Method for predicting resist deformation

Abstract

A method for determining deformation of a resist during patterning. The method involves: obtaining a resist deformation model of a resist having a pattern, the resist deformation model being configured to simulate fluid flow of the resist due to capillary forces acting on a profile of at least one feature of the pattern; and determining, via the resist deformation model, deformation of a resist pattern to be developed based on an input pattern to the resist deformation model.

Description

[0001] CROSS-REFERENCE TO RELATED APPLICATIONS

[0002] This application claims priority to U.S. Application No. 62 / 786,637, filed December 31, 2018, which is incorporated herein by reference in its entirety. Technical Field

[0003] The description herein relates to processes involving pattern formation on a substrate and more particularly to a method of determining resist deformation of a patterned layer on the substrate. Background Art

[0004] For example, a lithographic apparatus may be used in the manufacture of integrated circuits (ICs) or other devices. In this case, a patterning device (e.g., a mask) may contain or provide a pattern corresponding to a single layer of the device (a "design layout"), and the pattern may be transferred to a target portion (e.g., including one or more dies) on a substrate (e.g., a silicon wafer) that has been coated with a layer of radiation-sensitive material (a "resist"), such as by irradiating the target portion through the pattern on the patterning device. Typically, a single substrate contains multiple adjacent target portions to which the pattern is transferred successively, one at a time, by the lithographic apparatus. In one type of lithographic apparatus, the pattern on the entire patterning device is transferred to one target portion at a time; such an apparatus is generally referred to as a stepper. In an alternative apparatus, generally referred to as a stepper-scan apparatus, a projection beam is scanned across the patterning device in a given reference direction (the "scanning" direction) while the substrate is synchronously moved parallel or antiparallel to the reference direction. Different portions of the pattern on the patterning device are gradually transferred to one target portion. Typically, since the lithographic apparatus will have a magnification factor M (typically < 1), the speed F at which the substrate is moved will be a factor M times that of the projection beam scanning patterning device.

[0005] Before the device manufacturing process of transferring the pattern from the pattern forming device to the substrate of the device manufacturing process, the substrate may undergo various device manufacturing processes of the device manufacturing process, such as primer coating, resist coating and soft baking. After the pattern transfer, the substrate can undergo other device manufacturing processes of the device manufacturing process, such as post-exposure baking (PEB), development, hard baking and measurement / inspection of the transfer pattern. The array of the device manufacturing process is used as the basis for manufacturing a single layer of a device (such as an IC). Then, the substrate may undergo various device manufacturing processes of the device manufacturing process, such as etching, ion implantation (doping), metallization, oxidation, chemical mechanical polishing, metrology (such as using a scanning electron microscope (SEM)), etc., all of which are intended to complete a single layer of the device. If multiple layers are required in the device, the entire process or its variants are repeated for each layer. Ultimately, the device will be present in each target portion on the substrate. If there are multiple devices, the devices are separated from each other by techniques such as cutting or sawing, so that a single device can be mounted on a carrier, connected to pins, etc.

[0006] Therefore, manufacturing equipment such as semiconductor devices typically involves processing a substrate (e.g., a semiconductor wafer) using a number of fabrication processes to form various features and multiple layers of the device. Such layers and features are typically manufactured and processed using, for example, deposition, lithography, etching, chemical mechanical polishing, and ion implantation. Multiple devices can be made on multiple dies on a substrate and then separated into individual devices. The device manufacturing process can be considered a patterning process. The patterning process involves a patterning step, such as optical or nanoimprint lithography using a lithography device to provide a pattern on a substrate, and typically but optionally involves one or more related pattern processing steps, such as resist development by a developer, baking of the substrate using a baking tool, etching using a pattern using an etching device, etc. Further, one or more metrology processes are typically involved in the patterning process.

[0007] As semiconductor manufacturing processes continue to evolve, following a trend generally referred to as "Moore's Law", the size of functional elements has been continuously reduced, while the number of functional elements (such as transistors) per device has steadily increased for decades. At the current state of the art, the layers of a device are manufactured using a lithographic projection apparatus that projects a pattern corresponding to the design layout onto a substrate using illumination from a deep ultraviolet illumination source, thereby creating individual functional elements with dimensions well below 100 nm (i.e., less than half the wavelength of the radiation from the illumination source (e.g., a 193 nm illumination source)). This process in which features with dimensions less than the classical resolution limit of the lithographic projection apparatus are printed is generally referred to as low-k1 lithography, according to the resolution formula CD=k1×λ / NA, where λ is the wavelength of the radiation employed (currently 248 nm or 193 nm in most cases), NA is the numerical aperture of the projection optics in the lithographic projection apparatus, CD is the "critical dimension", typically the smallest printed feature size, and k1 is an empirical resolution factor. In general, the smaller the k1, the more difficult it is to reproduce on the substrate a pattern similar in shape and size to that planned by the circuit designer in order to achieve specific electrical functionality and performance. To overcome these difficulties, complex fine-tuning steps are applied to the lithographic projection apparatus and / or the pattern corresponding to the design layout. For example, these include but are not limited to optimization of NA and / or optical coherence settings, customized illumination schemes, use of phase-shifted patterning devices, optical proximity correction (OPC) in the pattern corresponding to the design layout (such as offsetting pattern features, adding assist features, applying serifs to pattern features, etc.) or other methods generally defined as "resolution enhancement techniques" (RET). Summary of the invention

[0008] In an embodiment, a method for determining deformation of a resist during patterning is provided. The method includes: obtaining a resist deformation model of a resist having a pattern, the resist deformation model being configured to simulate a fluid flow of the resist due to capillary forces acting on a profile of at least one feature of the pattern; and determining, via a processor and the resist deformation model, a deformation of a resist pattern to be developed based on an input pattern of the resist deformation model. The resist deformation model is based on linearized Navier-Stokes flow equations. The fluid flow is characterized as Stokes flow and / or Hull-Shaw flow.

[0009] In addition, in an embodiment, a method for determining parameters of a patterning process is provided. The method includes: obtaining (i) a patterning process model, including a resist deformation model of a resist having a pattern, the resist deformation model being configured to simulate a fluid flow of the resist due to capillary forces acting on the profile of at least one feature of the pattern, and (ii) a target pattern; determining the resist pattern via a processor based on a simulation of the patterning process model, wherein the target pattern is used as an input to the patterning process model, wherein there is a difference between the resist pattern and the target pattern; and determining via a processor a value of a parameter of the patterning process based on the simulation of the patterning process, the value of the parameter being determined so that the difference between the resist pattern and the target pattern is reduced. In an embodiment, the parameters of the patterning process include at least one of: dose, focus, and optical proximity correction. The method also includes: applying the value of the parameter of the patterning process to a lithographic device during the patterning process.

[0010] In addition, in an embodiment, a method for determining the deformation of a pattern to be formed in a patterning process is provided. The method includes: inputting pattern information about the pattern to be formed into a resist deformation model (e.g., a film-based model), the model being configured to simulate the deformation of a portion of the resist, the portion including a boundary liquid layer at a boundary between a developed region in the resist and a region of the resist surrounding the developed region, wherein the model is configured to determine a first deformation component of the boundary liquid layer caused by a fluid flow of the boundary liquid layer and a second deformation component of the boundary liquid layer caused by a fluid flow of the boundary liquid layer; and determining, via a processor, the deformation of the pattern to be formed in the resist based on the input pattern information, wherein the deformation includes a combination of the first deformation component and the second deformation component of the boundary liquid layer. The thickness of the boundary liquid layer is less than the length of the developed region in the resist at the boundary.

[0011] In an embodiment, the first deformation component is determined in the horizontal plane based on a horizontal component of the flow rate of the boundary liquid layer, and the second deformation is determined in the horizontal plane based on a vertical component of the flow rate of the boundary liquid layer.

[0012] In addition, a method for determining deformation of a pattern to be formed in a patterning process is provided. The method includes: inputting pattern information about the pattern to be formed into a resist deformation model, the model being configured to simulate deformation of a portion of the resist, the portion including a boundary liquid layer at a boundary between a developed region in the resist and a region of the resist surrounding the developed region, wherein the model is configured to determine deformation of the boundary liquid layer caused by a horizontal fluid flow of the boundary liquid layer; and determining, via a processor, deformation of the pattern to be formed in the resist by simulating the resist deformation model based on the input pattern information. The thickness of the boundary liquid layer is less than the length of the developed region in the resist at the boundary.

[0013] Furthermore, a non-transitory computer program product comprising machine-readable instructions for causing a processor to perform the steps of the aforementioned method is provided. BRIEF DESCRIPTION OF THE DRAWINGS

[0014] The accompanying drawings, which are incorporated in and constitute a part of the specification, illustrate one or more embodiments and, together with the description, explain these embodiments. Embodiments of the present invention will now be described, by way of example only, with reference to the accompanying schematic drawings, in which corresponding reference numerals indicate corresponding parts, and in which:

[0015] Figure 1 is a block diagram of various subsystems of a lithography system according to an embodiment.

