Method for obtaining a fingerprint model, computer program

Abstract

To solve the problem of the requirement to sample an entire substrate in order to accurately determine a correction model intended to describe performance parameter behavior.SOLUTION: A method for determining a sampling scheme includes: obtaining a first fingerprint model relating to a first spatial distribution of a performance parameter over a first portion of a semiconductor substrate, and a second fingerprint model relating to a second spatial distribution of the performance parameter over a second portion of the semiconductor substrate; and determining a sampling point corresponding to a measuring location on the semiconductor substrate for generating measurement data based on an expected reduction of a first uncertainty metric associated with evaluation of the first fingerprint model over the first portion and an expected reduction of a second uncertainty metric associated with evaluation of the second fingerprint model over the second portion.SELECTED DRAWING: Figure 5

Description

[Technical Field]

[0001] CROSS-REFERENCE TO RELATED APPLICATIONS

[0001] This application claims priority to European Patent Application Publication No. 19176024.8, filed May 22, 2019, and European Patent Application Publication No. 20173733.5, filed May 8, 2020, which are incorporated herein by reference in their entireties.

[0002] This specification relates to methods for generating efficient sampling schemes that can be used to measure and / or inspect specimens or substrates, such as semiconductor devices, and the results of which can be used to control a lithographic apparatus. [Background technology]

[0003] Lithography is a process in which a desired pattern is applied to a substrate, usually onto a target portion of the substrate. Lithographic apparatus (scanners) can be used, for example, in the manufacture of integrated circuits (ICs). In this case, a patterning device (alternatively called a mask or reticle) can be used to generate a product pattern, which will be formed on an individual layer of the IC. This pattern can be transferred onto a target portion (e.g. comprising part of one or several dies) on the substrate (e.g. a silicon wafer). Transfer of the pattern is typically by imaging onto a layer of radiation-sensitive material (resist) provided on the substrate. Stepping and / or scanning movements can be involved to repeat the pattern in successive target portions across the substrate. It is also possible to transfer the pattern from the patterning device to the substrate by imprinting the pattern onto the substrate. The pattern can be transformed into functional product features by further processing steps.

[0004]

[0004] A key performance parameter of a lithography process is overlay error. Often simply referred to as "overlay," this error is the error associated with correctly positioning a product feature relative to features formed in a previous layer. As product features become increasingly smaller, overlay specifications become increasingly stringent.

[0005] Currently, overlay errors are controlled and corrected using correction models, such as those described in U.S. Patent Application Publication No. 2013230797A1. Advanced process control techniques have been introduced in recent years that use measurements of metrology targets applied to substrates along with the device patterns. These targets allow overlay to be measured using high-throughput inspection tools such as scatterometers, and these measurements can be used to determine overlay fingerprints and generate corrections that are then fed back to the lithography tool during subsequent substrate patterning. Examples of advanced process control (APC) are described in U.S. Patent Application Publication No. 2012008127A1, for example. The inspection tool may be separate from the lithography tool. Instead of directly using measurements to define control actions, it is often desirable to first apply correction models to measurements corresponding to overlay targets applied to the substrate. Modeled measurement data is generally less noisy, allowing for the definition of control actions across the wafer, even at small spatial scales. Modern correction models typically include higher-order terms to correct for nonlinear distortions of the wafer. Nonlinear distortions can be associated with many effects. These effects can be, for example, processing-induced deformations of the substrate due to stress components introduced during the annealing, etching, or layer deposition steps of the patterning process. Correction models can also be extended to consider other measured and / or calculated effects, such as thermal deformations during patterning operations.

[0006]

[0006] While the use of higher-order models may allow for more effects to be taken into account, such models require more position measurements to be made. Furthermore, higher-order correction models require more computing power and / or more computation time. Thus, while the use of higher-order correction models may be theoretically feasible under certain circumstances, in practice they may not be economically viable due to the adverse impact on the throughput (i.e., number of wafers per hour) of the lithography process. In addition, higher-order correction models may be of limited use if the patterning device itself does not provide control at a spatial resolution corresponding to the spatial scale of the higher-order terms of the correction model.

[0007] As described in WO 2016 / 146217, certain components of the overlay fingerprint vary randomly from substrate to substrate. However, other components are systematic in nature, whether their causes are known or not. When similar substrates have similar patterns of performance parameters (e.g., overlay error), this pattern of performance parameters is sometimes called the "fingerprint" of the lithography process. Performance parameters such as overlay error can be roughly classified into two different groups: 1) Contributions that vary across the substrate (wafer) are known in the art as field-to-field fingerprints; 2) Contributions that vary similarly across each target portion (field) or subfield of a substrate or wafer are known in the art as intra-field fingerprints.

[0008] At cutting edge nodes, or in 3D-NAND on semiconductor wafers, repeating patterns of overlay error fingerprints induced by wafer processing steps such as etching are observed. These fingerprints can be compensated in terms of sub-field overlay correction, which allows and / or requires scanner correction at high spatial frequencies.

[0009]

[0009] WO 2016 / 146217, incorporated herein by reference, discloses a method in which one or more subfields of a field are defined based on available data. Performance parameter data relating to each subfield is processed to generate subfield correction information. Exposure of the subfield is corrected using the subfield correction information. By controlling the lithography apparatus with reference to performance data of a particular subfield within a field, overlay errors can be minimized with respect to the critical feature corresponding to that particular subfield, rather than averaged over the entire field. By controlling the lithography apparatus with reference to the subfield rather than only the entire field, residual errors can be reduced in one or more subfields. Performance parameter data relating to one or more subfields can be modeled using a correction model specifically designed down to the spatial scale of subfield-level performance parameter behavior.

[0010]

[0010] Methods such as those described in WO 2016 / 146217 require measuring performance parameters of the lithography process (such as overlay error) at a large number of locations on the substrate, particularly at a sufficiently dense distribution across one or more subfields, to enable sufficiently accurate determination of a subfield-specific correction model. However, in order to accurately determine a correction model that aims to represent the performance parameter behavior across the entire substrate, or at least across a portion larger than a subfield, it is often also necessary to sample the entire substrate. The object of the present invention is to determine suitable locations for measuring performance parameters that meet the above-mentioned requirements. Summary of the Invention

[0011]

[0011] The present specification aims to provide a method for determining a sampling scheme that can be used, for example, to measure and / or inspect a semiconductor substrate and / or to control a lithography process performed on a semiconductor substrate.

[0012] According to one embodiment of the present invention, there is provided a method for determining a sampling scheme, the method comprising: - obtaining a first fingerprint model of a first spatial distribution of a performance parameter across a first portion of a semiconductor substrate and a second fingerprint model of a second spatial distribution of said performance parameter across a second portion of the semiconductor substrate; - determining sampling points corresponding to measurement locations on the semiconductor substrate for generating measurement data based on an expected decrease in a first uncertainty metric associated with evaluation of the first fingerprint model over the first portion and an expected decrease in a second uncertainty metric associated with evaluation of the second fingerprint model over the second portion.

[0013]

[0013] A single sampling point determined according to this embodiment can be used to derive model parameters for both the first fingerprint model and the second fingerprint model, while reducing the number of measurements that need to be made to obtain the model parameters.

[0014] In one embodiment of the above method for determining a sampling scheme, the first portion and the second portion at least partially overlap each other.

[0015]

[0015] This is a practical implementation of the above embodiment, which allows for effective control of the lithographic apparatus.

[0016] In one embodiment of the above method for determining a sampling scheme, the first fingerprint model is a global fingerprint model and the second fingerprint model is a local fingerprint model.

[0017] For example, the first fingerprint model is a fingerprint model of the entire top surface of the semiconductor substrate (inter-field), for example, for a performance parameter whose value depends on a position within the entire semiconductor substrate surface, for example, a radial distance from the center of the substrate. For example, the second fingerprint model is a fingerprint model of a portion of the semiconductor substrate surface, for example, a field, die, or cell (intra-field), for example, for a performance parameter whose value depends on a position within the portion of the semiconductor substrate, for example, a distance in the x- or y-direction from the center of the field.

[0018]

[0018] This is a practical implementation of the above embodiment, which allows for effective control of the lithographic apparatus.

[0019] In one embodiment of the above method for determining a sampling scheme, the first portion and / or the second portion is at least a part of an irradiated layer in or on a top surface of the semiconductor substrate.

[0020]

[0020] This is a practical implementation of the above embodiment, which allows for effective control of the lithographic apparatus.

[0021]

[0021] In one embodiment of the above method for determining a sampling scheme, the second portion is one of one field, multiple fields, one subfield, multiple subfields, one die, multiple dies, part of one die, multiple parts of multiple dies, one cell, multiple cells, part of one cell, or multiple parts of multiple cells.

[0022]

[0022] This is a practical implementation of the above embodiment, which allows for effective control of the lithographic apparatus.

[0023]

[0023] In one embodiment of the above method for determining a sampling scheme, the performance parameters are overlay error, critical dimension (e.g., line width), alignment mark position, sidewall angle, edge roughness, edge placement error, and / or focus error.

[0024]

[0024] This is a practical implementation of the above embodiment, which allows for effective control of the lithographic apparatus.

[0025] In one embodiment of the above method for determining a sampling scheme, the first fingerprint model and / or the second fingerprint model further relates to the spatial distribution of at least one further parameter.

[0026] For example, the at least one further parameter is the prediction uncertainty of the first and / or second fingerprint model, for example the normalized model uncertainty, G-optimality, least squared error, and / or moving standard deviation.

[0027]

[0027] This is a practical implementation of the above embodiment, which allows for effective control of the lithographic apparatus.

[0028] In one embodiment of the above method for determining a sampling scheme, determining the locations of the sampling points comprises taking into account drift of the first and / or second spatial distributions of the performance parameter over time.

[0029]

[0029] This allows for precise control of the lithographic apparatus.

[0030]

[0030] In one embodiment of the method for establishing a sampling scheme, the method further comprises the step of determining additional sampling points.

[0031]

[0031] The number of additional sampling points is preferably less than the number of sampling points determined by any one of the above method embodiments.

[0032]

[0032] Thus, the sampling scheme may include only sampling points determined by any one of the above method embodiments. Alternatively, the sampling scheme may include sampling points determined by any one of the above method embodiments and further sampling points determined in another way.

[0033] In one embodiment of the above method of establishing a sampling scheme, the method comprises: - identifying one or more critical areas on a semiconductor substrate; - determining a greater number of sampling points per surface area for one or more critical areas of the semiconductor substrate than for other areas of the semiconductor substrate; Further includes:

[0034] This allows for precise control of the lithographic apparatus when applying a pattern to more critical areas of the substrate.

[0035] According to one embodiment of the present invention, there is provided a method for generating measurement data from a semiconductor substrate, the method comprising: - determining the locations of the sampling points according to any one of the embodiments of the method for determining a sampling scheme described above; - generating measurement data by measuring performance parameters at sampling point locations on a semiconductor substrate; Includes:

[0036] In this embodiment, the sampling points determined according to any one of the embodiments of the method for determining a sampling scheme described above are used to generate the measurement data.

[0037] According to one embodiment of the present invention, there is provided a method for generating measurement data from a semiconductor substrate, the method comprising: - receiving information about the locations of the sampling points, the locations being determined according to one of the embodiments of the method for determining a sampling scheme described above; - generating measurement data by measuring performance parameters at sampling point locations on a semiconductor substrate; Includes:

[0038]

[0038] Optionally, the method further comprises determining model parameters of the first fingerprint model and model parameters of the second fingerprint model based on the generated measurement data.

[0039]

[0039] In this embodiment, the sampling points determined according to any one of the embodiments of the method for determining a sampling scheme described above are used to generate measurement data, and optionally further to determine model parameters of a first fingerprint model and model parameters of a second fingerprint model based on the generated measurement data.

[0040] According to one embodiment of the present invention, there is provided a method for determining model parameters, the method comprising: - generating metrology data from a semiconductor substrate according to one embodiment of the method for generating metrology data from a semiconductor substrate described above; - determining model parameters of the first fingerprint model and model parameters of the second fingerprint model based on the generated measurement data; Includes: Optionally, the method is used in a method for controlling a lithographic apparatus, the method comprising the steps of: - placing a semiconductor substrate in a lithographic apparatus; - applying a pattern onto the semiconductor substrate in a lithographic apparatus; - controlling the applying of the pattern on the basis of control input data, the control input data being based at least in part on said model parameters of the first fingerprint model and / or model parameters of the second fingerprint model; Further includes:

[0041] For example, controlling the application of a pattern onto a semiconductor substrate includes controlling movement of the semiconductor substrate relative to exposure light from a patterning system of a lithographic apparatus and / or controlling movement of a patterning device, i.e., a reticle, relative to illumination light of an illumination system. Further examples of controlling the application of a pattern onto a semiconductor substrate include controlling the position and / or orientation of a lens in the patterning system, controlling the illumination settings of the illumination system, controlling the focus of the exposure light, and / or controlling the dose of the exposure light.

[0042] In this embodiment, the semiconductor substrate for which the measurement data is generated may be the same as the semiconductor substrate placed in the lithographic apparatus, or may be different.

[0043] In this embodiment, the lithographic apparatus is efficiently controlled by using measurements performed at the sampling points.

[0044]

[0044] According to one embodiment of the present invention, there is provided a computer program comprising one or more sequences of machine-readable instructions configured to perform one embodiment of any of the methods described above.

