncorrectable portion of the strain that can still cause crosstalk. If in-field correction is required at an in-field position different from the measurement position, the model can be fitted to an evaluation of the average field model on the in-field measurement grid. An additional advantage of this method is that it is possible to fit interpolation models such as thin plate splines (with or without regularization) to in-field strain.

[0141]

[0065] Intra-field regularization (i.e., as described in the paragraphs above relating to intra-field bending energy and the corresponding intra-field model regularization matrix) may also be used to regularize the fit of the intra-field model itself or in combination with different inter-field models. In some embodiments, it may be used for a combined fit of a polynomial inter-field model (such as HOWA3) and a polynomial intra-field model. In that case, the cost function to be minimized is given by the following equation:

[0142]

number

[0143] Here M and

[0144]

number

[0145] These are the polynomial inter-field model matrix and parameters, respectively. The solution is the parameter of the cost function.

[0146]

number

[0147] This can sometimes be obtained by setting the directional gradient to zero, as follows:

[0148]

number

[0149]

[0066] In some embodiments, the regularized model fit described above may be used to fit a model to data in which the model is inferior without using a regularization term included in the cost function. For example, a higher-order (e.g., third-order) polynomial in-field model may be fitted to data containing fewer than 10 (but more than 2) in-field position measurements (x and y positions) using bending energy regularization.

[0150]

[0067] As an alternative to pure in-field distortion (i.e., the same distortion in every field), the method may also be used in combination with other field distortion models. Such alternative in-field models may include field unit models that describe only the distortion of individual fields and are zero outside of those fields, scan-up scan-down in-field models in which fields exposed to move upwards are described by a different in-field model than fields exposed to move downwards, or trend in-field models in which the field distortion parameter is not constant across fields but is a function of the field sequence number (trend, e.g., linear).

[0151]

[0068] Although this text specifically refers to the use of lithography equipment in the manufacture of ICs, it should be understood that lithography equipment described herein has other applications. For example, this includes the manufacture of integrated optical systems, guidance and detection patterns for magnetic domain memory, flat panel displays, liquid crystal displays (LCDs), thin-film magnetic heads, and the like. In this context, the “substrate” being processed may be a semiconductor wafer or other substrates depending on the type of product being manufactured.

[0152]

[0069] While embodiments of the present invention are specifically referenced in relation to lithography apparatus, embodiments of the present invention can also be used in other apparatuses. Embodiments of the present invention may form part of a patterning device inspection apparatus, a metrology apparatus, or any apparatus for measuring or processing objects such as wafers (or other substrates) or masks (or other patterning devices). These apparatuses are sometimes commonly referred to as lithography tools. Such lithography tools may be used under vacuum conditions or ambient (non-vacuum) conditions.

[0153]

[0070] In this document, the terms “radiation” and “beam” are used to encompass all types of radiation, including ultraviolet radiation (e.g., 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 approximately 5 nm to 100 nm).

[0154]

[0071] The terms “reticle,” “mask,” or “patterning device,” as used herein, may be broadly interpreted to refer to a general-purpose patterning device that can be used to give an incoming radiation beam a patterned cross-section corresponding to a pattern generated on a target portion of a substrate. The term “light bulb” may also be used in this context. In addition to classic masks (transmissive or reflective masks, binary masks, phase-shift masks, hybrid masks, etc.), other examples of such patterning devices include programmable mirror arrays and programmable LCD arrays.

[0155]

[0072] The above has specifically referred to the use of embodiments of the present invention in relation to optical lithography, but it will be understood that the present invention may be used in other applications, such as imprint lithography, and is not limited to optical lithography as the context allows.

[0156]

[0073] As used herein, the term “optimize” means adjusting an apparatus (e.g., a lithography apparatus), process, etc., so that the result and / or process has desirable features such as improved accuracy of design pattern projection onto the substrate or an expanded process window. Accordingly, as used herein, the term “optimize” means the process of identifying one or more values ​​of one or more parameters that give an improvement, such as a local optimum, to an initial set of one or more values ​​of one or more parameters in at least one relevant metric. “Optimal” and other relevant terms should be interpreted accordingly. In one embodiment, the optimization step can be repeatedly applied to obtain further improvements in one or more metrics.