[0016] Figure 2 is a block diagram of a simulation model of computational lithography according to an embodiment.

[0017] Figure 3A An image of a wafer having a pattern of multiple features (eg, bars) printed on the wafer is illustrated in accordance with an embodiment.

[0018] Figure 3B According to the embodiment, the printed Figure 3A Aerial image of the pattern on the wafer.

[0019] Figure 3C The diagram shows a Figure 3B Example of contours obtained from a spatial image.

[0020] Figure 4A is an example flow chart for obtaining resist deformation using a resist deformation model according to an embodiment.

[0021] Figure 4B Illustrated are example features within a resist field according to an embodiment.

[0022] Figure 4Cis the force Fi at vertex i determined based on neighboring vertices i-1 and i+1, respectively, according to an embodiment.

[0023] Figure 4D The diagram illustrates vertical outflow (eg, along the y-axis) and horizontal inflow (eg, along the x-axis) generated on a resist domain according to an embodiment.

[0024] Figure 4E An example boundary condition is illustrated, wherein rotators R1 , R2 , R3 , and R4 are placed at boundary corner points around a feature, according to an embodiment.

[0025] Figure 4F An example of a boundary condition referred to as a no-flux condition is illustrated in accordance with an embodiment.

[0026] Figure 4G An example of a wall condition applied around a resist domain boundary according to an embodiment is illustrated.

[0027] Figure 4H An example of a periodic boundary condition applied around a resist domain boundary according to an embodiment is illustrated.

[0028] Figure 5 Another example of determining resist deformation according to an embodiment is illustrated.

[0029] Figure 6 A squeeze mode velocity field is illustrated according to an embodiment.

[0030] Figure 7 Examples of initial feature shapes and deformed features corresponding to the initial features according to embodiments are illustrated.

[0031] Fig. 8A is a flow chart of a method for determining resist deformation according to an embodiment.

[0032] Figure 8B is a simulation according to an embodiment Fig. 8A Flowchart of the method for using a resist deformation model.

[0033] Fig. 9 is a flow chart of a method for determining parameters of a patterning process based on resist deformation calculated based on a resist deformation model according to an embodiment.

[0034] Fig. 10A and 10B 2 are top and side cross-sectional views, respectively, of a substrate having resist and features therein according to an embodiment.

[0035] Fig. 10C Illustrated is a boundary liquid layer around a feature within a resist according to an embodiment.

[0036] Fig.11 A comparison between an initial profile, a deformation profile based on a full-fluid resist deformation model, and a thin-film based resist deformation model according to the present disclosure is shown.

[0037] Fig. 12A An example vertex and its corresponding vertex according to the present disclosure are shown. Fig. 10C The resist deformation model is the movement of the boundary liquid layer.

[0038] Fig. 12B The diagram shows the Fig. 10C Example symbols used in the determination of vertical flow in a boundary layer.

[0039] Fig.13 An example cross section after contraction due to vertical flow according to the present disclosure is illustrated.

[0040] Fig.14A is a flow chart of a method for determining deformation of a pattern to be developed using a boundary layer based resist deformation model according to the present disclosure.

[0041] Fig. 14B is an example flow chart of a method for determining deformation due to horizontal flow of a boundary layer according to the present disclosure.

[0042] Fig.15 The diagram shows the Fig.14A Example deformation of the resist model used in the method.

[0043] Fig.16 is a block diagram of an example computer system according to an embodiment.

[0044] Fig.17 is a schematic diagram of a lithography projection apparatus according to an embodiment.

[0045] Fig.18 is a schematic diagram of another lithography projection apparatus according to an embodiment.

[0046] Fig.19 According to the embodiment Fig.18 A more detailed view of the device.

[0047] Fig. 20 According to the embodiment Fig.18 and Fig.19 A more detailed view of the source collector module of the device. DETAILED DESCRIPTION

[0048] As background for the examples and turning to Figure 1, illustrates an exemplary lithographic projection apparatus 10A. The main components are a radiation source 12A, which may be a deep ultraviolet excimer laser source or other type of source, including: an extreme ultraviolet (EUV) source; an illumination optic that defines a partial coherence (denoted as σ) and may include optics 14A, 16Aa, and 16Ab that shape the radiation from source 12A; a support configured to hold a patterning device 18A; and projection optics 16Ac that projects an image of the patterning device pattern onto a substrate plane 22A. An adjustable filter or aperture 20A at a pupil plane of the projection optics may limit the range of beam angles that impinge on the substrate plane 22A, where the maximum possible angle defines the numerical aperture NA of the projection optics = sin(Θ max ). In an embodiment, the lithographic projection apparatus itself need not have a radiation source 12A.

[0049] Thus, in a lithographic projection apparatus, the optical device 16Ac directs an aerial image of the patterning device pattern onto the substrate (usually a reduced version). The aerial image (AI) is the radiation intensity distribution at the substrate level. The resist layer on the substrate is exposed and the aerial image is transferred to the resist layer as a latent "resist image" (RI) therein. The resist image (RI) can be defined as the spatial distribution of the solubility of the resist in the resist layer.

[0050] Now, it is often desirable to be able to determine by computation how a patterning process produces a desired pattern on a substrate. Thus, a simulation may be provided to simulate one or more parts of the process. For example, it is desirable to be able to simulate a lithography process that transfers a patterning device pattern onto a resist layer of a substrate and the pattern produced in the resist layer after the resist is developed.

[0051] An exemplary flow chart for simulating lithography in a lithographic projection apparatus is in Figure 2 . The illumination model 31 represents the optical characteristics of the illumination (including the radiation intensity distribution and / or the phase distribution). The projection optics model 32 represents the optical characteristics of the projection optics (including the changes in the radiation intensity distribution and / or the phase distribution caused by the projection optics). The design layout model 35 represents the optical characteristics of the design layout (including the changes in the radiation intensity distribution and / or the phase distribution caused by the given design layout), which is a representation of the feature arrangement on or by the pattern forming device. The aerial image 36 can be simulated using the illumination model 31, the projection optics model 32, and the design layout model 35. The resist image 38 can be simulated from the aerial image 36 using the resist model 37. The simulation of lithography can, for example, predict the profile and / or CD in the resist image.

[0052] More specifically, it is noted that the illumination model 31 can represent the optical characteristics of the illumination, including but not limited to the NA sigma (σ) setting and any specific illumination shape (e.g., off-axis illumination, such as annular, quadrupole, dipole, etc.). The projection optical model 32 can represent the optical characteristics of the projection optical device, including, for example, aberrations, distortions, refractive indices, physical sizes or dimensions, etc. The design layout model 35 can also represent one or more physical characteristics of a physical pattern forming device, such as described in U.S. Patent No. 7,587,704, which is incorporated herein by reference in its entirety. The optical characteristics associated with the lithographic projection apparatus (e.g., characteristics of the illumination, pattern forming device, and projection optical device) indicate the aerial image. Since the pattern forming device used in the lithographic projection apparatus can be changed, it is desirable to separate the optical characteristics of the pattern forming device from the optical characteristics of the rest of the lithographic projection apparatus, including at least the illumination and projection optical devices (and therefore including the design layout model 35).

[0053] The resist model 37 can be used to calculate the resist image from the aerial image, an example of which can be found in U.S. Pat. No. 8,200,468, which is incorporated herein by reference in its entirety. The resist model is typically only related to the properties of the resist layer (e.g., the effects of chemical processes occurring during exposure, post-exposure baking, and / or development).

[0054] The goal of the simulation is to accurately predict, for example, edge location, spatial image intensity slope, and / or CD, which can then be compared to the intended design. The intended design is typically defined as a pre-OPC design layout that can be provided in a standardized digital file format, such as GDSII, OASIS, or other file formats.

[0055] Through the design layout, one or more portions may be identified, which are referred to as "clips". In an embodiment, a collection of clips is extracted that represents complex patterns in the design layout (typically about 50 to 1000 clips, although any number of clips may be used). As will be appreciated by those skilled in the art, these patterns or clips represent small portions of the design (e.g., circuits, cells, etc.), and in particular clips represent small portions that require special attention and / or verification. In other words, a clip may be a portion of the design layout, or may be similar or have similar behavior to a portion of the design layout, where key features are identified through experience (including customer-provided clips), through trial and error, or by running full-chip simulations. Clips typically contain one or more test patterns or instrumentation patterns. An initial larger set of clips may be provided a priori by the customer based on known key feature areas in the design layout that require specific image optimization. Alternatively, in another embodiment, an initial larger set of clips may be extracted from the entire design layout using some automatic (such as machine vision) or manual algorithm that identifies key feature areas.