[0045]

[0045] According to one embodiment of the present invention, - receiving sampling points generated according to any one of the embodiments of the method for determining a sampling scheme described above; - controlling the measurement device to generate measurement data from the semiconductor substrate at measurement locations corresponding to the sampling points; - determining model parameters of the first fingerprint model and model parameters of the second fingerprint model based at least in part on the generated measurement data; A computer program product is provided that includes one or more sequences of machine-readable instructions configured to:

[0046]

[0046] According to one embodiment of the present invention, receiving measurement data from the semiconductor substrate at measurement locations corresponding to the sampling points determined according to any one of the embodiments of the method for determining a sampling scheme described above; - determining model parameters of the first fingerprint model and model parameters of the second fingerprint model based at least in part on the received measurement data; - controlling the dispensing of a pattern on a semiconductor substrate on the basis of control input data, the control input data being based at least in part on said model parameters of the first fingerprint model and / or model parameters of a second fingerprint model; A computer program product is provided that includes one or more sequences of machine-readable instructions configured to:

[0047]

[0047] According to one embodiment of the present invention, there is provided a semiconductor substrate measurement apparatus, the apparatus comprising: an input terminal configured to receive sampling points determined according to any one of the embodiments of the method for determining a sampling scheme described above; - a controller configured to control the measurement device to generate measurement data from the semiconductor substrate at measurement locations corresponding to the sampling points; - a processor configured to determine model parameters of the first fingerprint model and model parameters of the second fingerprint model based at least in part on the generated measurement data; Includes.

[0048]

[0048] According to an embodiment of the present invention, there is provided a lithographic apparatus comprising: - an input terminal configured to receive measurement data from the semiconductor substrate at measurement locations corresponding to the sampling points determined according to any one of the embodiments of the method for determining a sampling scheme described above; - a processor configured to determine model parameters of the first fingerprint model and model parameters of the second fingerprint model based at least in part on the received measurement data; - a controller configured to control the dispensing of a pattern onto a semiconductor substrate on the basis of control input data, the control input data being based at least in part on said model parameters of the first fingerprint model and / or model parameters of the second fingerprint model; and Includes.

[0049]

[0049] According to one embodiment of the present invention, there is provided a lithography system, which includes a semiconductor substrate measurement apparatus according to the embodiments described above, and a lithography apparatus according to the embodiments described above.

[0050]

[0050] According to one embodiment of the present invention, there is provided a device manufacturing method comprising transferring a pattern from a patterning device onto a substrate, the method comprising using a lithographic apparatus according to the embodiment described above.

[0051]

[0051] According to a further embodiment of the present invention, there is provided a method of generating a sampling scheme for a semiconductor substrate, comprising the steps of: identifying at least two fingerprints of the substrate, each fingerprint associated with a different spatial scale; generating a sampling scheme based on the accuracy required to determine at least two different fingerprints; A method is provided which includes:

[0052] In one embodiment of the method according to the invention, the step of generating the sampling scheme is based on at least two different parameters.

[0053] For example, one of the at least two different parameters is a prediction uncertainty of the model. Optionally, the prediction uncertainty of the model is a normalized model uncertainty, a G-optimality, a least squares error, and / or a moving standard deviation.

[0054]

[0054] In one embodiment of the above method according to the present invention, the step of generating a sampling scheme is based on at least two different parameters, and the at least two different parameters are not normalized model uncertainty, G-optimality, least squares error, and / or moving standard deviation.

[0055]

[0055] In one embodiment of the above method according to the present invention, at least one of the at least two different fingerprints is a fingerprint across the substrate, a fingerprint across the field, a fingerprint across the subfield, a fingerprint across the die, a fingerprint across a set of dies, a fingerprint across the cell, a fingerprint across a set of cells or a subfield of cells.

[0056] In an embodiment of the method according to the invention, the step of generating a sampling scheme is to use at least one model of at least two different fingerprints, optionally related to a profile correctable using a lithographic apparatus.

[0057] In an embodiment of the method according to the invention, the step of generating the sampling scheme is further based on the drift of at least one of the at least two different fingerprints.

[0058] In one embodiment of the method according to the invention, one of the at least two different fingerprints is at least partially related to the substrate context data.

[0059]

[0059] In one embodiment of the above method according to the present invention, the step of generating a sampling scheme includes the steps of generating at least two sub-sampling schemes for at least two different fingerprints, and combining the at least two sub-sampling schemes to generate the sampling scheme.

[0060] In one embodiment of the above method according to the invention, one of the at least two different fingerprints is a fingerprint over an area of ​​the substrate that is smaller than a minimum dimension associated with the correction potential of the lithographic apparatus.

[0061]

[0061] According to a further embodiment of the present invention, there is provided a method for determining a sampling scheme according to any one of the embodiments of the method for determining a sampling scheme described above, wherein the first fingerprint model or the second fingerprint model is obtained using a method for obtaining a fingerprint model for modeling the spatial distribution of a performance parameter across a portion of a substrate, the method comprising: defining an initial fingerprint model for the spatial distribution of the performance parameter across said portion as a parameterized combination of basis functions; determining parameter values ​​for the parameterized combination of basis functions based on prior knowledge; and determining the fingerprint model for modeling the spatial distribution of the performance parameter across said portion of the substrate based on a measurement set representative of performance parameters across said portion of a set of one or more semiconductor substrates and the initial fingerprint model, whereby the ratio of at least two parameter values ​​of the basis functions determined in the initial fingerprint model is kept constant or at least a part of the shape of the initial fingerprint model is maintained in the fingerprint model.

[0062]

[0062] According to a further embodiment of the present invention, there is provided a method for determining a sampling scheme according to any one of the embodiments of the method for determining a sampling scheme described above, wherein a first fingerprint model relates to a first spatial distribution of a performance parameter across a first portion of the substrate as a parameterized combination of a first set of basis functions, and a second fingerprint model relates to a second spatial distribution of the performance parameter across a second portion of the substrate as a parameterized combination of a second set of basis functions, and one or more of the first set of basis functions are substantially orthogonal to one or more of the second set of basis functions.

[0063]

[0063] Embodiments of the present invention will now be described, by way of example only, with reference to the accompanying schematic drawings, in which: [Brief explanation of the drawings]

[0064] [Figure 1]

[0063] A lithographic apparatus according to one embodiment of the present invention is depicted. [Figure 2] 1 shows diagrammatically stages in the measurement and exposure process of the apparatus of FIG. 1, according to known conventions. [Figure 3]

[0063] Computer system hardware useful in implementing the processes disclosed herein is illustrated. [Figure 4]

[0063] A schematic diagram of the area of ​​interest relative to the minimum size of a subfield is shown. [Figure 5]

[0063] A semiconductor substrate is shown schematically. [Figure 6A]

[0063] A first fingerprint model of the performance parameter P is shown schematically. [Figure 6B]

[0063] A second fingerprint model for the performance parameter P is shown schematically. [Figure 7]

[0063] A fingerprint model of the performance parameter P across subfields is shown schematically. DETAILED DESCRIPTION OF THE INVENTION

[0065]

[0064] Figure 1 depicts a schematic representation of a lithographic apparatus LA according to one embodiment of the present invention. an illumination system (illuminator) IL configured to condition a radiation beam B (e.g. UV radiation or EUV radiation), a support structure (e.g. mask table) MT constructed to support a patterning device (e.g. mask) MA and connected to a first positioner PM configured to accurately position the patterning device according to certain parameters; a substrate table (e.g., wafer table) WTa or WTb constructed to hold a substrate (e.g., a resist-coated wafer) W and connected to a second positioner PW configured to accurately position the substrate according to certain parameters; and a projection system (e.g. a refractive projection lens system) PS configured to project a pattern imparted to the radiation beam B by the patterning device MA onto a target portion C (e.g. comprising one or more dies) of the substrate W.

[0066]

[0065] The illumination system may include various types of optical components, such as refractive, reflective, magnetic, electromagnetic, electrostatic or other types of optical components, or any combination thereof, for directing, shaping or controlling radiation.

[0067] The mask support structure supports, i.e. bears the weight of, the patterning device. The support structure holds the patterning device in a manner that depends on the orientation of the patterning device, the design of the lithographic apparatus, and other conditions, such as for example whether or not the patterning device is held in a vacuum environment. The support structure can use mechanical, vacuum, electrostatic or other clamping techniques to hold the patterning device. The support structure may be, for example, a frame or a table, which may be fixed or movable as required. The support structure may ensure that the patterning device is at a desired position, for example with respect to the projection system. Any use of the terms "reticle" or "mask" herein may be considered as synonymous with the more general term "patterning device".

[0068]

[0067] The term "patterning device", as used herein, should be interpreted broadly as referring to any device that can be used to impart a radiation beam with a pattern in its cross-section so as to generate a pattern in a target portion of a substrate. It should be noted that the pattern imparted to the radiation beam may not exactly correspond to the desired pattern in the target portion of the substrate, for example if the pattern includes phase-shifting features or so-called assist features. Typically, the pattern imparted to the radiation beam will correspond to a particular functional layer in a device, such as an integrated circuit, being created in the target portion.

[0069]

[0068] A patterning device may be transmissive or reflective. Examples of patterning devices include masks, programmable mirror arrays, and programmable LCD panels. Masks are well known in lithography, and include mask types such as binary, levelson phase-shift, and attenuated phase-shift, as well as various hybrid mask types. An example of a programmable mirror array employs a matrix arrangement of small mirrors, each of which can be individually tilted so as to reflect an incoming radiation beam in different directions. The tilted mirrors impart a pattern to the radiation beam that is reflected by the mirror matrix.

[0070]

[0069] The term "projection system", as used herein, should be interpreted broadly as encompassing any type of projection system, including refractive, reflective, catadioptric, magnetic, electromagnetic and electrostatic optical systems, or any combination thereof, as appropriate for the exposure radiation used, or other factors such as the use of an immersion liquid or the use of a vacuum. Any use of the term "projection lens" herein may be considered as synonymous with the more general term "projection system".

[0071] As depicted here, the apparatus is of a transmissive type (e.g. employing a transmissive mask) Alternatively, the apparatus may be of a reflective type (e.g. employing a programmable mirror array of a type as referred to above, or employing a reflective mask).

[0072] The lithographic apparatus may be of a type having two (dual stage) or more substrate tables (and / or two or more mask tables). In such a "multi-stage" machine, the additional tables may be used in parallel, or preparatory steps may be performed on one or more tables while one or more other tables are being used for exposure. The invention disclosed herein may be used in stand-alone form, but may provide additional functionality, in particular in the pre-exposure measurement stage of a single or multi-stage apparatus.

[0073] The lithographic apparatus may be of a type in which at least a part of the substrate may be covered by a liquid having a relatively high refractive index, for example water, so as to fill a space between the projection system and the substrate. Immersion liquids may also be applied to other spaces in the lithographic apparatus, for example between the mask and the projection system. Immersion techniques are well known in the art for increasing the numerical aperture of projection systems. The term "immersion" as used herein does not imply that a structure such as the substrate has to be immersed in liquid, but merely that a liquid is located between the projection system and the substrate during exposure.

[0074]

[0073] Referring to Figure 1, the illuminator IL receives a radiation beam from a radiation source SO. The radiation source and the lithographic apparatus may be separate entities, for example when the radiation source is an excimer laser. In such cases, the radiation source is not considered to form part of the lithographic apparatus, and the radiation beam is passed from the source SO to the illuminator IL with the aid of a beam delivery system BD, which may include, for example, suitable directing mirrors and / or beam expanders. In other cases, the radiation source may be an integral part of the lithographic apparatus, for example when the radiation source is a mercury lamp. The radiation source SO and the illuminator IL, together with the beam delivery system BD if required, may be referred to as a radiation system.

[0075] The illuminator IL may include an adjuster AD for adjusting the angular intensity distribution of the radiation beam. Typically, at least the outer and / or inner radial extent (commonly referred to as σ-outer and σ-inner, respectively) of the intensity distribution in a pupil plane of the illuminator may be adjusted. In addition, the illuminator IL may include various other components, such as an integrator IN and a condenser CO. The illuminator may be used to adjust the radiation beam so that it has a desired uniformity and intensity distribution in its cross-section.

[0076] The radiation beam B is incident on a patterning device (e.g., mask MA), which is held on a support structure (e.g., mask table MT), and is patterned by the patterning device. After passing through the mask MA, the radiation beam B passes through the projection system PS, which focuses the beam onto a target portion C of the substrate W. With the aid of the second positioner PW and a position sensor IF (e.g., an interferometric device, a linear encoder or a capacitive sensor), the substrate table WTa / WTb can be moved accurately, for example to position different target portions C in the path of the radiation beam B. Similarly, the first positioner PM and further position sensors (not explicitly depicted in Figure 1) can be used to accurately position the mask MA with respect to the path of the radiation beam B, for example after a mechanical lookup in a mask library or during a scan. In general, movement of the mask table MT may be realized with the aid of a long-stroke module (coarse positioning) and a short-stroke module (fine positioning), which form part of the first positioner PM. Similarly, movement of the substrate tables WTa / WTb may be realized using a long-stroke module and a short-stroke module, which form part of the second positioner PW. In the case of a stepper (as opposed to a scanner) the mask table MT may be connected to a short-stroke actuator only, or may be fixed. The mask MA and substrate W may be aligned using mask alignment marks M1, M2 and substrate alignment marks P1, P2. Although the illustrated substrate alignment marks occupy dedicated target portions, the substrate alignment marks may be located in spaces between the target portions (these are known as scribe-lane alignment marks). Similarly, in situations in which more than one die is provided on the mask MA, the mask alignment marks may be located between the dies.