[0157]

[0074] Aspects of the present invention may be carried out in any convenient form. For example, one embodiment may be carried out by one or more suitable computer programs that can be carried on a suitable carrier medium which may be a tangible carrier medium (e.g., a disk) or an intangible carrier medium (e.g., a communication signal). Embodiments of the present invention may be carried out using a suitable apparatus which may specifically take the form of a programmable computer that runs a computer program configured to carry out the method described herein.

[0158]

[0075] In the block diagrams, the illustrated components are shown as separate functional blocks, but embodiments are not limited to systems in which the functions described herein are organized as illustrated. The functions provided by each component may be provided by software or hardware modules in configurations different from those currently depicted, for example, such software or hardware may be mixed, combined, duplicated, divided, distributed (e.g., within a data center or geographically), or otherwise organized. The functions described herein may be provided by one or more processors of one or more computers executing code stored in tangible, non-temporary, machine-readable media. In some cases, a third-party content distribution network may host some or all of the information transmitted over the network. In such cases, to the extent that information (such as content) is said to be supplied or provided, the information may be provided by sending instructions to retrieve that information from the content distribution network.

[0159]

[0076] Unless otherwise specified, as will be apparent from the discussion, throughout this specification, any discussion using terms such as “processing,” “calculating,” “calculating,” and “determining” is understood to refer to the actions or processes of a particular device, such as a dedicated computer or similar dedicated electronic processing / calculating device.

[0160]

[0077] Readers should understand that this application describes several inventions. Rather than separating those inventions into multiple separate patent applications, the applicant has grouped these inventions into a single document because the relevant subject matter is economical in the filing process. However, such clear merits and aspects of the invention should not be confused. In some cases, embodiments address all the defects described herein, but the inventions are useful independently, and it should be understood that some embodiments address only a subset of such problems or provide other unmentioned merits that would be obvious to those skilled in the art considering this disclosure. Due to cost constraints, some inventions disclosed herein may not be claimed at present and may be claimed in a later application, such as a continuation application, or by amending the current claims. Similarly, due to space constraints, neither the abstract section nor the summary section of this document should be considered to contain a comprehensive list of all such inventions or all aspects of such inventions.

[0161]

[0078] It should be understood that the description and drawings are not intended to limit the present invention to any particular form disclosed, but rather to encompass all modifications, equivalents, and substitutions that fall within the spirit and scope of the invention as defined by the appended claims.

[0162]

[0079] Various modifications and alternative embodiments of the present invention will be apparent to those skilled in the art in consideration of this description. Accordingly, this description and drawings should be construed as illustrative only and are intended to teach those skilled in the art a general way of carrying out the present invention. It should be understood that the forms of the present invention shown and described herein should be construed as examples of embodiments. Elements and materials can be replaced with those illustrated and described herein, parts and processes can be reversed, rearranged, omitted, certain functions can be used independently, or functions of embodiments or embodiments can be combined. Having benefited from this description of the present invention, those skilled in the art can make modifications to the elements described herein without departing from the spirit and scope of the present invention as set out in the following claims. The headings used herein are for organizational purposes only and are not intended to be used to limit the scope of the description.