[0056] In some examples, simulation and modeling can be used to configure one or more features of the patterning device pattern (e.g., perform optical proximity correction), one or more features of the illumination (e.g., change the spatial / angular intensity distribution of the illumination, such as changing the shape), and / or one or more features of the projection optics (e.g., numerical aperture, etc.). Such configurations may be generally referred to as mask optimization, source optimization, and projection optimization, respectively. Such optimizations may be performed individually or combined in different combinations. One such example is source mask optimization (SMO), which involves configuring one or more features of the patterning device pattern together with one or more features of the illumination. Optimization techniques may focus on one or more clips. Optimization may use the simulations described herein to generate values ​​for various parameters.

[0057] In the optimization process of the system, the quality factor of the system can be expressed as a cost function. The optimization process is reduced to the process of finding a set of parameters (design variables) of the system to minimize the cost function. Depending on the goal of the optimization, the cost function can have any suitable form. For example, the cost function can be the weighted root mean square (RMS) of the deviation of certain characteristics (evaluation points) of the system relative to the expected values ​​(e.g., ideal values) of these characteristics; the cost function can also be the maximum value (i.e., the worst deviation) of these deviations. The term "evaluation point" herein should be broadly interpreted as including any characteristics of the system. Due to the practicality of the implementation of the system, the design variables of the system can be limited to a limited range and / or interdependent. In the case of a lithographic projection device, the constraints are usually associated with the physical characteristics and features of the hardware, such as the tunable range and / or the pattern forming device manufacturability design rules, and the evaluation points can include physical points on the resist image on the substrate and non-physical features, such as dose and focus.

[0058] As mentioned above, the layer on the substrate may have a pattern transferred thereto. Such a layer is generally referred to as a resist layer and may have various chemical compositions. In an embodiment, the resist layer is a radiation-sensitive material layer. The resist layer generally has a very small but limited thickness, which may be comparable to the pattern imaged on the resist. The resist layer may undergo various treatments during the lithography process. For example, the resist may be exposed to radiation such as EUV or DUV, which may cause a chemical reaction in the resist. The resist may undergo post-exposure baking (PEB), development (e.g., positive development or negative development) and / or hard baking. Each of these treatments may cause the resist to deform in one, two or three dimensions, and the deformation may be site-dependent (e.g., three-dimensional site-dependent). The deformation of the resist may affect downstream treatments, such as material deposition and etching. In the lithography process using negative development, the effect of resist deformation on resist top loss and critical dimensions may be particularly significant. Therefore, a resist model 37 having the ability to predict resist deformation is beneficial to more accurate lithography and higher yields. The resist model 37 may also be able to predict the reaction of the resist layer to various other physical and chemical treatments in the lithography process. An exemplary resist model according to one aspect of the present disclosure is described later.

[0059] A pattern is formed in the resist layer, for example by exposing the resist layer to radiation via a pattern forming device. The resist layer may undergo a post-exposure treatment such as PEB and be deformed into a deformed resist layer having a deformed pattern therein. In an embodiment, a portion of the deformed resist layer that receives a sufficiently high dose during exposure remains on the substrate after negative development, and the remainder (e.g., portion) of the deformed resist layer dissolves after negative development. Alternatively, a portion of the deformed resist layer receives a sufficiently high dose during exposure to dissolve during positive development, and the remainder of the deformed resist layer remains on the substrate after positive development. Whether the portion remains or dissolves depends on the chemical composition of the resist and the chemical composition of the developer.

[0060] As mentioned above, simulation model attempts to accurately predict the pattern in resist properties.Resist is a viscoelastic material, and over a period of time, resist can show significant fluid behavior.Under this assumption, intermolecular force, pressure and / or other forces (generally referred to as fluid internal forces in this article) may cause the stress of producing viscous flow in fluid.Simulation model cannot explain this viscous flow in resist.And, when resist pattern is dense, the influence of the fluid internal forces acting on resist tends to be significant.Therefore, simulation model cannot predict the shape of the resist pattern (particularly dense pattern or pattern with high curvature) in resist with high accuracy, which may cause downstream effects, such as modeling errors.

[0061] The resist pattern includes a plurality of developed portions surrounded by the resist. One or more fluid internal forces (e.g., surface tension) typically act on the resist, thereby deforming the resist at the boundaries of the developed portions of the resist. Therefore, the shape of the resist pattern corresponding to the developed portion is deformed at multiple locations along the boundaries of the developed portion. In an embodiment, a resist deformation model is developed under the assumption that capillary and viscous flow effects dominate. In addition, the resist deformation model is developed under the assumption that a continuous description holds. In an embodiment, the driving force acting on the resist is a capillary force acting on the interface between two materials, such as the resist and an internal portion of the developed portion (e.g., air), such as with respect to the following Figure 3C The contours in the image 320 of FIG. In an embodiment, the contours are obtained via lithography simulations and contour extraction algorithms known to those skilled in the art.

[0062] In an embodiment, a mask pattern layout of a pattern to be printed on a wafer is obtained. The mask pattern layout is obtained, for example, via lithography simulation software such as Tachyon, HyperLith, or ProLith, or other software configured to generate an aerial image from the mask pattern layout. The aerial image (e.g., Figure 3B In a typical case, the aerial image is a grayscale image, where each pixel of the image corresponds to a different radiation intensity (i.e., before the radiation strikes the resist). Examples of aerial images (e.g., 310) and corresponding patterns printed on a wafer (e.g., 300) are shown in FIG. Figure 3B and 3A Medium picture.

[0063] Figure 3A An image 300 of a wafer having a pattern of multiple features (e.g., gratings) printed on a wafer is illustrated. It can be seen that the contours of the features are deformed due to the deformation of the resist on the wafer. It is beneficial to predict the deformation of such features of the pattern in the resist to adjust or control the parameters of the patterning process so that the desired pattern is printed without defects, thereby improving the yield of the patterning process. In an embodiment, based on the disclosure below (e.g., with respect to Figure 4A and Figure 5 ) discussed in detail above (e.g., a fluid dynamics model) is used to predict this deformation in the resist.

[0064] Figure 3B Is on the chip ( Figure 3A An aerial image 310 of a pattern printed on a resist (in an image). In an embodiment, the aerial image 310 is a grayscale image, and each pixel corresponds to a different light intensity (before the light hits the resist). Based on the aerial image 310, the outline of the pattern can be extracted at a certain threshold level. For example, the outline of the aerial image is extracted at an intensity value that breaks through a desired threshold.

[0065] Figure 3C An example of an image 320 is illustrated, which includes contours in a resist 323 obtained from an aerial image 310 (also referred to as a resist contour), which can be input into a deformation model of the present disclosure to determine the deformation of the resist. In an embodiment, the image 320 including the contours is a binary image. The image 320 includes multiple contours of features of a pattern (e.g., gate bars, contact holes, etc.) surrounded by the resist. The binary image represents the contour lines of each of the multiple features of the pattern.

[0066] In an embodiment, in contour 320, the inside of the extracted contours are all gas 321 (white areas), and outside these contours are resist 323 (grey areas). The resist 323 and gas 321 can then be simulated as different or the same fluids to determine the deformation of the contours. When using Stokes or Hull-Shaw flow, the fluids 323 and 321 are considered to have the same viscosity. In an embodiment, the simulation is based on a 2D resist deformation model, where the resist height or thickness is not considered. However, the present disclosure is not limited to 2D models and can be modified to also apply to 3D resists. An example of the resist deformation process is relative to Figure 4A Explained.

[0067] Figure 4A is an example flow chart for obtaining resist deformation using a resist deformation model, wherein the resist deformation model is a fluid dynamics model based on the Navier-Stokes equations, which captures the viscous flow in the resist caused by one or more fluid internal forces, and calculates the corresponding deformation of the resist. Therefore, the fluid dynamics model can accurately predict complex shapes such as resist patterns. In an embodiment, the fluid dynamics model is linearized, such as a Stokes flow (e.g., a 2D Stokes flow) represented by the linearized Navier-Stokes equations.

[0068] In an embodiment, obtaining the resist deformation involves: using an image including contours of feature(s) as input to simulate a fluid dynamics model. In an embodiment, the contours are obtained from above relative to Figure 3B and 3C The extracted contours (eg, in 320) are extracted from the discussed aerial image (eg, in 310). The extracted contours may be a grayscale image or a binary image. In an embodiment, the extracted contours refer to points where forces due to surface tension will be applied.