[0077] The depicted apparatus could be used in at least one of the following modes: 1. In step mode, the mask table MT and the substrate table WTa / WTb are kept essentially stationary while an entire pattern imparted to the radiation beam is projected onto the target portion C in one go (i.e. a single static exposure). The substrate table WTa / WTb is then shifted in the X and / or Y direction so that a different target portion C can be exposed. In step mode, the maximum size of the exposure field limits the size of the target portion C imaged in a single static exposure. 2. In scan mode, the mask table MT and the substrate table WTa / WTb are scanned synchronously while a pattern imparted to the radiation beam is projected onto a target portion C (i.e. a single dynamic exposure). The velocity and direction of the substrate table WTa / WTb relative to the mask table MT may be determined by the (de-)magnification and image reversal characteristics of the projection system PS. In scan mode, the maximum size of the exposure field limits the width (in the non-scanning direction) of the target portion in a single dynamic exposure, whereas the length of the scanning movement determines the height (in the scanning direction) of the target portion. 3. In another mode, the mask table MT is kept essentially stationary holding a programmable patterning device, and the substrate table WTa / WTb is moved or scanned while a pattern imparted to the radiation beam is projected onto a target portion C. In this mode, a pulsed radiation source is typically used, and the programmable patterning device is updated as required after each movement of the substrate table WTa / WTb, or between successive radiation pulses during a scan. This mode of operation is readily adaptable to maskless lithography employing a programmable patterning device, such as a programmable mirror array of the type referred to above.

[0078] Combinations and / or variations on the above described modes of use or entirely different modes of use may also be employed.

[0079] The lithographic apparatus LA in this example is of a so-called dual-stage type, having two substrate tables WTa and WTb and two stations (an exposure station and a measurement station) that allow the substrate tables to be swapped between them. While one substrate on one substrate table is being exposed at the exposure station EXP, another substrate can be loaded onto the other substrate table at the measurement station MEA so that various preparation steps can be performed. The preparation steps may include mapping the surface of the substrate using the level sensor LS and measuring the position of an alignment mark on the substrate using the alignment sensor AS. This allows for a significant increase in the throughput of the apparatus. If the position sensor IF is unable to measure the position of the substrate table when it is at the measurement station and the exposure station, a second position sensor may be provided to allow the position of the substrate table to be tracked at both stations. The present invention may be applied to apparatuses with only one substrate table or with three or more substrate tables.

[0080] The apparatus further includes a lithography apparatus control unit LACU, which controls all of the operations and measurements of the various actuators and sensors described. The LACU also includes signal processing and data processing capabilities for implementing desired calculations related to the operation of the apparatus. In practice, the control unit LACU is realized as a system of multiple subunits, each handling real-time data acquisition, processing, and control of a subsystem or component within the apparatus. For example, one processing subsystem may be specialized for servo control of the substrate positioner PW. Separate units may handle coarse and fine actuators, or even different axes. Another unit may be specialized for readout of the position sensor IF. Overall control of the apparatus may be controlled by a central processing unit, which communicates with these subsystem processing units, operators, and other devices involved in the lithography manufacturing process.

[0081] FIG. 2 shows known steps for exposing a target portion (e.g., a die) on a substrate W in the dual stage apparatus of FIG. 1. The left side of the dotted box shows steps taking place in the measurement station MEA, and the right side shows steps taking place in the exposure station EXP. At any given time, one of the substrate tables WTa, WTb is in the exposure station and the other is in the measurement station, as described above. For purposes of explanation, it is assumed that a substrate W is already loaded at the exposure station. In step 200, a new substrate W′ is loaded into the apparatus by a mechanism not shown. These two substrates are processed in parallel to increase the throughput of the lithography apparatus. When referring to a newly loaded substrate W′ first, this may be a previously unprocessed substrate that has been prepared with new photoresist for its first exposure in the apparatus. However, in general, the lithography process described is only one step in a series of exposure and processing steps, so that the substrate W′ may have already been through this and / or other lithography apparatus several times and may also have subsequent processes due to be undergone.

[0082]

[0081] The preceding and / or subsequent processes may be performed in other lithography apparatus, as described above, or even in different types of lithography apparatus. For example, some layers in a device manufacturing process that are very demanding in parameters such as resolution and overlay may be performed in a more advanced lithography tool than other layers that are less demanding. Thus, some layers may be exposed in an immersion lithography tool, while other layers are exposed in a "dry" tool. Some layers may be exposed in a tool operating at DUV wavelengths, while other layers are exposed using EUV wavelength radiation.

[0083] In 202, the alignment of the substrate relative to the substrate tables WTa / WTb is measured and recorded using substrate marks P1 etc. and alignment measurements using an image sensor (not shown). In addition, several alignment marks are measured across the substrate W' to establish a "wafer grid" that very accurately maps the distribution of marks across the substrate, including distortions relative to a nominal rectangular grid. In step 204, a map of substrate height versus XY position is also measured for use in accurate focusing of the exposure pattern.

[0084] When the substrate W′ is loaded, recipe data 206 is received, defining the exposure to be performed as well as the characteristics of the substrate, the previously created patterns, and the patterns to be created thereon. Measurements of the substrate position, the substrate grid, and the height map generated in 202, 204 are added to these recipe data so that a complete set of recipe and measurement data 208 can be passed to the exposure stage. Measurements of alignment data include, for example, the X and Y positions of alignment targets formed in a fixed or nominally fixed relationship to the product pattern that is the product of the lithography process. These alignment data, acquired immediately before exposure, are combined and interpolated to provide parameters for an alignment model. These parameters and the alignment model are used during the exposure operation to correct the position of the pattern applied in the current lithography step. Conventional alignment models may include four, five, or six parameters that together define the translation, rotation, and scaling of an “ideal” grid in different dimensions. As explained further below, advanced models using more parameters are known. In 210, substrate W' and substrate W are swapped so that the measured substrate W' becomes the substrate W entering exposure station EXP. This swap is performed by exchanging supports within the apparatus so that substrates W, W' remain precisely clamped and positioned on supports WTa and WTb to maintain relative alignment between the substrate tables and the substrates themselves. Thus, once the tables have been swapped, all that is required to utilize measurement information 202, 204 for substrate W (formerly W') in controlling the exposure step is to determine the relative position between projection system PS and substrate table WTb (formerly WTa). In step 212, reticle alignment is performed using mask alignment marks M1, M2. In steps 214, 216, 218, scanning movements and radiation pulses are applied at multiple sequential target locations across substrate W to complete exposure of several patterns.By using the alignment data and height maps acquired in the metrology station during the exposure steps, these patterns are precisely aligned to the desired locations, and in particular to features previously provided on the same substrate. The exposed substrate, now labeled W", is removed from the apparatus in step 220 to undergo etching or other processes depending on the exposure pattern.

[0085] 5 schematically shows a semiconductor substrate W. The semiconductor substrate W has a substantially circular shape with a center 10.

[0086]

[0085] The semiconductor substrate W (or "substrate W" for short) comprises a plurality of fields 11.1, 11.2, two of which are shown schematically in Figure 5. A field is, for example, an area of ​​the substrate W that is illuminated during subsequent movement of a stage relative to the projection system. In this example, each field 11.1, 11.2 is formed by a plurality of dies 12.1, 12.2, also shown schematically in Figure 5. In this example, the dies are therefore smaller than the fields. After processing of the substrate, the dies form individual microchips. In another example, the dies may comprise a plurality of fields. In that example, the dies are therefore larger than the fields. Each die optionally comprises a plurality of cells 13.1, 13.2, also shown schematically in Figure 5. A cell is therefore an area within a die. A cell is therefore smaller than a die. Figure 5 shows two fields 11.1, 11.2. In Figure 5, each field comprises four dies, for example 12.1, 12.2. In Figure 5, each die includes two cells, for example 13.1, 13.2. The number of fields, dies and cells may vary.

[0087] The value of the performance parameter may vary with the position of the points 14.1, 14.2 on the substrate W (e.g. the radial distance from said points to the centre 10 of the substrate W). It is known that in some cases this variation can be represented using a mathematical model (e.g. a polynomial), although the model parameters may differ between individual substrates W.

[0088] For example, it can be seen that the value of the performance parameter at points 14.1, 14.2 depends on the radial distance to the center 10 of the substrate, and that the value follows a parabolic curve. In this case, the mathematical model for representing the parameter value is a second order polynomial, which is: p1=aR 2 +bR+c, where p1 represents the value of the performance parameter P for the substrate W, R is the radial distance of a point to the center 10 of the substrate W, and a, b, and c are model parameters. The mathematical model may or may not include periodicity. An example of a mathematical model that includes periodicity is a sine wave.

[0089] The model parameters for each individual substrate may be determined by performing measurements at sample points. Sample points are locations on the substrate W where measurements are made with the aim of providing measurement data from which the model parameters a, b, and c can be determined. A plurality of sampling points together form a sampling scheme. Once the model parameters a, b, and c have been determined, the distribution of the performance parameters can be predicted for locations where no measurements have been made, and the performance parameter measurement data can therefore be densified. By using the densified performance parameter measurement data, the lithographic apparatus can be controlled with improved accuracy, as knowledge of the values ​​of the densified performance parameter measurement data and / or the model parameters of the model for the performance parameters allows errors or other deviations to be compensated for.

[0090]

[0089] When sampling scheme includes a large number of sampling points, it can accurately determine the performance parameter distribution. However, it takes a long time to perform the measurements required to determine model parameters, which reduces the throughput of the lithography process. Therefore, it is advantageous to be able to reduce the number of sampling points of sampling scheme.

[0091] The value of a performance parameter may vary not only with the position of a point 14.1, 14.2 on the substrate W, but also with the position of said point 14.1, 14.2 within a field 11.1, 11.2, die 12.1, 12.2 or cell 13.1, 13.2, e.g. with the distance in the x or y direction from the corner 15.1, 15.1 of the field 11.1, 11.2 in which the point 14.1, 14.2 is located. The mathematical model describing the variation of the value of a performance parameter may be the same as or different from the mathematical model describing the variation of that performance parameter as a whole across the surface of the substrate surface. For example, the variation of a performance parameter within fields 11.1, 11.2 depending on the distance in the x direction from the corner points 15.1, 15.2 of fields 11.1, 11.2 can be expressed as p2 = fY + g, where p2 represents the value of the performance parameter P for fields 11.1, 11.2, Y is the distance in the y direction from points 14.1, 14.2 to the corner points 15.1, 15.2 of the respective fields 11.1, 11.2, and f and g are model parameters.

[0092] 5, the value of the performance parameter P (which depends on the position of points 14.1, 14.2 on the substrate W) will generally be different between points 14.1 and 14.2 because these points are located at different radial distances from the centre 10 of the substrate W. However, the value of the performance parameter P will be the same or at least similar for points 14.1 and 14.2 because these points are located at the same distance in the y direction from points 15.1, 15.2 at the corners of fields 11.1, 11.2, respectively. Taking these considerations into account in the control of processes in a lithographic apparatus (e.g. in the control of movement of the substrate relative to the projection system of the lithographic apparatus) can help to improve the accuracy of the lithographic apparatus.

[0093] The spatial distribution of a performance parameter across a semiconductor substrate or across a portion thereof (i.e., the variation in values ​​of the performance parameter across a semiconductor substrate or across a portion thereof) is called a "fingerprint." A fingerprint may also change over time, for example due to process changes over time. A fingerprint model is a mathematical model that models the fingerprint across a substrate W or across a portion of a substrate W. The fingerprint model includes several model parameters (e.g., parameters a, b, c, f, and g in the example above). The same fingerprint model may be used for a group of substrates W (e.g., substrates W from one or more lots, or all substrates W processed in the same lithographic apparatus). However, the model parameters may differ for each individual substrate W.

[0094] 6A and 6B illustrate this. Figure 6A shows a graph representing the value p1 of a performance parameter P depending on the radial distance R from the center 10 of the substrate. Line p1-1 represents the variation of the value of P for a first substrate, and line p1-2 represents the variation of the value of P for a second substrate. The shapes of lines p1-1, p1-2 can be represented by the same type of polynomial ("fingerprint model"), the parameters of which differ between lines p1-1 and p1-2. For example, the polynomial representing both lines p1-1 and p1-2 can be represented by aR 2 +bR+c, the values ​​of the model parameters a, b, and c are different between line p1-1 and line p1-2.

[0095] 6B shows a graph representing the value p2 of a performance parameter P depending on the distance Y in the y direction from the corners 15.1, 15.2 of the fields 11.1, 11.2. The line p2-1 represents the variation of the value of P for the first substrate, and the line p2-2 represents the variation of the value of P for the second substrate. The shapes of the lines p2-1, p2-2 can be represented by the same type of polynomial ("fingerprint model"), but the parameters of this polynomial are different between the lines p2-1 and p2-2. For example, if the polynomial representing both lines p1-1 and p1-2 can be expressed as fY+g, the values ​​of the model parameters f and g will be different between the lines p2-1 and p2-2.

[0096]

[0095] A fingerprint model can be obtained, for example, by first obtaining a fingerprint and then converting the obtained fingerprint into a mathematical model that represents the obtained fingerprint. A fingerprint can be obtained, for example, by measuring a performance parameter on a semiconductor substrate or a group of semiconductor substrates at multiple measurement locations. Generally, to obtain a fingerprint, measurements must be made at multiple measurement locations. Alternatively or additionally, a fingerprint can be obtained by simulating a performance parameter in a computer model and obtaining a spatial distribution of the values ​​of that performance parameter across the entire semiconductor substrate or across a portion thereof.