[0163]

[0080] The term “possibly” as used throughout this application is used in an allowable sense (i.e., a potential possibility) rather than an obligatory sense (i.e., a must). Words such as “include,” “including,” and “include” mean “include but not limit.” As used throughout this application, the singular forms “a,” “an,” and “the” include multiple referents unless the content explicitly indicates otherwise. Thus, for example, a reference to an “an” element or an “a” element includes a combination of two or more elements, despite the use of other terms and phrases for one or more elements, such as “one or more.” The term “or” is non-exclusive unless otherwise indicated, i.e., encompasses both “and” and “or.” Terms describing conditional relationships, such as "depending on X and Y," "due to X and Y," "if X and Y," and "if X and Y," indicate that the antecedent is a necessary causal condition, the antecedent is a sufficient causal condition, or the antecedent is a contributing causal condition to the consequent. For example, "State X occurs when condition Y is met" includes "X occurs only with Y" and "X, Y and Z occur." Such conditional relationships are not limited to results that immediately follow the meeting of the antecedent condition, as some results may be delayed. In conditional statements, the antecedent is connected to the result, for example, the antecedent is connected to the possibility of what might happen as a result. A statement in which multiple attributes or functions are mapped to multiple objects (for example, one or more processors that perform steps A, B, C, and D) includes, unless otherwise specified, all such attributes or functions mapped to all such objects, and attributes or functions mapped to a subset of those attributes or functions (for example, all processors each perform steps A through D, processor 1 performs step A, processor 2 performs steps B and part of step C, and processor 3 performs part of step C and step D). Furthermore, unless otherwise specified, a statement that one value or action "based on" another condition or value includes both cases where the condition or value is the sole factor and cases where the condition or value is one of several factors.Unless otherwise specified, a statement that "each" instance of a set has certain properties should not be interpreted as meaning that other identical or similar members of a larger set do not have those properties, i.e., that each does not necessarily mean all of them. A reference to a selection from a range includes the endpoint of that range.

[0164]

[0081] In the foregoing description, any process, description, or block in a flowchart should be understood as representing a module, segment, or portion of code containing one or more executable instructions for implementing a particular logical function or step in the process, and alternative implementations that allow the functions to be performed in an order different from the illustrated or considered order, including substantially simultaneous or reverse order depending on the functions involved, are included within the scope of the exemplary embodiments of the present application, as will be understood by those skilled in the art.

[0165]

[0082] Although specific embodiments of the present invention have been described above, it will be understood that the present invention can be practiced in ways other than those described. The above description is for illustrative purposes only and is not limiting. Accordingly, it will be obvious to those skilled in the art that the present invention can be modified as described without departing from the claims set out below.

Claims

1. A method for modeling measurement data related to a substrate area in a lithography process, To obtain measurement data relating to the aforementioned substrate, This includes performing a coupled fitting to adapt at least a first inter-field model describing the strain of the substrate and a field strain model describing the strain within the exposure field to the measurement data. The method described above is The strain residual of the aforementioned coupling fitting is A method further comprising fitting a radial basis function model that describes the strain of the substrate with respect to radial basis functions, or an elastic energy minimization spline model that describes the strain of the substrate with respect to basis functions that minimize a functional of an elastic energy minimization spline model.

2. The method according to claim 1, wherein the radial basis function model includes a multiple harmonic spline model.

3. The method according to claim 2, wherein the multi-harmonic spline model includes a thin-plate spline model.

4. The method according to any one of claims 1 to 3, comprising including a regularization term in the cost function minimized in the fit of the radial basis function model or the elastic energy minimizing spline model, wherein the regularization term depends at least on the parameters of the radial basis function model or the elastic energy minimizing spline model.

5. The method according to claim 4, wherein the regularization term is equal to the functional that minimizes the regularization term.

6. The method includes performing the combined fitting to adapt the radial basis function model and the field strain model describing the strain in the exposure field to the measurement data, The method according to any one of claims 1 to 5, further comprising fitting a polynomial inter-field model to the measurement data.

7. The method according to claim 6, wherein the polynomial inter-field model includes a linear model.

8. The method according to any one of claims 1 to 6, wherein the method comprises fitting the radial basis function model or the elastic energy minimizing spline model to the strain residual of the coupled fitting of the inter-field model and the field strain model, and the inter-field model includes a higher-order polynomial inter-field model.

9. The method according to any one of claims 1 to 8, further comprising including a regularization term dependent on the parameters of the field strain model in the cost function minimized in the combined fitting.