[0069] On the extracted profile, the force (also referred to as capillary force) at different positions along the profile is determined, thereby generating a force field. The force or force field along the profile can cause the deformation of the feature. In an embodiment, the force acting on other features may cause the deformation of the desired feature. These forces are also used to determine the velocity field (or flow field) at and around the characteristic profile of the resist. In an embodiment, the velocity field imitates the movement of the edge of the simulated profile over time, which eventually causes the final deformation of the characteristic profile in the resist.

[0070] In an embodiment, the simulation is an iterative process. In the iteration, the first step P41 involves: determining the force (F) at the location (i) along the desired contour i In an embodiment, the force is calculated for a location (i) along the desired contour based on the force at the adjacent location (j) (e.g., as Figure 4B As shown in the figure), where j is different from i. Based on other parts along the contour, this force (F i ) is calculated for each part along the profile. In an embodiment, the parts (i) and / or (j) are represented by vertices or points. The equations and methods for determining the forces are explained later in this disclosure (e.g., Equations 3 to 5). In an embodiment, the forces may be calculated using the Stokes equations, which are linearized versions of the Navier-Stokes equations obtained by neglecting the inertial terms, as given below (Equation 1).

[0071]

[0072] In Equation 1, the pressure P causes the divergence of the velocity field to vanish in the resist domain. Thus, the velocity The force acting on the liquid Determined. The equation is linear. Thus, the flow field (interchangeably referred to as velocity or velocity flow field) is the convolution of the response to the point force and the applied force. In an embodiment, the velocity field is driven by capillary forces determined using equations (3 to 5) discussed later.

[0073] Once the force at each location along the contour is determined, the second step P43 involves: calculating the force (F) at all other points except point i by i ) to determine the total velocity field (u). In an embodiment, by evaluating the equation u j =ΣG ij ·F i ∨i,j+regularization, to determine the velocity field at point j, where G jj=0 (the Einstein summation convention is used in the notation of this article). In an embodiment, the fluid dynamics model may need to be regularized because solving the velocity field using the fluid dynamics model may be an ill-posed problem, which may lead to unstable simulations (for example, due to singularities that may occur at vertices or points along the contour). Thus, regularization can provide stability to the simulation. In an embodiment, regularization can be performed to address problems such as singular behavior of Stokes particles. This singular behavior disappears when the applied force is distributed over the line segment rather than applied at a single point, which is equivalent to applying a pair of rotators of equal magnitude and opposite sign at the ends of the line segment (see, for example, Figure 4E ).

[0074] According to an embodiment, the position x at the Velocity response of the point force at is called the Stokes subson and is given by the following equation (2):

[0075]

[0076] In an embodiment, the flow may be a 2D Stokes flow, and G (in the above equation) represents the Green's function. The equations and methods for determining the velocity (e.g., using Stokes sub- and boundary conditions) will be explained later in this disclosure. In an embodiment, the boundary conditions on the resist or resist domain may be implemented in different ways, such as Figures 4D to 4F As illustrated and explained in detail later in this disclosure, these boundary conditions are applied to the resist domain, which enables determination of the resist deformation when equations such as 1 and 2 are simulated.

[0077] In an embodiment, the iteration involves a third step P45, in which the velocity at point i is determined. In an embodiment, the velocity at point i is given by the equation Σ(r i ) i .u i = 0, where r is the neighboring point x of point i i-1 and x i+1 , n is the normal vector, and u i is the velocity at point i. In an embodiment, this step is applied to conserve the surface area of ​​the desired polygon, or any other polygon for that matter.

[0078] Further, the first P41, second P43 and / or third P45 steps are repeated and the velocity fields at all locations along the profile and the corrections corresponding to each such location are superimposed to determine the velocity at the time step t n Further, in step P47, the position difference can be equated with the velocity to determine the final deformation. For example, the differential equation Can be targeted at position x i Solved to determine the final deformation of the generally desired feature or resist. For example, the differential equations can be solved using an ODE solving method (such as Euler's explicit method), which will require a very small time step to obtain an accurate solution, or using a higher order Runge-Kutta method.

[0079] Relative to Figures 4B to 4F An example of a process for determining resist deformation and associated conditions capable of simulating a resist deformation model is explained with respect to desired feature 452 .

[0080] Figure 4B An example feature 452 within the resist domain 450 is illustrated. The feature 452 is interchangeably referred to as a contour 452. As mentioned earlier, the contour 452 can be extracted from the corresponding aerial image. In an embodiment, the contour 452 is divided into vertices or points, such as 452a, 452b, 452c, 452d, etc., representing a set of locations within the resist domain 450. Such vertices or points (e.g., 452a to 452d) can be arranged or distributed along the contour 452 so that the area within the contour 452 is conserved. The forces and velocities at these vertices are further calculated.

[0081] In the embodiments, reference Figure 4C , the force Fi at vertex i is determined based on vertex i and adjacent vertices i-1 and i+1, respectively. In an embodiment, the force Fi is calculated based on the interfacial tension (σ) of the interface between the two liquids and the tangent vectors τ before and after vertex i. An example equation for calculating Fi is as follows:

[0082]

[0083]

[0084]

[0085] In Equation 3, σ is the surface tension at the interface (i.e., at the contour of the feature), which is a property of the resist used to print the pattern on the substrate. The tangent vector is calculated using Equations 4 to 5. Once the force is determined, the velocity generated by the force is determined based on, for example, Stokes ions. Further, velocity determination involves assigning or specifying certain boundary conditions on the resist domain, such as with respect to Figures 4D to 4F Boundary conditions shown in the figure.

[0086] In an embodiment, force points are defined along the wall outside the resist domain to generate additional velocity fields. In an embodiment, the force value at each force point is such that the total superposition velocity field adds up to zero velocity (e.g., no flux) at the boundary point, or is equal to the velocity of the boundary node at the opposite position. All force points (e.g., along the wall) together give a system of equations with an equal number of unknowns. In an embodiment, the number of force points is approximately the number of velocity nodes at the boundary of the resist domain (or the same in an embodiment). In an embodiment, the position of the force point is ideally close to the velocity node but not the same as the velocity node. The spacing of the force point is less than or equal to the minimum distance from the feature to the wall.

[0087] In an embodiment, the individual forces along the resist profile 452 result in an effective force acting at the resist domain boundary (e.g., 450) that results in a squeeze pattern in the resist domain. This squeeze pattern refers to the deformation or movement of a desired feature due to the effective force from an adjacent feature. In an embodiment, the squeeze pattern is eliminated by applying one or more boundary conditions, such as relative to Figures 4D to 4F Discussed.

[0088] In an embodiment, Figure 4D The diagram illustrates the horizontal inflow (e.g., along the x-axis) and the vertical outflow (e.g., along the y-axis) generated on the resist domain 450. In an embodiment, the resist domain 450 is treated as an infinite domain for simulation purposes. Further, no boundary conditions are defined. Thus, multiple features together may result in an overall squeeze pattern flow. In order to eliminate the squeeze pattern and retain only the motion of interest (e.g., the motion of feature 452), rotators R1, R2, R3, and R4 are placed at the boundary corner points around feature 452, such as Figure 4E As shown. Rotors R1 to R4 generate flow opposite to the flow created by the force along feature 452. In the example, to counteract, Figure 4E The rotators R1 to R4 drawn in FIG. 4 change direction (ie, clockwise to counterclockwise and vice versa).

[0089] In an embodiment, the squeezing mode is eliminated by applying a rotator of strength S defined based on making the first moment of all forces at the vertices of the contour zero. For example, using the equation S = Σ(F xn .x n -F yn .Y n ), where Fxn and Fyn are forces in the x and y directions, respectively, n refers to a plurality of points along the resist domain 450 where the forces are determined (e.g., points 450a, 450b, 450c, 450d, etc.), and xn and Yn refer to the position of the nth node.

[0090] In another embodiment, referring to Figure 4F, the boundary condition may be a no-flux condition achieved by adding n force points 460a, 460b, 460c, etc. on an external portion 460 (i.e., at a certain distance from the domain 450) to counteract the outflow and / or inflow at the n boundary points 450a, 450b, 450c, 450d, etc., respectively, to impose a no-flux condition. The n boundary points 450a, 450b, 450c, 450d, etc. may also be referred to as velocity nodes. Velocity nodes are nodes on the domain boundary where the velocity field is determined. It is at these velocity nodes that the boundary conditions may be applied.

[0091] In some cases, periodic boundary conditions or wall boundary conditions are required. In an embodiment, implementing these boundary conditions in the singularity method (e.g., if the Green's function is singular when x=0) involves using a fundamental solution that satisfies the boundary conditions. Unfortunately, deriving a fundamental solution for 2D Stokes flow in a periodic domain requires a lot of work, and closed form solutions, i.e., closed solutions, are not found in the prior art. Instead of using a fundamental solution that satisfies the boundary conditions, the present method directly imposes the boundary conditions.