[0097] From the acquired fingerprints, a fingerprint model can be obtained, for example, by determining a pattern in the spatial distribution of the measured performance parameters, for example, by curve fitting. For example, WO 2013 / 092106 describes in paragraphs

[0108] -

[0109] how a correction model can be applied to measured data to calculate model parameters. Thus, a fingerprint model is a mathematical model determined based on domain knowledge. Measurements are then performed at the sampling locations. A fingerprint model including model parameters is fitted to the measurements. The model parameters are then determined by fitting. The fitted fingerprint model (i.e., a fingerprint model in which the model parameters are fitted to the measurement data) can predict performance parameter values ​​at every location on the substrate W, typically at locations where no measurements have been performed. In this way, the fitted fingerprint model makes it possible to "model" or "densify" measurements across the entire substrate W while keeping the number of measurements fairly low. This allows for a good balance between throughput and accuracy to be achieved.

[0098] The control architecture of a lithographic apparatus is configured to control performance (e.g., overlay, CD, focus) at both the intra-field and inter-field levels. Recent developments (WO2016146217A1, WO2018121921A1) have enabled control at spatial scales smaller than the field. The control architecture of modern scanners also allows control at the subfield level. The definition of a subfield in terms of size and / or shape can generally be selected to correspond to the spatial variation of one or more performance parameters depending on the layout of the fields, dies, and cells across the substrate. A subfield-based control architecture provides the lithographic apparatus with increased control flexibility. For example, a particular layout (distribution) of dies / cells across a field may result in a corresponding stress distribution within the field, which causes an in-plane wafer deformation of the field, which induces a corresponding fingerprint of the performance parameter (error) fingerprint. If the stress distribution is periodic, the fingerprint may show a repeating (periodic) pattern across the field. This type of fingerprint can be efficiently corrected by the lithographic apparatus if a subfield-based control interface is available, where the subfields are defined to cover a single or discrete number of periods contained within the repeating pattern, such that the fingerprint corrections for each subfield are generally substantially identical (e.g., each correction is intended to correct the same periodic performance parameter fingerprint).

[0099]

[0098] When fingerprints at different spatial scales (e.g., the entire substrate and fields, fields and subfields, etc.) are identified, it is beneficial to correct these fingerprints by controlling the lithographic apparatus based on the identified fingerprints.

[0100]

[0099] The object of the present invention is to obtain a sampling scheme that allows accurate modeling of two or more fingerprints at different spatial scales while preventing the number of measurements required to identify the model parameters from becoming too large.

[0101] [000100] Model parameters of a fingerprint model can be determined based on (normalized) model uncertainty. For example, International Publication Nos. 2015110191A1 and 2016162231A1 describe this method. When noise is present in the data, the data can be modeled differently depending on the nature of the noise, the measurement (e.g., sampling) scheme used, etc. Thus, model uncertainty provides a measure of the noise sensitivity of a mathematical model when its parameters are estimated for a given metrology scheme using noisy measurements. Therefore, model uncertainty can be interpreted as a noise amplification / suppression factor from the noise present in the measurement to the variation of the model prediction. Model uncertainty is a function of the mathematical model used, the location of the measurement point, where the model is evaluated (interpolation / extrapolation), and the number of substrates measured. Normalized model uncertainty (NMU) is a unitless version of model uncertainty and does not change as a function of noise level. An NMU<1 implies noise suppression, while an NMU>1 implies noise amplification. Thus, the normalized model uncertainty represents the amount of variation in the modeled value scaled with the amount of noise in the measurement. A low NMU (<1) helps ensure that the combination of the sampling scheme and mathematical model results in a consistent fit, i.e., a fit that is robust to noise (although it may not guarantee that the model accurately represents the actual measurement). In some embodiments, for good noise suppression, the maximum NMU should be less than 0.6, 0.5, 0.4, or 0.3. The product of the in-substrate noise (e.g., the 3-sigma value) and the NMU is the output noise (e.g., the residual noise) based on the in-substrate noise. It is the theoretical output noise and is an indicator of the impact of the selected model and the sample scheme used.Thus, in one embodiment, the sampling scheme optimizer may optimize to reduce or minimize the NMU, and then, taking into account the measurement data, the remaining uncertainty for further use in evaluating one or more mathematical models and sampling schemes may be determined by multiplying the applicable NMU value by the in-substrate noise of the measurement data (e.g., a 3 sigma value).

[0102] [000101] A sampling scheme for determining sampling point locations for obtaining performance parameter measurements to determine model parameters of two or more fingerprint models may be determined based on the (normalized) model uncertainties of the fingerprint models. The sampling scheme may be determined to perform co-optimization to simultaneously reduce or minimize the NMU of the two or more fingerprint models. The sampling scheme thus determined provides sampling point locations useful for determining model parameters of two or more fingerprints with minimized or reduced model uncertainty.

[0103] In an embodiment of the present invention, the sampling scheme is determined by obtaining a first fingerprint model for a first spatial distribution of a performance parameter across a first portion of a semiconductor substrate and a second fingerprint model for a second spatial distribution of said performance parameter across a second portion of the semiconductor substrate, and determining sampling points corresponding to measurement locations on the semiconductor substrate for generating measurement data based on an expected decrease in a first uncertainty metric associated with evaluation of the first fingerprint model across the first portion and an expected decrease in a second uncertainty metric associated with evaluation of the second fingerprint model across the second portion. The number of fingerprint models may be three or more, for example, three, four, or more.

[0104] [000103] In an embodiment of the present invention, the sampling scheme is further determined by obtaining a third fingerprint model for a third spatial distribution of the performance parameter across a third portion of the semiconductor substrate; Determining sampling points corresponding to measurement locations on the semiconductor substrate for generating the measurement data is further based on an expected decrease in a third uncertainty metric associated with evaluating the third fingerprint model over the third portion.

[0105] [000104] In an embodiment of the invention, the sampling scheme is further determined by obtaining a fourth fingerprint model for a fourth spatial distribution of the performance parameter across a fourth portion of the semiconductor substrate; Determining sampling points corresponding to measurement locations on the semiconductor substrate for generating the measurement data is further based on an expected reduction in a fourth uncertainty metric associated with evaluating the fourth fingerprint model over the third portion.

[0106] [000105] A single sampling point determined according to this embodiment can be used to derive model parameters for both the first fingerprint model and the second fingerprint model, while reducing the number of measurements that need to be made to obtain the model parameters.

[0107] [000106] Not all possible measurement locations on the substrate W will provide useful information for both fingerprint models. For example, if the fingerprint model is a periodic function, locations where the performance parameter is zero will not provide information about the peak height of the periodic function. To obtain this information, further measurements at other locations are required. According to an embodiment, only locations that provide useful information for both the first and second fingerprint models qualify as sampling points.

[0108] [000107] These measurement locations that qualify as sampling points can be determined in a variety of ways.

[0109] [000108] For example, first, the first and second fingerprint models are determined based on measurements performed at a large number of locations on the sample substrate W. Thus, a large number of sampling points are included in the measurements. Model parameters of the first and second fingerprints of the sample substrate are determined based on this large number of measurements. Then, for example by a numerical approach or other computer-implemented approach, the model parameters of the first and second fingerprint models are determined again, but this time based on measurements performed at a smaller number of sampling points. Then, it is checked whether the deviation between the first run (using all sampling points) and the second run (using a smaller number of sampling points) is acceptable. If not, too many sampling points or too many important sampling points have been omitted in the second run. If the deviation is acceptable, perhaps more sampling points can be omitted. In this way, the sampling scheme can be optimized to allow all, or at least the majority, of the sampling points to determine the model parameters of the first and second fingerprint models.

[0110] [000109] Another optimization method that can be used in addition to or as an alternative to the method described in the previous paragraph is to check the effect of omitting or changing a measurement at a point and determine what effect it has on the model parameters that were originally determined based on the first run (thus using a large number of sampling points). If there is a significant effect on both the model parameters of the first fingerprint model and the model parameters of the second fingerprint model, then this is a sampling point that qualifies as a sampling point for implementing an embodiment of the present invention.

[0111] [000110] Other additional or alternative methods are possible.

[0112] [000111] Optionally, the first fingerprint model is a global fingerprint model, and the second fingerprint model is a local fingerprint model.

[0113] [000112] For example, the first portion of the substrate to which the first fingerprint model relates is the top surface of the substrate W (i.e. the portion of the top surface of the substrate W to be irradiated or which has been irradiated that does not include any focus out-of-focus areas, if any), and the second portion of the substrate W to which the second fingerprint model relates is, for example, a field.

[0114] [000113] Alternatively, for example, the first portion of the substrate to which the first fingerprint model relates is the top surface of the substrate W (i.e. the portion of the top surface of the substrate W to be irradiated or which has been irradiated that does not include any focus out-of-focus areas, if any), and the second portion of the substrate W to which the second fingerprint model relates is, for example, a die.

[0115] [000114] Alternatively, for example, the first portion of the substrate to which the first fingerprint model relates is the top surface of the substrate W (i.e. the portion of the top surface of the substrate W to be irradiated or which has been irradiated that does not include any focus out-of-focus areas, if any), and the second portion of the substrate W to which the second fingerprint model relates is, for example, a cell.

[0116] [000115] Alternatively, for example, the first fingerprint model or the second fingerprint model is a lithographic apparatus correction capability model.

[0117] [000116] The lithographic apparatus correction capability model may be a model that represents the correction capabilities that a lithographic apparatus can apply to a substrate. For example, intra-die fingerprints may be more difficult to correct compared to slowly changing whole-wafer fingerprints. By using such a lithographic apparatus correction capability model as the first or second fingerprint model, the sampling points determined in accordance with the present invention can identify only fingerprint components that can be corrected by the lithographic apparatus. This allows the selection of sampling locations to be more effective, as the sampling locations chosen in this way identify only fingerprint components that can be corrected by the lithographic apparatus and do not identify other fingerprint components that cannot be corrected by the lithographic apparatus.

[0118] [000117] Alternatively, for example, the first fingerprint model and / or the second fingerprint model may be a physical model, such as a stress profile model or other process model.

[0119] [000118] For example, the stress profile model may be a model representing the stress distribution across the entire substrate or across a portion of the substrate (e.g., a field, a subfield, a die, a cell, or multiple fields, subfields, dies, or cells). When a particular semiconductor manufacturing process (e.g., coating, baking, CMP, developing, or etching) introduces a stress distribution across a portion or the entire substrate, the corresponding fingerprint caused by the stress is effectively identified by measurements made at sampling locations determined by the present invention, where at least one of the first fingerprint model and / or the second fingerprint model is a stress profile model. Similarly, other types of fingerprint components induced by semiconductor manufacturing processes can be identified by the present invention, where the first fingerprint model and / or the second fingerprint model is a process model corresponding to a semiconductor manufacturing process model.

[0120] [000119] Alternatively, for example, the first fingerprint model and / or the second fingerprint model are related to a process fingerprint.

[0121] [000120] For example, the first fingerprint model and / or the second fingerprint model may be associated with a process fingerprint for a portion or the entire substrate induced by a semiconductor manufacturing process. The process fingerprint is effectively identified by measurements made at sampling locations determined by the present invention to which the first fingerprint model and / or the second fingerprint model are associated with the process fingerprint. The process fingerprint may be represented, for example, by a non-zero offset between after development inspection (ADI) and after etch inspection (AEI) overlay errors measured on one or more substrates of the same / different wafers or lots.

[0122] [000121] Alternatively, for example, the first portion of the substrate to which the first fingerprint model relates is a field of the substrate W, and the second portion of the substrate W to which the second fingerprint model relates is, for example, a die.

[0123] [000122] Alternatively, for example, the first portion of the substrate to which the first fingerprint model relates is a field of the substrate W, and the second portion of the substrate W to which the second fingerprint model relates is, for example, a cell.

[0124] [000123] Alternatively, for example, the first portion of the substrate to which the first fingerprint model relates is a die of the substrate W, and the second portion of the substrate W to which the second fingerprint model relates is, for example, a cell.

[0125] [000124] Alternatively, for example, the first portion of the substrate to which the first fingerprint model relates is a field of the substrate W, and the second portion of the substrate W to which the second fingerprint model relates is, for example, a subfield.

[0126] [000125] Alternatively, for example, the first fingerprint model is configured to represent the behavior of the performance parameter across an interior portion of the substrate, and the second fingerprint model is configured to represent the behavior of the performance parameter at the edge of the substrate.

[0127] [000126] Optionally, the first portion and the second portion at least partially overlap each other.

[0128] [000127] Optionally, the first portion and / or the second portion are at least a part of an irradiated layer in or on top of the semiconductor substrate.

[0129] [000128] Optionally, the second portion is one of a field, multiple fields, a subfield, multiple subfields, a die, multiple dies, a portion of a die, multiple portions of multiple dies, a cell, multiple cells, a portion of a cell, or multiple portions of multiple cells.

[0130] [000129] Optionally, the first fingerprint model is a global model, and the second portion is one of a field, multiple fields, a subfield, multiple subfields, a die, multiple dies, a portion of a die, multiple portions of multiple dies, a cell, multiple cells, a portion of a cell, or multiple portions of multiple cells. Even more optionally, the first fingerprint model is a global model, and the second portion is one of a field, multiple fields, a subfield, multiple subfields, a die, multiple dies, a portion of a die, multiple portions of multiple dies, a cell, multiple cells, a portion of a cell, or multiple portions of multiple cells.

[0131] [000130] Optionally, the first portion is a top surface of the substrate W (i.e. a portion of the top surface of the substrate W to be irradiated or which does not include any out-of-focus areas, if any exist), and the second portion is one of a field, multiple fields, a subfield, multiple subfields, a die, multiple dies, part of a die, multiple parts of multiple dies, a cell, multiple cells, part of a cell, or multiple parts of multiple cells.

[0132] [000131] Optionally, the first portion is a plurality of fields, and the second portion is one of a field, a subfield, a plurality of subfields, a die, a plurality of dies, a portion of a die, a plurality of portions of a plurality of dies, a cell, a plurality of cells, a portion of a cell, or a plurality of portions of a plurality of cells.