10. The method according to claim 9, wherein the regularization term, which depends on the parameters of the field strain model, relates to the bending energy of the field strain model.

11. The method according to claim 9, wherein the regularization term, which depends on the parameters of the field strain model, relates to the integral of the square of the field strain model over the field or the entire wafer.

12. The method according to claims 9 to 11, wherein the combined fitting, which includes the regularization term in the cost function being minimized, is used to fit data in which the combined fitting without regularization is inferior.

13. The method according to any one of claims 1 to 12, wherein the coupling fitting is performed on a coupling measurement layout that includes various in-field positions for each exposure field where the measurement positions distributed on the substrate are located.

14. The method according to any one of claims 1 to 13, wherein the field strain model includes a polynomial intra-field model.

15. The method according to any one of claims 1 to 13, wherein the field strain model includes an average field model that describes the contents of the field of the measurement data using translation parameters for each position within the field.

16. The method according to claim 15, comprising fitting an interpolation model to describe the field strain model.

17. The aforementioned field strain model is A field-unit model that describes the distortion of individual fields, where the values ​​are zero outside of that field. Scan-up scan-down in-field model in which the field exposed in the first direction is described by a field distortion model different from the field exposed in the second direction, The method according to any one of claims 1 to 14, wherein the parameter describing the distortion for each field includes one of the in-trend-field models which is a function of the field sequence number.

18. A method for modeling measurement data related to a substrate area in a lithography process, To obtain measurement data relating to the said substrate, and A method comprising fitting a field strain model describing strain in an exposure field to measurement data by minimizing a cost function that includes a regularization term dependent on the parameters of the field strain model, the regularization term relating to the bending energy of the field strain model, A method comprising a coupled fitting, in which the fitting is performed in combination with fitting an inter-field model that describes the strain of the substrate.

19. The aforementioned field strain model is A field-unit model that describes the distortion of individual fields, where the values ​​are zero outside of that field. Scan-up scan-down in-field model in which the field exposed in the first direction is described by a field distortion model different from the field exposed in the second direction, The method according to any one of claims 1 to 18, wherein the parameter describing field-specific distortion includes one of the in-trend-field models which is a function of the field sequence number.

20. The method according to any one of claims 1 to 19, comprising using the results of the fitting to define a grid in the positioning operation of one or more substrate stages in a lithography process.

21. The method according to any one of claims 1 to 20, comprising measuring the substrate to obtain the measurement data.

22. A computer program comprising program instructions operable to perform the methods of claims 1 to 21 when operating on a suitable device.

23. A non-temporary computer program carrier comprising the computer program of claim 22.

24. A non-temporary computer program carrier according to claim 23, and A processing unit comprising a processor capable of operating to execute the computer program contained on the non-temporary computer program carrier.

25. Alignment.com, Support for patterning devices, A substrate support for supporting the substrate, and A lithography apparatus equipped with the processing apparatus of claim 24.

26. The lithography apparatus according to claim 25, wherein the alignment sensor is operable to measure the substrate in order to obtain the measurement data.

27. The lithography apparatus according to claim 25 or 26, wherein the processing apparatus is further operable to determine a correction for controlling the patterning device and / or substrate support based on the results of the fitting.

28. A method for modeling measurement data related to a substrate area in a lithography process, To obtain measurement data relating to the aforementioned substrate, This includes performing a coupled fitting to adapt at least a first inter-field model describing the strain of the substrate and a field strain model describing the strain within the exposure field to the measurement data. The at least first inter-field model includes a radial basis function model that describes the strain of the substrate with respect to radial basis functions, or an elastic energy minimizing spline model that describes the strain of the substrate with respect to basis functions that minimize a functional of an elastic energy minimizing spline model, The method further comprises including a regularization term in the cost function minimized in the fit of the radial basis function model or the elastic energy minimizing spline model, wherein the regularization term depends at least on the parameters of the radial basis function model or the elastic energy minimizing spline model.

Images