[0092] Figure 4G An example of a wall condition is illustrated. The accompanying figure shows the velocity field due to a force along the contour of multiple features (or a general single feature). At the wall 475 (outside the resist domain), the liquid adjacent to the wall 475 (e.g., the resist within the boundary 470) moves at the same velocity as the wall. If the wall 475 were stationary, both components of the velocity of the liquid at the wall 475 would be zero. By applying a force on the liquid 470, the wall 475 keeps the velocity of the liquid 470 equal to zero. In the proposed method, the force is applied to the liquid along its entire length, keeping the velocity zero along its entire length. In an embodiment, the force is applied at a finite number of points 475a, 475b, 475c, etc. (also called nodes) along the wall 475. This also means that the velocity can be set to zero at a finite number of points. These points cannot coincide with the points where the force is applied because the underlying solution is singular unless we use some regularization. In an embodiment, the nodes (similar to Figure 4F The nodes in the domain) are placed at the domain boundary 470, and force points 475a, 475b, 475c, etc. are placed outside the domain, and the distance between the force points is approximately similar to the spacing of the nodes at the domain boundary 470.

[0093] In an embodiment, the number of domain boundary points (or nodes) should be large enough so that the truncation error (due to approximations in the model) is small enough, but small enough so that the computation time is acceptable. In an embodiment, the spacing between nodes is not large relative to the minimum distance from the feature to the boundary. In an embodiment, the spacing is equal to the minimum distance from the feature to the boundary. Based on this wall condition, the resulting deformation and velocity field of the 2D Stokes flow with wall boundary conditions is as follows Figure 4GForces are applied along wall 475 at locations marked by dots (e.g., 475a, 475b, 475c, etc.) outside domain boundary 470. These forces cause the velocities at the dots along domain boundary 470 to be zero.

[0094] In an embodiment, Figure 4H As illustrated, periodic boundary conditions are applied around the domain boundaries. The accompanying drawings show the velocity field caused by the force along the contour of a single feature (or a single feature in general). The periodic boundary conditions have additional constraints. The first constraint is that the velocities at the opposite sides of the domain must be equal. The second constraint is that the stress on the boundary must be opposite to the stress at the opposite side. In an embodiment, this periodic constraint is applied as a pair of equal and opposite forces at any point and its connection point. This ensures that the total force is zero. Since the force to be resisted does not apply torque, the applied force will also have zero torque. If torque is applied somewhere in the domain, the periodic boundary condition algorithm will cancel it. The sum of the forces in any pair of forces is zero. Therefore, the difference of a pair of forces is determined. The criterion for the magnitude of this force difference is that the velocity difference at the corresponding nodes of the resist domain must disappear. The velocity at the node caused by a pair of forces is obtained by evaluating the difference of the basic solution. The requirement that at each node the sum of these velocities must oppose the velocity due to the feature constitutes a system of linear equations that can be solved using scientific software (such as Matlab) configured to solve a set of linear equations, differential equations, or other mathematical calculations.

[0095] In an embodiment, the resulting deformation and velocity field for a 2D Stokes flow with periodic boundary conditions is calculated. Forces are applied at the locations (dots) marked along the outer boundary 485. These forces make the velocity at each dot along the domain boundary 480 equal to the velocity at the opposite location on 480.

[0096] Figure 5 Another example of determining resist deformation is shown. Figure 3C and 4A As discussed in , the contour or extracted contour of a feature from an aerial image is converted to a polygon. The polygon represents the shape of the contour and can be constrained so that the area of ​​the polygon is conserved. In an embodiment, the polygon includes a boundary or edge represented by vertices (e.g., represented by dots) in the input 500. In an embodiment, a plurality of vertices are associated with position information along the contour. For example, as shown, the input 500 includes vertices represented by dashed lines (e.g., 501) of the feature, wherein each vertex is associated with position information within the input 500. The position of the vertex (e.g., 501) is represented in terms of Cartesian coordinates, polar coordinates, relative position relative to another feature, and the like.

[0097] Input 500 may include multiple features (e.g., bars, holes, lines, etc.) converted to polygons or representing vertices of polygons. The multiple vertices of the feature and the location of each vertex thereby represent the geometry of the contour of the feature. In an embodiment, multiple vertices are redistributed so that they are approximately evenly (or uniformly) spaced apart while conserving the area (or volume in the case of a 3D resist deformation process) of the geometry of each contour. Thus, in an embodiment, each feature (e.g., 501, 503, 505, etc.) of input 500 may be converted to multiple vertices, and further, these vertices may be redistributed or constrained so that the area of ​​the corresponding feature or the area representing the contour of the feature is conserved.

[0098] According to an embodiment, during the simulation of resist deformation, the redistribution of multiple vertices may be desired to maintain stability, because the simulation involves the calculation of a matrix between force and velocity, which uses, for example, Stokes sub-(Equation 2 discussed above) to convert force into velocity. For example, Stokes sub-dots do not have to be evaluated at the source vertex, because once the velocity at other vertices is known, the velocity at the vertex can be calculated by continuity. However, this method causes truncation errors to be collected into the velocity of the vertex. For some vertices, this will result in a force-to-velocity matrix that is not diagonally dominant (i.e., not diagonally dominant), which will result in an unstable solution when velocity is used to advance the position. Therefore, in an embodiment, redistributing the vertices so that the spacing is uniform may be sufficient to maintain stability in the simulation of resist deformation.

[0099] Once the input 500 is obtained, the fluid dynamics of the resist are determined in terms of the forces and velocities representing the movement of a contour of a feature in the resist. In an embodiment, the force or force field on the capillary surface, such as the contour of a polygon or feature, is calculated and the movement of multiple vertices along the contour is tracked.

[0100] In an embodiment, the flow of a body of liquid (e.g., resist) is considered to be a 2D Stokes flow driven by capillary forces in a compact region or selected region (also referred to as a resist domain). According to an embodiment, the force on each vertex of the contour is the sum of the tensions on either side of the vertex. The force is applied at different locations along the contour of the liquid, so a Stokes sub-wavelength for each vertex is determined. The strength of the Stokes sub-wavelength depends on the interfacial tension (σ) of the interface between the two liquids, and the tangent vectors before and after the vertex, as in Figure 4A3 to 5. In an embodiment, not only can the curvature between two adjacent vertices of each vertex be calculated, but also the curvature can be calculated from more adjacent vertices. In an embodiment, such a high-order difference scheme (i.e., determined based on the curvature of more than two adjacent vertices) for calculating capillary forces is possible. However, such a high-order difference scheme makes it (too) difficult to determine a scheme that conserves and stabilizes the volume during the simulation. Moreover, the formula becomes more complicated, increasing the chance of error. Existing methods do not use the low-order method described in this article because it is considered impossible.

[0101] In an embodiment, equations 3 to 5 are applied to each feature (e.g., 501, 503, 505) of the input 500 and the vertices corresponding to the features of the input 500. Thus, the input 500 is transformed into a velocity field for a given time step. For example, a velocity field 510 at a first time step t1 is represented. In the velocity field 510, flow vectors 513, 515, 511 are obtained inside and around the features (e.g., 503 and 505).

[0102] In an embodiment, by taking these forces as coefficients of the Stokes phonon, the forces are applied to the liquid (e.g., resist) and thus the velocity is known everywhere except at the vertex. The vertex is excluded because at the vertex the Stokes phonon is exist is singular at a vertex. To obtain the velocity at a vertex, the flow due to the force at that vertex requires special treatment. The flow due to the force at all other vertices does not require special treatment. For example, to compute the velocity at a vertex, only the flow field due to that same vertex needs to be regularized.

[0103] When further simulating, the velocity field 520 of the second time step t2 is further obtained with respect to the velocity field 510. It can be seen that, as the flow progresses, features such as 513 (or 515) are deformed into deformed features 523 (or 525). Compared with the original elongated shape of the feature 503 (or 505) of the input 500, the deformed feature 523 (or 525) looks curved or rounded. However, as mentioned earlier, the constraints applied to the resist domain and the boundary conditions retain, i.e., maintain the area of ​​the feature 503 (or 505) when the feature is deformed. Thus, the area of ​​the deformed feature 523 (or 525) is approximately similar to the area of ​​the feature 503 (or 505) of the input 500, however, the shape can be significantly changed. Figure 5The velocity fields 510 and 520 shown and the deformation of the features are just examples of simulations of the resist deformation model. For example, the system of equations (1 to 5) at different vertices and the boundary conditions discussed above are simulated. In an embodiment, different instances of deformation can be obtained based on the time instance when the deformation feature must be determined. For example, the simulation may run for 10 seconds, and the result is a deformed resist and corresponding features at 10 seconds. Similarly, the simulation can be run for 20 seconds, 50 seconds, etc. to obtain different deformation instances.