[0133] [000132] Optionally, the first portion is a field, and the second portion is one of a subfield, multiple subfields, a die, multiple dies, a portion of a die, multiple portions of multiple dies, a cell, multiple cells, a portion of a cell, or multiple portions of multiple cells.

[0134] [000133] Optionally, the first portion is a plurality of subfields, and the second portion is one of a subfield, a die, a plurality of dies, a portion of a die, a plurality of portions of a plurality of dies, a cell, a plurality of cells, a portion of a cell, or a plurality of portions of a plurality of cells.

[0135] [000134] Optionally, the first portion is a subfield, and the second portion is one of a die, multiple dies, a portion of a die, multiple portions of multiple dies, a cell, multiple cells, a portion of a cell, or multiple portions of multiple cells.

[0136] [000135] Optionally, the first portion is a plurality of dies, and the second portion is one of a die, a portion of a die, portions of dies, a cell, a plurality of cells, a portion of a cell, or portions of cells.

[0137] [000136] Optionally, the first portion is a die, and the second portion is one of: a portion of a die, portions of multiple dies, a cell, multiple cells, a portion of a cell, or portions of multiple cells.

[0138] [000137] Optionally, the first portion is portions of a plurality of dies, and the second portion is one of a cell, a plurality of cells, a portion of a cell, or portions of a plurality of cells.

[0139] [000138] Optionally, the first portion is a plurality of cells, and the second portion is one of a cell, a portion of a cell, or portions of a plurality of cells.

[0140] [000139] Optionally, the first portion is one cell, and the second portion is a portion of the one cell or one of portions of multiple cells.

[0141] [000140] Optionally, the first portion is a portion of a plurality of cells, and the second portion is a portion of one cell.

[0142] [000141] The first and second portions have a larger area than the sampling point has.

[0143] [000142] Optionally, the first fingerprint model is a global model, the second fingerprint model is a substrate edge model, the third portion is a field, and the fourth portion is a subfield.

[0144] [000143] The performance parameters are, for example, overlay error, critical dimension (eg, line width), alignment parameters (alignment mark position), sidewall angle, edge roughness, edge placement error, and / or focus (error).

[0145] [000144] Optionally, the first fingerprint model and / or the second fingerprint model further relate to the spatial distribution of at least one further parameter.

[0146] [000145] For example, the at least one further parameter is the prediction uncertainty of the model, such as the normalized model uncertainty, G-optimality, least squared error, and / or moving standard deviation.

[0147] [000146] In addition to providing a method for establishing a sampling scheme, one aspect of the present invention relates to establishing or determining an appropriate model for modeling a performance parameter. In general, a performance parameter fingerprint, or more specifically, a performance parameter fingerprint model, can be defined by a combination of basis shapes or functions. Such basis shapes or functions may include, for example, polynomials, splines, or other mathematical functions. In general, modeling a phenomenon measured along a trajectory or over an area may involve determining a best fit of a linear or nonlinear combination (e.g., a weighted combination) of basis shapes of functions along that trajectory or over that area. As will be appreciated, the more basis shapes considered for fitting, the more measurement data must be available. For example, if one wishes to fit a fifth-order polynomial using a set of measurements representing a performance parameter P measured along the X direction, there are six variables, namely, p0 through p5, that need to be determined. P=p0+p1.x+p2.x 2 +p3.x 3 +p4.x 4 +p5.x 5 (1)

[0148] In a similar manner, the performance parameters can be modeled using a set of spline functions Ci. P=p1.C1+p2.C2+p2.C3(2)

[0149] Therefore, when a large number of basis shapes or functions are involved in the fitting process, it may require at least an equal number of measurements to determine the variables or parameters of the applied model. It should be noted that the variables or parameters applied to achieve the fitting of the fingerprint model to the measured data may also be called weighting coefficients. Thus, fitting a model or fingerprint model that includes a combination of basis shapes or functions corresponds to determining a weighted combination of the basis shapes or functions that corresponds as well as possible to the measured data set.

[0150] In this case, the weighted combination of basis shapes can be referred to as a fingerprint model of the measurement data, eg, a fingerprint model or fingerprint of a particular performance parameter such as overlay error, critical dimension, sidewall angle, etc.

[0151] With regard to modeling or building a model, it can be pointed out that using a large number of basis functions provides great fitting freedom, i.e., it allows the model to be adjusted to a wide variety of shapes, but has the disadvantage that it creates some complexity in building the model, since it requires the determination of a large number of variables, parameters, or weighting coefficients.

[0152] It will be appreciated that if the fingerprint model represents performance parameters of a subfield of a substrate, it may require multiple measurements to determine the fingerprint model for each subfield of the substrate. Requiring multiple measurements to determine the model may also increase noise sensitivity.

[0153] In an embodiment of the present invention, a technique is proposed to reduce the number of measurements required to obtain a fingerprint model for several fields or subfields of a substrate. The proposed technique is based on the insight that although the performance parameter P across the subfields of a substrate may vary from subfield to subfield, the overall shape of the performance parameter in different subfields may show strong similarities. This led the inventors to devise that it may be sufficient to at least partially retain a previously determined fingerprint model, for example, the fingerprint model of a particular subfield, in order to derive a fingerprint model for another subfield.

[0154] Thus, in such embodiments, rather than fitting an entire new model, i.e., a model in which all variables or weighting coefficients have not yet been determined, certain aspects of a previously determined model, called an initial fingerprint model, can be maintained or preserved, thereby reducing the number of variables that need to be determined for the new model.

[0155] This can be shown as follows:

[0156] Referring to the fifth order polynomial fitting described above, assume that for a particular subfield S0 of the substrate, a fit has already been performed to obtain a model or fingerprint for that subfield that can be expressed as follows: P(S0)=a0+a1.x+a2.x 2 +a3.x 3 +a4.x 4 +a5.x 5 (3)

[0157] a0, a1, a2, a3, a4, and a5 are fixed parameters that result in the best fit of the performance parameter P measured over subfield S0. Subfield S0 may be referred to as the reference subfield used to determine the fitting parameters of the model in equation (1), resulting in fingerprint model P(S0), which may be referred to as the initial fingerprint model or reference fingerprint model for modeling the performance parameter P over the subfields of the substrate, for example.

[0158] According to an embodiment of the invention, the fingerprint model of the performance parameter P for another sub-field, for example sub-field S1, may now be determined as follows. P(S1)=p S .(a0+a1.x+a2.x 2 +a3.x 3 +a4.x 4 +a5.x 5 ) (4)

[0159] p S is the model (a0+a1.x+a2.x) for the measurement of the performance parameter P of the subfield S1 2 +a3.x 3 +a4.x 4 +a5.x 5 ) are the variables or parameters that are determined to obtain the best fit of p S is also referred to herein as the shape parameter.

[0160] In the presented example, the shape of the model used to model the performance parameter P of subfield S1 is fixed, and only the amplitude of the model is determined by the variable p, which can be determined by fitting the model of equation (4) to the measured data of the performance parameter P of subfield S1. S Expressed more mathematically, it can be observed that by fixing the parameters a0 to a5 of the initial fingerprint model, the ratios between the different basis functions of the model are fixed, or the relative weights of the different basis functions used are fixed or kept constant. This fixes or maintains the shape of the fingerprint model.

[0161] As will be appreciated by those skilled in the art, the fingerprint model P(S1) for subfield S1, specifically the shape of the model, is substantially fixed, apart from the amplitude, so that it is not necessary to have a large measurement data set for fitting the model. As can be seen from the example provided, to completely determine the fingerprint model for subfield S1, only one variable p S In principle, the shape parameters or variables p S To determine , it is sufficient to have one measurement of the performance parameter taken for subfield S1.

[0162] Therefore, in the example presented, the shape of the applied model is substantially fixed so that only the amplitude of the fingerprint model, representing the performance parameter P, can be varied or modeled for a particular subfield.

[0163] Such a fingerprint model developed by the above process may be advantageously applied when a measured performance parameter, e.g., a measured overlay error, is observed to have substantially the same shape across different sub-fields.

[0164] If it is observed that the actual shape of the fingerprint or model representing the performance parameters across subfields does not remain substantially constant across the substrate, it may be necessary to introduce additional flexibility or variability into the applied fingerprint model. In such cases, rather than completely fixing or freezing the shape of the applied model, the shape may also be allowed to vary, albeit more restrictedly, compared to using a general model according to equation (1).

[0165] Examples of such models, where the shape of the model is not completely fixed but can vary to some extent, are as follows: P(S1)=p S1 .(a0+a1.x+a2.x 2 )+p S2 .(a3.x 3 +a4.x 4 +a5.x 5 ) P(S1)=p S1 .(a0+a1.x+p S2 .a2.x 2 +a3.x 3 +a4.x 4 +a5.x 5 ) P(S1)=p S1 .(a0+a1.x+a2.x 2 +a3.x 3 +a4.x 4 )+p S2 .x 5 (5)

[0166] It can be observed that in the top equation of equation (5), the relative weights of the 3rd, 4th, and 5th order basis functions, as well as the relative weights of the 0th, 1st, and 2nd order basis functions, are fixed or held constant. This allows the shape parameters or variables p Si The number of functions is reduced from 6 to 2. In general, it can be said that by fixing the relative weights of two of the applied basis functions, or by keeping the ratio of those two basis functions constant, one variable or parameter that needs to be determined can be eliminated.

[0167] In a similar manner, a model or fingerprint based on a spline function can be determined for a particular subfield, and this model can be used to model other subfields, where the shape of the model is somewhat maintained or fixed.

[0168] In such an embodiment, a fingerprint or model for a particular subfield S0 of the substrate can be determined by taking multiple measurements across that subfield, resulting in a model or fingerprint for said subfield S0 that can be expressed as follows: P(S0)=a1.C1+a2.C2+a2.C3(6)

[0169] Thereby, a1, a2 and a3 are now fixed parameters that result in the best fit of the performance parameter P measured over the subfield S0.

[0170] According to an embodiment of the invention, a model of the performance parameter P for subfield S1 may now be determined as follows: P(S1)=p S .(a1.C1+a2.C2+a2.C3) (7)

[0171] p Sare the variables determined to obtain the best fit of the model (a1.C1+a2.C2+a2.C3) to the measurements of the performance parameter P of the subfield S1.

[0172] In the example given by Equation 7, the shape of the model is substantially fixed, so that when modeling another subfield, for example subfield S1, only the amplitude of the applied model is considered as a variable.

[0173] Similar to the approach described for polynomial models, spline-based models can also be adapted to account for specific variations in the model across the substrate. Thus, if it is observed that the actual shape of the fingerprint or model representing the performance parameters across subfields does not remain substantially constant across the substrate, additional flexibility or variability can be introduced into the applied model. In such cases, rather than completely fixing or freezing the shape of the applied model, it can be allowed to vary, albeit more limitedly, compared to using a general model according to equation (1).

[0174] An example of such a spline-based model, where the shape of the model is not completely fixed but can vary to some extent, is as follows: P(S1)=p S1 .(a1.C1+p S2 .a2.C2+a2.C3) P(S1)=p S1 .(a1.C1+a2.C2)+p S2 .C3) (8)

[0175] Thus, for both polynomial models and spline-based models, or other models defined as linear or nonlinear combinations of basis functions, the process of modeling performance parameters for multiple subfields on a substrate can begin by determining a model or fingerprint for a particular subfield or set of subfields. In some embodiments of the invention, a set of reference subfields is used, which can be measured across multiple locations on the substrate and / or across multiple substrates. Measurements across multiple subfields can be averaged. In some embodiments, these reference subfields are selected to provide a good representation of the expected shape. Because this initial fingerprint only needs to be determined once and is heavily used to model other subfields, it can be worthwhile to perform detailed measurements of fingerprints or performance parameters across subfields. This initial fingerprint or reference fingerprint can be determined during an initial setup phase, for example, using available measurement data, such as training and validation data.

[0176] Once an initial or reference fingerprint or fingerprint model is determined, it can be used to determine a fingerprint model for another subfield, which requires only a limited set of measurements, taking into account the constraint that at least part of the shape of the initial fingerprint model is maintained or fixed in the fingerprint model for the other subfield.

[0177] This process is illustrated in Figure 7. Figure 7 shows schematically a fingerprint 500 representing a performance parameter P across sub-fields, for example along the X direction, determined based on measurements made on the sub-fields.

[0178] Therefore, fingerprint 500 can be considered a reference fingerprint, and its shape is considered to correspond to the shapes of fingerprints of other subfields. Thus, assuming that the fingerprints of other subfields of the substrate have substantially the same shape, the fingerprints of other subfields may correspond, for example, to graph 510 or graph 520 (graphs 510 and 520 differ from the reference fingerprint only in amplitude). As will be appreciated by those skilled in the art, to determine the actual fingerprint for a subfield (which predetermines the fingerprint shape), only one measurement needs to be made. Referring to FIG. 7 , it is sufficient to know the actual amplitude of the fingerprint model at a particular location, e.g., amplitude P1 at x=x1, and scale fingerprint 500 so that fingerprint 500 includes measurement value P1 at x=x1.

[0179] Because only a limited set of measurements is needed to determine the actual fingerprint of a subfield, it is important to ensure that the selected measurements provide sufficient information, or preferably the most relevant information, to determine the actual fingerprint of the subfield. As an example, it may be preferable to sample or measure a subfield performance parameter at a location where a relatively high amplitude of the performance parameter is expected. Taking measurements at such a location may correspond, for example, to a location where the signal-to-noise ratio is low. In contrast, if measurements are taken at a location where a low amplitude of the performance parameter is expected, the precision with which the fingerprint can be determined may be low. With respect to the example presented in FIG. 7, it is clear that taking measurements near x=x2 or x=x3 does not provide measurements that provide information about the amplitude of the fingerprint sought.