[0104] As mentioned earlier, during the simulation process, the problem of singular behavior of Stokes subatomic particles disappears when the applied force is distributed over a line segment rather than applied at a single point. In an embodiment, regularization of the velocity field (eg, based on Stokes subatomic particles) may be performed, for example, based on the following equation (6).

[0105]

[0106] Equation 6 gives the velocity at the center of the line segment, since the force F is distributed over the line segment with a constant force density. For any line segment of finite length, the velocity is finite, which allows the velocity to be evaluated anywhere.

[0107] As mentioned earlier, the polygons (also referred to as outlines or features) in the resist are approximations of the true interface shape. The presence of sharp corners in this representation causes capillary forces to be concentrated in a limited number of points. In an embodiment, the entire interface is curved, so force is applied to the entire interface. When a vertex is considered, it is relatively unimportant that the force is distributed throughout a certain distant region. When the size of the region is comparable to the distance of the region, the distribution of force becomes relevant. At least, the force caused by the section directly adjacent to the vertex of the interface must be distributed as being distributed throughout the line segment as calculated above.

[0108] In another embodiment, the Stokes sub may not be evaluated at the source vertex, since the velocity at the vertex can be calculated from continuity once the velocities at other vertices are known. For each vertex, the curve length is specified, i.e., assigned. And the normal vector n along the x and y axes is assigned according to the following equations 7 and 8 respectively x and n y :

[0109] n x =r y …(7)

[0110] n y =-r x …(8)

[0111] By the continuity and divergence theorems, the total outward flux should be zero, which can be calculated based on the following equation (9)

[0112] Using equation (9) above, the regularization task involves: i≠j When u is known, determine j , so that the continuity equation is satisfied. In other words, the velocity is determined at all other vertices except at i. Equation (9) directly yields the desired velocity. Further, using continuity-based regularization, the points are redistributed so that the spacing is smooth to maintain stability during simulation.

[0113] According to an embodiment, the limitation of 2D Stokes flow as a model of resist deformation is domain size dependent. For a given force, the velocity field depends on the location of the walls surrounding the resist domain. In an infinite domain, the velocity diverges. Therefore, in an embodiment, the mitigation or reduction of velocity divergence will only consider force distributions that sum to zero. Therefore, the flow field due to the distribution of features across most of the space under the action of capillary forces is calculated. , The contribution to the velocity field due to features far away should be small. This needs to be reduced to r -2 Thus, the velocity field due to capillary forces on the feature is decomposed. In an embodiment, the velocity field is decomposed into the squeeze flow (see Figure 7 ) and decreases to r -2 or faster part. The extrusion mode velocity field is given by the moment In an embodiment, to remove the squeeze mode, a constraint is defined such that the sum of the diagonal parts of the squeeze mode flow field and the moments of the Stokes sub-forces sum to zero.

[0114] Figure 6 The squeeze mode velocity field 600 is illustrated. The velocity field is calculated by summing the Stokes subfields at the vertical domain boundaries. The force per unit length on the left boundary is constant and opposite to the force per unit length on the right boundary, which is also constant.

[0115] In different methods, such as the earlier relative Figure 4F As mentioned, by adding the expanded simple flow, a zero total flux condition is imposed through the horizontal walls of the resist domain, and thus a zero flux through the vertical walls of the resist domain.

[0116] In a straightforward implementation of the Stokes-based computation, every boundary point directly affects every other boundary point. Therefore, in the limit of a large number of boundary points N, the number of floating point operations (FLOPS) required to compute the velocity field is quadratic (i.e., N 2 ).

[0117] In an embodiment, to improve scaling in Stokes-based calculations, the velocity fields due to distant features are replaced by their truncated multipole expansions. In the simplest form of multipole expansion, only the squeeze mode is retained and all higher-order terms are ignored. This is because the effects of all squeeze mode terms will not converge due to the inclusion of farther terms, and the higher-order terms caused by features at distances between R and R+dR become zero as R increases. The squeeze mode flow due to each distance feature is calculated, and each feature is replaced by a combination of the 4 rotators described earlier. These rotators are placed at the corners of rectangles that are not large relative to the replaced feature (such as the bounding rectangle or constraint rectangle of the feature). Thus, for each feature whose impact is simplified in this way, the number of points N in the global calculation is reduced by n-4, at the expense of summing n forces. This is because the computational cost scaling N^2 in the global calculation is significantly improved. Thus, in an embodiment, the scaling of the velocity calculations for n points per feature and m features is reduced from n to n^2. 2 m 2 Improved to n 2 +nm 2 For example, suppose n = 10 3 And m = 10 9 , which means 10 3 In an embodiment, the scaling can be further improved by collecting multipole expansions of the features in relatively distant regions (eg, using a fast multipole method).

[0118] Furthermore, in an embodiment, further improvements in computation time can be obtained by completely ignoring even more distant features. In this approximation, only local interactions exist. The distance at which interactions are ignored can be specified as a tolerance for the relative magnitude. As a relative magnitude, the ratio of the velocity at a given distance to the velocity at a distance equal to the feature size can be taken. For example, if a tolerance of 14% is set for the relative error due to this approximation and a tolerance of 5% is set for the relative error due to the point spacing, the total relative error is below 20%. The cutoff distance for each feature is calculated as the distance at which the relative magnitude of the multipole expansion of the feature drops below the tolerance. This gives a computation time scaling that is linear in terms of the number of features and in terms of the number of points per feature. In other words, FLOPS is approximately nm.

[0119] After the velocity field is calculated, the final deformation is determined at the desired location. The desired location can be any point in the resist, a point on the contour of the desired feature, or multiple points on the desired contour. In an embodiment, the final position at any time "t" is determined based on the integral of the velocity field at or around the desired location. For example, the final displacement can be determined using the displacement equation 11 as follows:

[0120]

[0121] In the above equation, is the displacement of point i at time t, and u is the velocity field determined, for example, using Stokes neutrons, as discussed earlier. In an embodiment, the velocity field at time t0 (ie, the start time) and t e (i.e., the end time of the simulation) In an embodiment, input (e.g., in Figure 4A 500 in the figure), the velocity field (e.g., 510, 520) can be represented as a pixelated image. Thus, in an embodiment, the pixel value at any position at time t0 can be determined by interpolation. The pixel value should be constant along a space-time curve (or a space-time curve) that is described by points moving with the fluid (e.g., resist). Thus, point i can correspond to a pixel value, and the displacement of the pixel corresponds to the displacement of the resist.

[0122] In an embodiment, the characteristic shapes are obtained from the image by a contour extraction algorithm. In an embodiment, it may be desirable to change the value of the contour without re-doing the force and velocity calculations. Thus, in an embodiment, the input image may be calculated according to the 2D Stokes flow after deformation. This reverse calculation from the deformed input image also allows for further analysis that may be required. For example, it may be desirable to have the option of calculating only a portion of the image, such as the immediate neighborhood of the deformed contour, such as Figure 7 As shown.

[0123] exist Figure 7 , initial feature shapes 703 and 705 (rounded elongated rectangular features) are shown for reference purposes, and deformed features 713 and 715 correspond to initial features 703 and 705. Figure 4A and 5 The resist deformation model and simulation process discussed above obtains the deformed features 713 and 715 from the simulation of the forces and velocities corresponding to the initial features 703 and 705. In an embodiment, not only the deformation of the initial features 703 and 705 is determined, but also the deformation of the immediate vicinity of the resist represented by several instances of 721 can be determined, for example, based on the forces and velocities acting on the contours of the deformed features 713 and 715.

[0124] In an embodiment, the deformation of the entire resist domain may be obtained, for example, using the position information and the above displacement equation 11. Based on equations (eg, equations 2 to 11), the deformation may be determined for each pixel of the velocity field, thereby resulting in a deformation of the entire resist domain.

[0125] Those skilled in the art will appreciate that the above methods and examples are explained relative to 2D Stokes flow for conveying concepts. However, the above methods are not limited to 2D Stokes flow, and any other flow can be used to represent the fluid dynamics model, and the relevant boundary conditions can be applied to determine the force and velocity field. In an embodiment, the 2D Stokes flow driven by capillary force can be local in nature for the desired feature, so for any desired accuracy, since the feature edge is farther than this distance, there is a distance at which the velocity field can be ignored. For example, the fluid dynamics model can be based on Hull-Shaw flow, which can provide a more global solution. In Hull-Shaw flow, the velocity flow is solenoid-shaped, which is useful for obtaining the velocity of the force point generated by its own force. The basic solution of (depth average) Hull-Shaw flow is only a replacement of Stokes. Thus, the velocity response caused by the point force of Hull-Shaw flow can be determined by replacing the function G (e.g., of equation 2) associated with the 2D Stokes flow with a different function G associated with the Hull-Shaw flow.