[0180] It may be desirable to perform more measurements than exactly needed to mitigate the effects of noise or to include some form of redundancy in the measurements. Performing additional measurements also provides the possibility of checking whether the predetermined shape of fingerprint 500 is valid or still valid. Thus, redundant measurements provide the possibility of detecting whether the predetermined shape of the fingerprint is still valid or should be adjusted. With reference to FIG. 7 , it can be observed that measurements made at x=x4 and x=x5 for fingerprint model 500 provide the same performance parameter P2. Therefore, if measurements are made on another subfield at locations x=x4 and x=x5, one would expect to obtain the same performance parameter measurement. If such measurements yield different values, this may be considered an indication that the shape of the other subfield does not correspond to the shape of the fingerprint model used as a reference. Therefore, this may be an indication that the shape may need to be reconfigured or adjusted.

[0181] The examples presented above illustrate how a fingerprint model can be predetermined for a particular subfield and further used to determine fingerprint models for other subfields, thereby maintaining at least part of the shape of the initial fingerprint model.

[0182] [000147] Those skilled in the art will appreciate that the same approach can also be applied to modeling more global phenomena, i.e., phenomena that extend across the entire surface of the substrate. In such embodiments, for example, a global fingerprint model characterizing a particular performance parameter can be determined based on an extensive set of measurement data. Such a global fingerprint model can then be used to model performance parameters of additional substrates, whereby at least part of the shape of the initial global fingerprint model is maintained or fixed, similar to the manner described above.

[0183] [000148] Similar to the approach described above, the global fingerprint model can utilize a set of basis shapes or functions, such as polynomials or spline functions. With respect to the use of spline functions, it can be said that models based on spline functions offer great fitting freedom, allowing a flexible way of representing a particular shape when the particular analytical shape is unknown.

[0184] [000149] It has been mentioned above that a particular fingerprint model can be developed or determined for a particular subfield and further used to model other subfields, thereby maintaining the shape of the model of the particular fingerprint model to some extent. For example, it can be mentioned that the shape of the fingerprint model used for the performance parameter P(S1) is assumed to be fixed, i.e., a0, a1, a2, a3, a4, and a5 are fixed parameters obtained by best-fitting the performance parameter P measured over the subfield S0.

[0185] [000150] As mentioned above, the use of such a fixed model assumes that the shape of the local fingerprint model used to model the sub-field performance parameters remains substantially constant.

[0186] [000151] However, in some embodiments of the present invention, it is possible to take into account variations in the fingerprint model used across the substrate. In such embodiments, the variables or shape parameters P Siis not constant, but rather a function that can vary across the substrate. Rather than keeping these parameters constant, one can consider them to vary across the substrate, and an extensive set of measurement data can be used to fit a global model to the variations observed in these parameters. In other words, each of the parameters used to represent the subfield fingerprint model can be considered a function, e.g., a two-dimensional function, that can vary across the substrate. Such variations can be modeled in a manner substantially similar to that described for the subfield model, i.e., using multiple basis functions such as polynomials or splines.

[0187] [000152] In the example given above for determining a local or global fingerprint model, the following steps can be recognized:

[0188] [000153] In a first step, an initial fingerprint model for the spatial distribution of the performance parameter across a portion of the semiconductor substrate is defined as a parameterized combination of multiple basis functions. Examples of such models can be found, for example, in equations (1) and (2).

[0189] [000154] In a second step, the parameters of the initial fingerprint model are determined using a set of measurements representing performance parameters across an example of the portion of the semiconductor substrate.

[0190] [000155] In a third step, a fingerprint model for modeling the performance parameters over the portion is determined based on the initial fingerprint model, whereby a ratio of at least two basis functions used in the initial fingerprint model is kept constant.

[0191] [000156] This third step may also be referred to as a step in which a fingerprint model for modeling performance parameters over the portion is determined based on the initial fingerprint model, whereby at least part of the shape of the initial fingerprint model is maintained in the fingerprint model.

[0192] [000157] In some embodiments of the present invention, Legendre polynomials are used to model the fingerprints of performance parameters for modeling local fingerprint models, global fingerprint models, or both.

[0193] [000158] As already mentioned above, currently, performance parameters such as overlay are determined and modeled for the entire substrate, resulting in what is called a global fingerprint model, which is subsequently used in so-called advanced process control (APC). Such models used for modeling performance parameters across the entire substrate may, for example, be polynomial models, i.e., models in which the basis functions used are polynomials (e.g., low-order polynomials). Examples of advanced process control (APC) are described, for example, in U.S. Patent Application Publication No. 2012008127A1.

[0194] [000159] A local fingerprint model, as described above, can be developed to apply corrections or controls at the subfield level. According to certain aspects of the present invention, an optimal sampling scheme can be determined that can determine both a global fingerprint model, e.g., used for APC, and a local fingerprint model (i.e., at the die or subfield level), without the need for a high-density measurement or sampling scheme. As described above, fingerprint models, e.g., subfield fingerprint models, used to model local variations in the performance parameter P can also be determined using polynomial basis functions. The inventors have observed that crosstalk can occur when both the global fingerprint model and the local fingerprint model utilize polynomial functions or basis functions. If both models are determined using Cartesian polynomials or basis functions, the local fingerprint model may be indistinguishable from the global fingerprint model, e.g., used for APC. As a result, changes made to the global fingerprint model may affect the local fingerprint model, and vice versa, requiring revision or remeasurement of the local fingerprint model parameters.

[0195] [000160] To avoid, or at least mitigate, such crosstalk, an embodiment of the present invention proposes to model the local fingerprint model using different basis functions than the global fingerprint model.

[0196] [000161] Such embodiments of the invention include, for example: defining a first fingerprint model for a first spatial distribution of a performance parameter across a first portion of the substrate as a parameterized combination of a first set of basis functions; defining a second fingerprint model for a second spatial distribution of the performance parameter across a second portion of the substrate as a parameterized combination of a second set of basis functions; Including, It may be described as a method of obtaining a fingerprint model in which one or more of the first set of basis functions are substantially orthogonal to one or more of the second set of basis functions.

[0197] [000162] Specifically, it has been derived that the use of Legendre polynomials for modeling local fingerprints can enable a reduction in crosstalk occurring between global and local fingerprints. Specifically, it can be shown that in the case of an infinite number of uniformly distributed measurements, crosstalk is reduced to zero. The application of a finite number, e.g., a relatively small number of measurements, can still result in a significant reduction in crosstalk. The application of Legendre polynomials requires defining position boundary conditions. In one embodiment, the position boundary conditions of the applied Legendre polynomials are set between -1 and 1. This makes it possible to define a set of Legendre polynomials in which the zeroth-order and first-order polynomials are the same as for Cartesian polynomials, as shown in the table below.

[0198] [Table 1]

[0199] This reduces crosstalk between the zeroth-order and first-order polynomials of the global fingerprint model used for APC and the higher-order parameters defined by Legendre polynomials. This allows for a separation of the components of the global fingerprint model and the local fingerprint model. Even when Cartesian polynomials are used with the same position boundary condition [-1,1], Legendre polynomials can be easily converted to Cartesian polynomials, and vice versa. As a result of the separation of the components of the used fingerprint models, modifications or corrections of the parameters of one model can be made without substantially affecting the other model. Specifically, the used fingerprint model can be decomposed into a model with lower-order parameters and a model with higher-order parameters. As an example, a fingerprint model used for APC control may include, for example, zeroth-order and first-order parameters, while a fingerprint model for a subfield may include higher-order (>1) parameters. Such a decomposition can be advantageously applied to determine a sampling strategy for determining, modifying, or correcting the applied model. When advanced process control is based on a fingerprint model with only low-order parameters, these low-order parameters can be updated using fewer measurements, while using a model with higher-order parameters may require more measurements for updating. Decomposition can also be used to consider certain characteristics of the model, such as stability. It has been found that the less crosstalk there is between model parameters, the easier it is to monitor the parameters individually. As such, it has been observed that subfield fingerprint models, and in particular their higher-order components, are relatively stable over time. For example, based on this, a sampling and measurement strategy can be developed in which higher-order parameters, such as those associated with higher-order Legendre polynomials, require denser measurements but lower frequency modeling, thereby reducing measurement costs.A lower order fingerprint model, for example, can be updated with fewer measurements but at a higher frequency.

[0200] [000163] Optionally, determining the locations of the sampling points comprises taking into account drift of the first and / or second spatial distributions of the performance parameter over time, such drift may occur due to warming up of the lithographic apparatus, for example after start-up and during exposure of several substrates W.

[0201] [000164] The method according to any one of the embodiments described above can of course be used to determine multiple sampling points.

[0202] [000165] According to a further embodiment, the above embodiment provides a method of establishing a sampling scheme, comprising: - determining a plurality of sampling points according to any one of the embodiments of the method for determining a sampling scheme described above; - establishing a sampling scheme including the determined sampling points; The method may be used in a method comprising:

[0203] [000166] In the sampling scheme established according to this embodiment, the sampling points are used effectively, and the model parameters of both the first fingerprint model and the second fingerprint model can still be obtained, but the number of sampling points is reduced.

[0204] [000167] For example, the number of sampling points in the sampling scheme established by the embodiments described above is at least 50% less than the number of sampling points that were required or used to establish the first fingerprint model and / or the second fingerprint model.

[0205] [000168] Optionally, the number of sampling points in the sampling scheme established by the embodiments described above is at least 75% less than the number of sampling points that were required or used to establish the first fingerprint model and / or the second fingerprint model.

[0206] [000169] The established sampling scheme may also include additional sampling points that are determined in different ways.

[0207] [000170] Optionally, when establishing the sampling scheme, the location of critical areas on the substrate W is taken into account, for example where a higher accuracy is required or where a smaller deviation is tolerated than elsewhere on the substrate W.

[0208] [000171] This can be achieved by one embodiment of the method for establishing a sampling scheme, wherein the method comprises: - identifying one or more critical areas on the semiconductor substrate W; - determining a higher number of sampling points per surface area for one or more critical areas of the semiconductor substrate W than for other areas of the semiconductor substrate W; Further includes:

[0209] [000172] Embodiments of the method for determining a sampling scheme, determining a plurality of sampling points and / or establishing a sampling scheme may be implemented in a method for generating metrology data from a semiconductor substrate, a method for controlling a lithographic apparatus, a computer program product, a semiconductor substrate measurement apparatus, a lithographic apparatus, a lithography system, and a device manufacturing method.

[0210] [000173] In one embodiment of a method for generating metrology data from a semiconductor substrate, a single sampling point, multiple sampling points, and / or a sampling scheme determined by one of the method embodiments described above is used.

[0211] [000174] For example, one embodiment of a method for generating measurement data from a semiconductor substrate W includes the following steps: - determining the locations of the sampling points according to any one of the embodiments of the method for determining a sampling scheme described above; generating measurement data by measuring performance parameters at sampling point locations on the semiconductor substrate W; Includes:

[0212] [000175] By way of further example, one embodiment of a method for generating measurement data from a semiconductor substrate W may include the following steps: - receiving information about the locations of the sampling points, the locations being determined according to one of the embodiments of the method for determining a sampling scheme described above; generating measurement data by measuring performance parameters at sampling point locations on the semiconductor substrate W; Includes:

[0213] [000176] Optionally, this embodiment further comprises determining model parameters of the first fingerprint model and model parameters of the second fingerprint model based on the generated measurement data.

[0214] [000177] In an embodiment of a method for controlling a lithographic apparatus, a sampling point, a plurality of sampling points and / or a sampling scheme determined by one of the method embodiments described above is used.

[0215] [000178] For example, such an embodiment may include the following steps: - generating and / or receiving measurement data from a semiconductor substrate according to an embodiment of the method for generating measurement data from a semiconductor substrate as described above; - determining model parameters of the first fingerprint model and model parameters of the second fingerprint model based on the generated measurement data; - placing a semiconductor substrate in a lithographic apparatus; - applying a pattern onto the semiconductor substrate in a lithographic apparatus; - controlling the applying of the pattern on the basis of control input data, the control input data being based at least in part on said model parameters of the first fingerprint model and / or model parameters of the second fingerprint model; Includes.

[0216] [000179] The lithographic apparatus is for example a lithographic apparatus according to FIG.

[0217] [000180] For example, controlling the application of a pattern onto a semiconductor substrate includes controlling movement of the semiconductor substrate relative to exposure light from a patterning system of a lithographic apparatus and / or controlling movement of a patterning device, i.e., reticle, relative to illumination light of an illumination system. Further examples of controlling the application of a pattern onto a semiconductor substrate include controlling the position and / or orientation of a lens in the patterning system, controlling the illumination settings of the illumination system, controlling the focus of the exposure light, and / or controlling the dose of the exposure light.

[0218] [000181] In this embodiment, the semiconductor substrate for which the measurement data is generated may be the same as the semiconductor substrate placed in the lithographic apparatus, or may be different.

[0219] [000182] According to one embodiment of the present invention, there is provided a computer program comprising one or more sequences of machine-readable instructions configured to perform an embodiment of any of the methods described above.