[0126] In the embodiments, reference Fig. 8A and 8B , a method 800 for determining resist deformation and further applying a resist deformation model to adjust parameters of the patterning process is provided. In an embodiment, the resist deformation model may be obtained and simulated to determine the deformation.

[0127] In process P82, the method involves: obtaining a resist deformation model 801 of a resist having a pattern. In an embodiment, the resist deformation model is configured to simulate fluid flow of the resist due to capillary forces acting on a feature profile of the pattern. Further, in an embodiment, an input pattern 803 (e.g., Figure 3C In the embodiment, some other quantity indicating the presence of a feature at the location is thresholded to obtain a feature boundary. Examples of other quantities include, but are not limited to, a red component of a color image, a blue component of a color image, a green component of a color image, a hue or chroma of an image, a saturation of an image, a pixel value of an image, etc. The input pattern 803 may also be an output from a convolution filter or another edge detection filter.

[0128] In an embodiment, for example as discussed above with respect to Figure 4A and 5 As discussed above, a resist deformation model may be obtained. In an embodiment, the resist deformation model is a fluid dynamics model. In an embodiment, obtaining the resist deformation model involves generating the model, including defining a fluid dynamics model and defining boundary conditions, such as in Figure 4A and 5In an embodiment, obtaining the resist model involves receiving the resist model via a network. In an embodiment, the resist model may be received from a database, or the process may be configured to communicate with another process that implements the resist model. In an embodiment, the fluid dynamics model is based on linearized Navier-Stokes flow equations, as in Figure 4A and Figure 5 For example, fluid flow can be characterized as 2D Stokes flow (an example of Stokes flow) or Hull-Shaw flow.

[0129] In process P84, the method involves: via a processor (e.g., processor 104) and a simulation of a resist deformation model (e.g., based on equations 2 to 11 and 2D Stokes flow) Figures 4E to 4H The deformation of the developed resist pattern of the input pattern 803 for the resist deformation model is determined by using the corresponding boundary conditions in . Depending on the boundary conditions and the velocity field, the obtained deformation may be different. For example, as discussed below, Figure 4G 4H shows the deformation corresponding to the wall boundary condition, and 4H shows the deformation when the periodic boundary condition is applied. Figure 7 Deformation of a portion of the resist domain is illustrated.

[0130] In an embodiment, simulation also involves the following processes, such as Figure 8B In an embodiment, in process P842, the simulation involves defining initial vertices along a characteristic profile. In an embodiment, the initial vertices may be rearranged to satisfy certain conditions, thereby stabilizing the simulation process. For example, process P844 involves redistributing vertices so that they are evenly spaced while conserving the area or volume of the characteristic profile of the pattern.

[0131] Further, in process P846, the simulation process involves determining the capillary force at a given vertex along the feature profile of the pattern. In an embodiment, the capillary force acting on a given vertex is the sum of the tension forces on either side of the given vertex. For example, equations 3 to 5 may be used to determine the force. In an embodiment, the force may produce a squeeze flow, which is a resist flow caused by a net inward flux or a net outward flux through a vertical resist domain boundary, resulting in a large-scale migration of the feature. This may be achieved by applying a relative Figure 4A Appropriate boundary conditions are discussed to eliminate this squeeze flow (e.g. Figure 6 shown).

[0132] Process P848 involves applying boundary conditions to a fluid dynamics model, for example, to eliminate squeeze flow. Examples of boundary conditions and how to implement such boundary conditions are described earlier in this disclosure with respect to Figures 4A to 4H Discussion.

[0133] Once the forces and boundary conditions are applied, process P850 involves: determining the velocity field 850 of the fluid flow due to capillary forces based on the superposition of the Stokes sub- and boundary conditions ( Figure 5 520 and Figure 7 721 example). In an embodiment, process P850 involves: obtaining a velocity at a given vertex along a feature profile based on the velocities of all other vertices due to capillary forces at the given vertex. This is achieved by multiplying the force at the given vertex with a Stokes subcenter at the given vertex and evaluating it at the other vertices. The velocity at the given vertex is such that the velocity conserves the area or volume of the feature due to the force at the first vertex. In an embodiment, the velocity is in the normal direction of the force.

[0134] In an embodiment, the squeeze flow may be determined by decomposing the velocity field and determining appropriate boundary conditions to eliminate such squeeze flow. For example, process P852 involves decomposing the velocity field 850 into squeeze flows (e.g. Figure 6 600) and higher order velocity flows (not shown). Process P854 then involves eliminating the squeeze flow from the velocity field by applying the boundary conditions. In an embodiment, the boundary conditions include: setting the flow rate across the boundary of the resist to zero; and / or setting the velocity across the boundary of the resist to a no flux condition. As discussed earlier, by providing rotators of appropriate strength at the corners of the boundary of the resist domain (see, for example, Figure 4E ) (e.g. using Equation 10, and as relative to Figure 4E ), then the flow rate into the domain across the vertical boundary is set to zero. In an embodiment, rotators of equal magnitude and alternating sign are placed at the four corners of the resist domain.

[0135] Return to Fig. 8A In process P86, the method may optionally involve: determining forces at local regions within the resist, and using the forces at the local regions to obtain deformation of the entire region of the resist based on simulation of a resist deformation model, such as relative to Figure 7 Discussed.

[0136] In an embodiment, in process P88, the method may involve: performing a simulation for obtaining the resist deformation at the desired time instance via a simulation of the resist deformation model until the desired time instance. In an embodiment, the contribution to the velocity field due to features far from the region where the capillary force is applied is negligible.

[0137] As discussed earlier, the input pattern is provided to the resist deformation model in the form of an image of the input pattern (e.g. Figure 3C). The image may be a binary image. In an embodiment, the input pattern is a design pattern, a resist image, a mask pattern and / or an aerial image. In an embodiment, obtaining the input pattern includes generating the binary image. The binary image may be generated by: obtaining a pattern forming device pattern corresponding to the input pattern; generating an aerial image based on the pattern forming device pattern through simulation of a patterning process; and extracting a boundary of the pattern in the aerial image to generate the binary image.

[0138] In an embodiment, the above method 400, 500 or 800 may also involve: using a resist deformation model to calculate a critical dimension between a pair of locations disposed on a boundary of a developed resist pattern, and calculating an error between the calculated critical dimension and a measured critical dimension of the actual developed resist pattern. Such CD and error values ​​may also be used to perform optimization of a patterning process, such as OPC, mask optimization, source optimization, or a combination thereof.

[0139] In an embodiment, the deformation is determined at a plurality of locations, each location corresponding to a point located on a boundary of a developed portion of a developed resist pattern of the input pattern.In an embodiment, the resist is a chemically amplified or non-chemically amplified negative resist or a positive resist.

[0140] Fig. 9 900 is a flow chart of a method for determining parameters of a patterning process based on a resist deformation calculated according to the deformation model discussed above. In process P92, method 900 involves: obtaining (i) a patterning process model 901 including a resist deformation model of a resist having a pattern (e.g., relative to method Figure 4A and Figure 5 Discussion), and (ii) a target pattern 903 (e.g., a design pattern). The resist deformation model is configured to simulate a fluid flow of the resist due to a capillary force acting on a profile of at least one feature of the pattern. In an embodiment, the resist deformation model is a fluid dynamics model configured to simulate a fluid flow of the resist due to a capillary force acting on a feature profile of the pattern.

[0141] Process P94 involves: determining, via a processor (e.g., processor 104), a resist pattern 904 based on a simulation of a patterning process model 901, wherein a target pattern 903 is used as an input to the patterning process model, wherein there is a difference between the resist pattern and the target pattern. Example input pattern (e.g., Figure 5 An example of a resist pattern within a resist deformation (e.g., 520) of 500 in FIG. Figure 5 and 7 The deformation process based on the force and velocity of 2D Stokes flow is relative to Figure 4A The method 400 is discussed.

[0142] Process P96 involves: determining, via a processor, a value of a parameter of a patterning process based on a simulation of the patterning process, the value of the parameter being determined so that the difference between the resist pattern and the target pattern is reduced. In an embodiment, the parameter of the patterning process includes at least one of: dose, focus, optical proximity correction. For example, a resist deformation model may be included in a resist model of a resist process of the patterning process. Using such a resist model to simulate the patterning process may allow optimization of the patterning process, including determining optimal values ​​of parameters such as dose, focus, optical parameters, OPC, etc. The optimization process may involve reducing a cost function including the difference between the resist pattern and the input pattern.