[0220] [000183] According to one embodiment of the present invention, - receiving sampling points generated according to any one of the embodiments of the method for determining a sampling scheme described above; - controlling the measurement device to generate measurement data from the semiconductor substrate at measurement locations corresponding to the sampling points; - determining model parameters of the first fingerprint model and model parameters of the second fingerprint model based at least in part on the generated measurement data; A computer program product is provided that includes one or more sequences of machine-readable instructions configured to:

[0221] [000184] According to one embodiment of the present invention, receiving measurement data from the semiconductor substrate at measurement locations corresponding to the sampling points determined according to any one of the embodiments of the method for determining a sampling scheme described above; - determining model parameters of the first fingerprint model and model parameters of the second fingerprint model based at least in part on the received measurement data; - controlling the dispensing of a pattern on a semiconductor substrate on the basis of control input data, the control input data being based at least in part on said model parameters of the first fingerprint model and / or model parameters of a second fingerprint model; A computer program product is provided that includes one or more sequences of machine-readable instructions configured to:

[0222] [000185] According to one embodiment of the present invention, there is provided a semiconductor substrate measurement apparatus comprising: an input terminal configured to receive sampling points determined according to any one of the embodiments of the method for determining a sampling scheme described above; - a controller configured to control the measurement device to generate measurement data from the semiconductor substrate at measurement locations corresponding to the sampling points; - a processor configured to determine model parameters of the first fingerprint model and model parameters of the second fingerprint model based at least in part on the generated measurement data; Includes:

[0223] [000186] According to an embodiment of the present invention, there is provided a lithographic apparatus comprising: - an input terminal configured to receive measurement data from the semiconductor substrate at measurement locations corresponding to the sampling points determined according to any one of the embodiments of the method for determining a sampling scheme described above; - a processor configured to determine model parameters of the first fingerprint model and model parameters of the second fingerprint model based at least in part on the received measurement data; - a controller configured to control the dispensing of a pattern onto a semiconductor substrate on the basis of control input data, the control input data being based at least in part on said model parameters of the first fingerprint model and / or model parameters of the second fingerprint model; and Includes:

[0224] [000187] According to one embodiment of the present invention, there is provided a lithography system, the system including a semiconductor substrate measurement apparatus according to any of the above-described embodiments, and a lithography apparatus according to any of the above-described embodiments.

[0225] [000188] According to an embodiment of the present invention, there is provided a device manufacturing method comprising transferring a pattern from a patterning device onto a substrate, the method comprising using a lithographic apparatus according to the embodiments described above.

[0226] [000189] Although specific reference may be made herein to the use of lithographic apparatus in the manufacture of ICs, it should be understood that the lithographic apparatus described herein may have other applications, including integrated optical systems, guidance and detection patterns for magnetic domain memories, flat panel displays, liquid crystal displays (LCDs), thin film magnetic heads, etc.

[0227] [000190] Although specific reference may be made herein to embodiments of the invention in the context of lithographic apparatus, embodiments of the invention may also be used in other apparatus. Embodiments of the invention may form part of a mask inspection apparatus, a metrology apparatus, or an apparatus for measuring or processing objects such as wafers (or other substrates) or masks (or other patterning devices). These apparatus may be generally referred to as lithography tools. Such lithography tools may use vacuum conditions or ambient (non-vacuum) conditions.

[0228] [000191] While specific reference may have been made above to the use of embodiments of the invention in the context of optical lithography, it will be appreciated that the invention is not limited to optical lithography and may, where circumstances permit, be used in other applications, for example imprint lithography.

[0229] [000192] Where circumstances permit, embodiments of the present invention may be implemented in hardware, firmware, software, or any combination thereof. Embodiments of the present invention may also be implemented as instructions stored on a machine-readable medium, which may be read and executed by one or more processors. A machine-readable medium may include any mechanism for storing or transmitting information in a form readable by a machine (e.g., a computing device). For example, machine-readable media may include read-only memory (ROM), random-access memory (RAM), magnetic storage media, optical storage media, flash memory devices, electrical, optical, acoustic, or other forms of propagated signals (e.g., carrier waves, infrared signals, digital signals, etc.). Furthermore, firmware, software, routines, or instructions may be described herein as performing certain operations. However, it should be understood that such description is merely for convenience, and that such operations actually result from a computing device, processor, controller, or other device executing the firmware, software, routines, instructions, etc., and in doing so may cause actuators or other devices to interact with the physical world.

[0230] [000193] While specific embodiments of the invention have been described above, it will be understood that the invention may be practiced otherwise than as described. The above description is intended to be illustrative, not limiting. Thus, it will be apparent to those skilled in the art that changes can be made to the invention as described without departing from the scope of the claims set forth below.

[0231] [000194] The steps of the above method may be automated in a lithographic apparatus control unit LACU as shown in Figure 1. This unit LACU may include a computer assembly as shown in Figure 3. The computer assembly may be a dedicated computer in the form of a control unit in an embodiment of the assembly according to the invention, or alternatively may be a central computer controlling the lithographic projection apparatus. The computer assembly may be arranged to load a computer program comprising computer-executable code. This may enable the computer assembly, when downloaded, to control the above-mentioned use of the lithographic apparatus with embodiments of the level and alignment sensors AS, LS.

[0232] [000195] Memory 1229 coupled to processor 1227 may include several memory components such as a hard disk 1261, a read-only memory (ROM) 1262, an electrically erasable programmable read-only memory (EEPROM) 1263, and a random access memory (RAM) 1264. Not all of the memory components described above need be present. Furthermore, it is not necessary for the memory components described above to be in close physical proximity to processor 1227 or to each other. They may be located remotely.

[0233] [000196] The processor 1227 may also be connected to some type of user interface, such as a keyboard 1265 or a mouse 1266. A touch screen, trackball, voice converter, or other interface known to those skilled in the art may also be used.

[0234] [000197] The processor 1227 may be connected to a reading unit 1267 arranged to read data, for example in the form of computer executable code, from and under some circumstances store data on a data carrier such as a floppy disk 1268 or a CDROM 1269. DVDs or other data carriers known to those skilled in the art may also be used.

[0235] [000198] The processor 1227 may also be connected to a printer 1270 for printing out output data on paper, and a display 1271, for example a monitor or LCD (liquid crystal display) or other type of display known to those skilled in the art.

[0236] [000199] Processor 1227 may be connected to a communications network 1272, such as a public switched telephone network (PSTN), a local area network (LAN), a wide area network (WAN), etc., by a transmitter / receiver 1273 responsible for input / output (I / O). Processor 1227 may be arranged to communicate with other communications systems via communications network 1272. In some embodiments of the invention, an external computer (not shown), such as an operator's personal computer, may log into processor 1227 via communications network 1272.

[0237] [000200] The processor 1227 may be implemented as an independent system or as several processing units operating in parallel, each arranged to perform a subtask of a larger program. The processing unit may also be divided into one or more main processing units having several sub-processing units. Some processing units of the processor 1227 may even be located remotely from other processing units and communicate via a communications network 1272. The connections between modules may be wired or wireless.

[0238] [000201] The computer system may be any signal processing system using analog and / or digital and / or software technology arranged to perform the functions described herein.

[0239] Although specific reference may be made herein to the use of lithography apparatus in the manufacture of ICs, it should be understood that the lithography apparatus described herein may have other applications, such as the manufacture of integrated optical systems, guidance and detection patterns for magnetic domain memories, flat panel displays, liquid crystal displays (LCDs), thin-film magnetic heads, and the like. Those skilled in the art will recognize that, in the context of such alternative applications, any use of the terms “wafer” or “field” / “die” herein may be considered synonymous with the more general terms “substrate” or “target portion,” respectively. Substrates referred to herein may be processed, before or after exposure, in, for example, a track (a tool that typically applies a resist layer to a substrate and develops the exposed resist), a metrology tool, and / or an inspection tool. Where applicable, the disclosure herein may apply to such and other substrate processing tools. Furthermore, a substrate may be processed more than once, for example, to produce a stacked IC, and therefore the term substrate as used herein may also refer to a substrate that already includes multiple processed layers.

[0240] Although specific reference has been made above to the use of embodiments of the invention in the context of optical lithography, it will be appreciated that the invention may be used in other applications, for example imprint lithography, and that, where the context permits, the invention is not limited to optical lithography. In imprint lithography, a topography in a patterning device defines the pattern to be created on a substrate. The topography of the patterning device may be pressed into a layer of resist supplied to the substrate, after which the resist is cured by applying electromagnetic radiation, heat, pressure, or a combination thereof. The patterning device is then moved away from the resist, leaving a pattern in the resist, after the resist has been cured.

[0241]

[0204] As used herein, the terms "radiation" and "beam" include all types of electromagnetic radiation, including ultraviolet (UV) radiation (e.g., having a wavelength of 365, 248, 193, 157 or 126 nm or thereabouts) and extreme ultraviolet (EUV) radiation (e.g., having a wavelength in the range of 5 to 20 nm), as well as particle beams such as ion beams or electron beams.

[0242]

[0205] The term "lens", where the context allows, may refer to any one or combination of various types of optical components, including refractive, reflective, magnetic, electromagnetic and electrostatic optical components.

[0243]

[0206] While specific embodiments of the invention have been described above, it will be appreciated that the invention may be practiced otherwise than as described. For example, the invention may take the form of a computer program comprising one or more sequences of machine-readable instructions describing the methods disclosed above, or a data storage medium (for example a semiconductor memory, a magnetic disk or an optical disk) having such a computer program stored thereon.

[0244]

[0207] The above description is intended to be illustrative, not limiting. Thus, it will be apparent to those skilled in the art that modifications may be made to the invention as described without departing from the scope of the claims set forth below. In addition, it should be understood that structural features or method steps shown or described in any one embodiment herein may be used in other embodiments.

[0245] [000208] Currently, to perform sub-field correction, high density sampling data is required as input for sub-field (cell or die based) correction. Furthermore, post-etch sampling layouts are not optimized to have optimal sampling for APC control (based on whole wafer fingerprint control) and individual die / average die based (sub-field) correction over time. To date, sampling schemes aimed at achieving both goals have been intensive and expensive.

[0246] [000209] According to one embodiment of the present invention, an optimal sampling scheme for sub-field correction aims to reduce the number of sampling points while maintaining sufficient coverage across the wafer to enable APC control.

[0247] [000210] According to an embodiment, fingerprints across a substrate, within a field, across a subfield, across a group of dies, across a die, across a group of cells, across a cell, and / or across a subset of cells can be used to determine an optimal sampling scheme for subfield correction. In an embodiment, the fingerprints may be fingerprints that represent the distribution of a performance parameter, such as overlay error, across multiple locations on a substrate.

[0248] According to an embodiment of the present invention, a fingerprint of a substrate, e.g., a wafer, may be divided into multiple (at least two) different fingerprints having different spatial scales. For example, these fingerprints may be fingerprints across a substrate, a group of substrates, an intra-field fingerprint, a sub-field fingerprint, a group of dies, a die, a group of cells, a cell, and / or a subset of features within a cell. An optimal sampling scheme may be determined by selecting sample locations across the substrate with the goal of minimizing a measure of prediction uncertainty of a model configured to capture at least two different fingerprints. The measure of prediction uncertainty may be normalized model uncertainty, G-optimality, least squares error, moving standard deviation, or other suitable measure.

[0249] [000212] Alternatively, the optimal sampling scheme can be determined by obtaining at least two sub-sampling schemes each for one of the above fingerprints, and then integrating the sub-sampling schemes. When integrating the sub-sampling schemes, some sampling points may be removed from the optimal sampling scheme or weighted in the optimal sampling scheme. This can be done, for example, by considering the temporal / distribution / process characteristics of the fingerprints.

[0250] [000213] The generation of the sub-sampling scheme can be done by optimizing one, two, or more parameters, such as the predictive uncertainty of a model configured to capture the fingerprint of interest, which may be normalized model uncertainty, G-optimality, least squared error, moving standard deviation, etc.

[0251] [000214] In one embodiment of the present invention, an optimal sampling scheme for a substrate may be generated using a model. The model may relate to a profile that can be corrected by using a lithography apparatus, such as an exposure apparatus. The model may also consider time effects, such as drift in the fingerprint of the substrate, and / or context effects, such as process history. The optimal sampling scheme may be determined using two or more models, each for at least one of the fingerprints having a different spatial scale.

[0252] 4 illustrates an embodiment of the present invention. An optimal sampling scheme may be generated for an area of ​​interest 1 that is smaller than the minimum size of a subfield (the correctable field of a lithographic apparatus) 2. For example, area 1 may include only the critical cells of a 3D-NAND, excluding non-critical features such as wiring around the cells.

[0253] [000216] According to an embodiment of the present invention, the optimal sampling scheme may be applied to other sampling layouts, such as post-etch sampling layouts. According to an embodiment of the present invention, the optimal sampling scheme may be applied to other measurement data, such as focus measurements.

[0254] [000217] The present invention can be further described using the following clauses. 1. A method for determining a sampling scheme, comprising: - obtaining a first fingerprint model of a first spatial distribution of a performance parameter across a first portion of a semiconductor substrate and a second fingerprint model of a second spatial distribution of said performance parameter across a second portion of the semiconductor substrate; - determining sampling points corresponding to measurement locations on the semiconductor substrate for generating measurement data based on an expected decrease in a first uncertainty metric associated with evaluation of the first fingerprint model over the first portion and an expected decrease in a second uncertainty metric associated with evaluation of the second fingerprint model over the second portion; A method comprising: 2. The method of clause 1, wherein the first portion and the second portion at least partially overlap one another. 3. The method of clause 1 or 2, wherein the first fingerprint model is a global fingerprint model and the second fingerprint model is a local fingerprint model. 4. A method according to any one of the preceding clauses, wherein the first portion and / or the second portion is at least part of an irradiated layer within or on top of a semiconductor substrate. 5. A method according to any one of the preceding clauses, wherein the second fingerprint model comprises a lithographic apparatus correction capability model. 6. The method of any one of the preceding clauses, wherein the first fingerprint model and / or the second fingerprint model comprises a physical model. 7. The method of clause 6, wherein the physical model includes a stress profile model. 8. The method of clause 6 or 7, wherein the physical model comprises a process model. 9. The method of any one of the preceding clauses, wherein the second fingerprint model is associated with the process fingerprint. 10. The method of clause 9, wherein the process fingerprint includes a non-zero offset. 11. The method of any one of the preceding clauses, wherein the second portion is one of a field, multiple fields, a subfield, multiple subfields, a die, multiple dies, a portion of a die, multiple portions of multiple dies, a cell, multiple cells, a portion of a cell, or multiple portions of multiple cells. 12. The method of clause 11, wherein the first fingerprint model is a global model. 13. The first part is a field, and 10. The method of any one of the preceding clauses, wherein the second portion is a subfield. 14. The first fingerprint model is a substrate interior model; and 10. The method of any one of the preceding clauses, wherein the second fingerprint model is a substrate edge model. 15. Further comprising obtaining a third fingerprint model for a third spatial distribution of the performance parameter across a third portion of the semiconductor substrate; The method of any one of the preceding clauses, wherein determining sampling points corresponding to measurement locations on the semiconductor substrate for generating measurement data is further based on an expected decrease in a third uncertainty metric associated with evaluation of the third fingerprint model over the third portion. 16. The method of clause 15, wherein the third fingerprint model comprises a lithographic apparatus correction capability model. 17. Further comprising obtaining a fourth fingerprint model for a fourth spatial distribution of the performance parameter across a fourth portion of the semiconductor substrate; 16. The method of clause 15, wherein determining sampling points corresponding to measurement locations on the semiconductor substrate for generating measurement data is further based on an expected decrease in a fourth uncertainty metric associated with evaluating the fourth fingerprint model over the fourth portion. 18. The first fingerprint model is a global model; the second fingerprint model is a substrate edge model; The third part is a field, and 18. The method of clause 17, wherein the fourth portion is a subfield. 19. The method of any one of the preceding clauses, wherein the performance parameter is an overlay error, a critical dimension, an alignment parameter, a sidewall angle, a line edge roughness, an edge placement error, and / or a focus error. 20. The method of any one of the preceding clauses, wherein the first fingerprint model and / or the second fingerprint model further relate to the spatial distribution of at least one further parameter. 21. The method of clause 20, wherein the at least one further parameter is the prediction uncertainty of the first fingerprint model and / or the second fingerprint model. 22. The method of clause 21, wherein the prediction uncertainty is or includes normalized model uncertainty, G-optimality criterion, least squared error, and / or moving standard deviation. 23. A method according to any one of the preceding clauses, wherein determining the locations of the sampling points includes taking into account drift of the first and / or second spatial distributions of the performance parameter over time. 24. Further comprising the step of determining additional sampling points; The method of any one of the preceding clauses, wherein the number of additional sampling points is preferably less than the number of sampling points determined by the method of any one of the preceding clauses. 25. Further comprising identifying one or more critical areas on the semiconductor substrate; 10. The method of any one of the preceding clauses, wherein determining the sampling points is further based on one or more critical areas. 26. Follow these steps: - determining the locations of the sampling points using a method according to any one of the preceding clauses; configuring the metrology device to generate measurement data of the performance parameter at the location of the sampling point on the semiconductor substrate based on the determined location; 10. A method of configuring a metrology apparatus, comprising: 27. Follow these steps: - receiving information about the locations of the sampling points, the locations being determined using the method according to any one of the preceding clauses 1 to 25; configuring the metrology device to generate measurement data of the performance parameter at the locations of the sampling points on the semiconductor substrate based on the received information; 10. A method of configuring a metrology apparatus, comprising: 28. A method of generating measurement data from a semiconductor substrate, comprising the steps of: - determining the locations of the sampling points using the method described in any one of the preceding clauses; - generating measurement data by measuring performance parameters at sampling point locations on a semiconductor substrate; A method comprising: 29. A method of generating measurement data from a semiconductor substrate, comprising the steps of: - receiving information about the locations of the sampling points, the locations being determined using the method according to any one of the preceding clauses 1 to 25; and - generating measurement data by measuring performance parameters at sampling point locations on a semiconductor substrate; A method comprising: 30. Follow these steps: - generating measurement data from a semiconductor substrate by using a method according to any one of clauses 28 or 29; - determining model parameters of the first fingerprint model and model parameters of the second fingerprint model based on the generated measurement data; A method for determining model parameters, including: 31. - Determining model parameters of a first fingerprint model and model parameters of a second fingerprint model by using the method described in clause 30; - configuring a lithographic apparatus based at least in part on said model parameters of the first fingerprint model and / or on model parameters of the second fingerprint model; A method for configuring a lithographic apparatus. 32. Follow these steps: - determining model parameters of the first fingerprint model and model parameters of the second fingerprint model by using the method described in clause 30; - applying a pattern onto a semiconductor substrate using a lithographic apparatus based at least in part on said model parameters of the first fingerprint model and / or model parameters of the second fingerprint model; 1. A method of applying a pattern onto a semiconductor substrate using a lithographic apparatus, comprising: 33. The method of clause 32, wherein applying a pattern onto a semiconductor substrate comprises one or more of the following steps: controlling movement of the semiconductor substrate relative to exposure light from a patterning system of a lithographic apparatus, and / or controlling movement of a patterning device, i.e., a reticle, relative to illumination light of an illumination system, and / or controlling the position and / or orientation of a lens in the patterning system, and / or controlling illumination settings of the illumination system, and / or controlling the focus of the exposure light, and / or controlling the dose of the exposure light. 34. A method of obtaining a fingerprint model for modeling the spatial distribution of a performance parameter across a portion of a substrate, comprising: - defining an initial fingerprint model for the spatial distribution of performance parameters over said portion as a parameterized combination of basis functions; - determining parameter values ​​for a parameterized combination of basis functions based on prior knowledge; - determining a fingerprint model for modeling the spatial distribution of the performance parameters across a set of portions of one or more semiconductor substrates based on a measurement set representative of performance parameters across said portions and an initial fingerprint model, whereby a ratio of at least two parameter values ​​of the basis functions determined in the initial fingerprint model is kept constant or at least a portion of the shape of the initial fingerprint model is maintained in the fingerprint model; A method comprising: 35. Prior knowledge, a further set of measurements representing performance parameters across a further set of the portions of the one or more semiconductor substrates; The simulation results show that Process context information, or As a result of physical simulation, 35. The method of claim 34, comprising: 36. The method of clause 35, wherein the physical simulation includes simulating performance parameters based on stress distribution across the portion of the substrate and based on etcher tilt across the portion of the substrate. 37. The method of any one of clauses 34 to 36, wherein the basis functions are polynomial functions or spline functions. 38. The method of any one of clauses 34 to 37, wherein the shape of the initial fingerprint model is maintained in the fingerprint model. 39. The method of any one of clauses 34 to 38, wherein the fingerprint model is a local fingerprint model. 40. The method of any one of clauses 34 to 38, wherein the fingerprint model is a global fingerprint model. 41. A method according to any one of clauses 1 to 29, wherein the first fingerprint model or the second fingerprint model is obtained using a method according to any one of clauses 34 to 40. 42. The method of clause 41, wherein the first fingerprint model or the second fingerprint model is based on Legendre polynomials. 43. Defining a first fingerprint model for a first spatial distribution of a performance parameter across a first portion of a substrate as a parameterized combination of a first set of basis functions; defining a second fingerprint model for a second spatial distribution of the performance parameter across a second portion of the substrate as a parameterized combination of a second set of basis functions; Including, A method of obtaining a fingerprint model, wherein one or more of a first set of basis functions are substantially orthogonal to one or more of a second set of basis functions. 44. The method of clause 43, wherein the first part includes the second part. 45. The method of clause 43 or 44, wherein the first fingerprint model is a global fingerprint model and the second fingerprint model is a local fingerprint model. 46. ​​The method of any one of clauses 43-45, wherein the first set of basis functions are Cartesian polynomial functions and the second set of basis functions are Legendre polynomial functions. 47. The first fingerprint model relates to a first spatial distribution of the performance parameter across a first portion of the substrate as a parameterized combination of a first set of basis functions; a second fingerprint model relating to a second spatial distribution of the performance parameter across a second portion of the substrate as a parameterized combination of a second set of basis functions; and 30. The method of clauses 1-29, wherein one or more of the first set of basis functions are substantially orthogonal to one or more of the second set of basis functions. 48. The method of clause 47, wherein the first part includes the second part. 49. The method of clause 47 or 48, wherein the first fingerprint model is a global fingerprint model and the second fingerprint model is a local fingerprint model. 50. The method of any one of clauses 47-49, wherein the first set of basis functions are Cartesian polynomial functions and the second set of basis functions are Legendre polynomial functions. 51. A computer program comprising one or more sequences of machine-readable instructions configured to carry out a method according to any one of the preceding clauses. 52. - receiving sampling points generated according to the method according to any one of clauses 1 to 25, the method according to clause 41 when subordinate to any one of clauses 1 to 25, or the method according to any one of clauses 47 to 50; - controlling the measurement device to generate measurement data from the semiconductor substrate at measurement locations corresponding to the sampling points; - determining model parameters of the first fingerprint model and model parameters of the second fingerprint model based at least in part on the generated measurement data; 1. A computer program comprising one or more sequences of machine-readable instructions configured to perform the steps of: 53. - receiving measurement data from the semiconductor substrate at measurement locations corresponding to sampling points determined by the method according to any one of clauses 1 to 25, the method according to clause 41 when dependent on any one of clauses 1 to 25, or the method according to any one of clauses 47 to 50; - determining model parameters of the first fingerprint model and model parameters of the second fingerprint model based at least in part on the received measurement data; - generating control input data for controlling disposing a pattern on a semiconductor substrate, the control input data being based at least in part on the model parameters of the first fingerprint model and / or the model parameters of the second fingerprint model; 1. A computer program comprising one or more sequences of machine-readable instructions configured to perform the steps of: 54. A semiconductor substrate measuring device comprising: - an input terminal configured to receive sampling points determined according to the method of any one of clauses 1 to 25, the method of clause 41 when dependent on any one of clauses 1 to 25, or the method of any one of clauses 47 to 50; - a controller configured to control the measurement device to generate measurement data from the semiconductor substrate at measurement locations corresponding to the sampling points; - a processor configured to determine model parameters of the first fingerprint model and model parameters of the second fingerprint model based at least in part on the generated measurement data; An apparatus comprising: 55. A lithographic apparatus comprising: - an input terminal configured to receive measurement data from the semiconductor substrate at measurement locations corresponding to sampling points determined by the method according to any one of clauses 1 to 25, the method according to clause 41 when dependent on any one of clauses 1 to 25, or the method according to any one of clauses 47 to 50; - a processor configured to determine model parameters of the first fingerprint model and model parameters of the second fingerprint model based at least in part on the received measurement data; - a controller configured to control the dispensing of a pattern onto a semiconductor substrate on the basis of control input data, the control input data being based at least in part on said model parameters of the first fingerprint model and / or model parameters of the second fingerprint model; and 1. A lithography apparatus comprising: 56. A lithography system comprising a semiconductor substrate measurement apparatus according to clause 54 and a lithography apparatus according to clause 55. 57. A device manufacturing method comprising using a lithographic apparatus according to clause 55 to transfer a pattern from a patterning device onto a substrate.

[0255] [000218] The foregoing description of specific embodiments sufficiently reveals the general nature of the invention so that others, by applying knowledge within the skill of the art, can readily modify such specific embodiments and / or adapt such specific embodiments for various uses without departing from the general concept of the invention and without undue experimentation. Such adaptations and modifications are therefore intended to be within the meaning and range of equivalents of the disclosed embodiments, based on the teaching and guidance presented herein. It is to be understood that the terms or phrases herein are intended to be illustrative by example, not limiting, as they would be interpreted by one of ordinary skill in the art in light of the teaching and guidance.

[0256] [000219] The breadth and scope of the present invention should not be limited by any of the above-described exemplary embodiments, but should be defined only in accordance with the following claims and their equivalents.

Claims

1. 1. A method for obtaining a fingerprint model, comprising: defining a first fingerprint model for a first spatial distribution of a performance parameter across a first portion of the substrate as a parameterized combination of a first set of basis functions; defining a second fingerprint model for a second spatial distribution of the performance parameter across a second portion of the substrate as a parameterized combination of a second set of basis functions different from the basis functions; Including, A method wherein one or more of the first set of basis functions are substantially orthogonal to one or more of the second set of basis functions.

2. The method of claim 1 , wherein the first portion comprises the second portion.

3. The method of claim 1 or 2, wherein the first fingerprint model is a global fingerprint model and the second fingerprint model is a local fingerprint model.

4. 4. The method of claim 1, wherein the first set of basis functions are Cartesian polynomial functions and the second set of basis functions are Legendre polynomial functions.

5. 5. The method according to claim 1, wherein the first fingerprint model and the second fingerprint model are decomposed into a model having lower-order parameters and a model having higher-order parameters.

6. The method of claim 1 , wherein the first fingerprint model and / or the second fingerprint model comprises a physical model.

7. The method of claim 6 , wherein the physical model comprises a stress profile model.

8. The method of claim 6 or 7, wherein the physical model comprises a process model.

9. The method of claim 1 , wherein the second fingerprint model is associated with a process fingerprint.

10. the first portion is a field; and 10. The method of claim 1, wherein the second portion is a subfield.

11. 11. A computer program comprising one or more sequences of machine-readable instructions configured to perform the method of any one of claims 1 to 10.

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