[0143] Process P98 involves: during the patterning process, applying the values ​​of the parameters of the patterning process to the lithographic equipment. The wafer printed based on this parameter value can also be measured (for example, using an optical tool or SEM) and used to verify the results of the deformation model. For example, an SEM image of the printed wafer is obtained via a metrology tool. The SEM image can also be processed to determine the resist pattern, which can be compared with the deformation obtained in the expected instance of the simulation of the deformation model (for example, the EPE between the patterns of the simulated image and the SEM image). Based on the comparison, the fluid dynamics model can be verified or modified. For example, additional errors or penalty terms can be included in the deformation model. The error or penalty term can explain this difference in the simulation and actual measurement values ​​of the substrate.

[0144] Although the embodiments have been described in terms of a binary image as a resist pattern image, in embodiments, the input resist pattern image may be grayscale and / or the output deformed resist pattern image may be grayscale. Further, although the embodiments have been described with respect to the use of images, it is to be understood that the resist pattern may be more generally characterized with respect to data (such as CD, coordinate position, vector, etc.), and thus the input resist pattern data and / or the output deformed resist pattern resist deformation data may be in a non-image form, such as CD values, coordinate positions, vectors, etc.

[0145] In an embodiment, by employing binarization of the resist pattern image data, the fluid dynamics model may be relatively fast. Additionally or alternatively, simplification of the Navier-Stokes equations towards a full chip solution for faster applications may be accomplished by discretizing the equations and representing them by a sum of kernel functions.

[0146] Therefore, in summary, the influence of fluid internal forces (such as surface tension) on resist patterns is included in the prediction of resist patterns using computational fluid dynamics models. Data about resist patterns (such as optical images (or images derived therefrom) produced using, for example, ASML's Tachyon products) are used as inputs to the model. In an embodiment, in order to accelerate processing, the region and the remainder assumed to be fully developed are binarized. The remaining resist is regarded as a fluid on a two-phase flow field of laminar flow, and has a constant or non-constant viscosity. Then, the fluid internal forces are substantially modeled, such as surface tension applied on the developed resist boundary. The model determines, for example, fluid velocity and pressure, and therefore the deformation of the resist is calculated by solving fluid dynamics equations (such as Navier-Stokes equations). Thus, fluid dynamics is used to realize resist distribution prediction, and the fluid dynamics model can effectively include, for example, surface tension effects. Specifically, in an embodiment, strain and curvature effects of highly dense patterned shapes are captured by using fluid dynamics.

[0147] In order to adapt the model to a specific patterning process where the values ​​of physical and material parameters are unknown (e.g., viscosity, density, etc.), the model can be fitted (e.g., by regression) to measured values ​​on actual deformation features (e.g., experimental CD values ​​in the X and Y directions).

[0148] Thus, in an embodiment, a method of determining resist deformation is provided that balances accuracy and speed to allow resist deformation prediction to be relatively easily integrated into existing algorithms for patterning process configuration. For example, in an embodiment, a fluid dynamics model is provided that can provide better predictions of the deformation of a resist pattern than past modeling. Additionally or alternatively, the fluid dynamics model can provide faster deformation predictions than past modeling.

[0149] During the development phase, a solvent diffuses into the resist 1010 or features therein, which softens the outer layer of the resist (e.g., 1020), such as Fig. 10A and 10B One way to model this behavior is based on the assumption that (i) capillary and viscous flow effects dominate and (ii) a continuous description within the resist and / or liquid layer associated with the resist holds.

[0150] Capillary forces cause this outer layer 1020 to deform elastically or via viscous flow until the solvent has evaporated again. Since both deformation processes are approximately governed by similar equations for small deformations, the flow of the thin layer is considered to be a viscous flow. According to an embodiment, the thin liquid layer 1020 of the resist 1010 around the feature boundary (also referred to as the boundary liquid layer 1020) will be driven by capillary forces to flow. Since the feature boundary and the liquid layer around the feature are related to each other, and any deformation in the liquid layer will result in a change in the shape of the feature, the terms "boundary liquid layer", "thin film" and "contour" or "boundary" of the feature may be used interchangeably herein. Therefore, the deformation of the boundary liquid layer refers to the deformation of the boundary / contour of the feature. In an embodiment, the features may be defined by their position at a specified height H relative to the substrate surface (e.g., as Fig. 10B shown) (e.g. Fig. 10A In Chinese, it is indicated.

[0151] In an embodiment, the fluid flow of the boundary liquid layer 1020 is decomposed into two components: horizontal flow and vertical flow, respectively. Fig. 10A and 10B Thus, the deformation of the boundary liquid layer (and / or features associated therewith) caused by the fluid flow may be the result of a first deformation component caused by horizontal flow components (e.g., 1011, 1012, 1013, 1015, and 1016) and a second deformation component caused by vertical flow components (e.g., 1021, 1022, 1023). In an embodiment, the boundary of the feature may have a net inward motion at a given height (e.g., H) above the substrate.

[0152] like Fig. 10A As shown, horizontal flow components (e.g., 1011, 1012, 1013, 1015, and 1016) in the horizontal plane act along the boundaries of the layout or geometric shape (e.g., line or circle) of the features. The horizontal flow components cause a change in this shape of the boundary liquid layer 1020, but do not cause a change in the total area of ​​the features. In an embodiment, the horizontal flow depends on the curvature of the features in the horizontal plane.

[0153] like Fig. 10B As shown, the vertical flow component in the vertical plane acts along the shape of the cross section in the vertical plane of the boundary liquid layer (or the feature associated therewith). The vertical flow can change the total area. The vertical flow depends on the curvature in the vertical direction.

[0154] Thus, the resist deformation model (eg, a thin film model) is configured to determine a first deformation due to a horizontal flow component of the fluid flow, and also adjust the first deformation to account for a second deformation due to a vertical flow component.

[0155] In an embodiment, the resist deformation model is configured to capture different aspects of horizontal flow and vertical flow components (e.g., flow rate, deformation, conservation of mass and / or volume, etc.). In addition, according to the thin film approximation, the deformation of any feature is independent of the deformation of other adjacent features. This results in the computation time being linear in the number of features, which is the best possible scaling. Finally, the direct implementation of the model is implemented in parallel very well because any point (along the contour of the feature) only interacts with its nearest neighbor and the next nearest neighbor. In addition, the resist deformation model is configured to determine the deformation of the feature (caused by contraction) based on the lubrication approximation or asymptotic approximation. In the lubrication approximation or asymptotic approximation, vertical shear stress is ignored. If the deformation only changes gradually in the horizontal direction at a distance much larger than the layer thickness, the vertical shear stress can be ignored.

[0156] Fig. 10C Depicted is a schematic top view of a boundary liquid layer adjacent to a feature in a resist according to the present disclosure. Fig. 10C 9 is a top view of a model of an exposed resist with a defined boundary liquid layer (i.e., a gel-type layer). In this model, a boundary liquid layer 930 with a width 950 (also referred to as layer thickness δ) is defined for developed or open regions 910 (e.g., trenches). Developed or open regions 910 (such as trenches) between portions of the resist 900 are formed by development. In this example, regions 910 will have gas therein.

[0157] The resist 900 is specified to deform very little or not at all by being specified as a solid with a relatively high elastic modulus or a liquid with high viscosity (e.g., 30% or greater, 50% or greater, 75% or greater, 100% or greater, 200% or greater, 500% or greater, or 1000% or greater) compared to the boundary liquid layer 930. In fact, it may not even be necessary to specify the resist 900 in the model, and instead, boundary conditions are applied to the boundary liquid layer 930, which is equivalent to specifying adjacent areas of high viscosity or adjacent areas that cannot undergo any or much deformation. Thus, the boundary liquid layer 930 exhibits all or most of the deformation into the region 910. In this example, the extension of the boundary liquid layer 930 on top of the resist 900 is not shown. However, all or part of the resist 900 may be covered by the boundary liquid layer 930.

[0158] The resist deformation model is characterized by a boundary liquid layer 930 located at the boundary between the developed or open area of ​​the resist pattern and the resist 900. The boundary liquid layer 930 has a width that is smaller than the width of the resist at the boundary, and the parameters of the boundary liquid layer or parameters associated with the boundary liquid layer can be varied according to appropriate boundary conditions at the boundary of the feature (at the open area) and at the side of the boundary liquid layer opposite to the developed or open area. For example, the boundary conditions applied to the boundary liquid layer can be equivalent to specifying adjacent areas of high viscosity or adjacent areas that cannot undergo any or much deformation.

[0159] In an embodiment, under the assumption that capillary and viscous flow effects dominate, the resist deformation model is developed. In addition, the resist deformation model is developed under the assumption that continuous description is established. Therefore, the applicability of the model to physical results is limited to length scale and / or time scale (such as width, the period of simulation of the deformation, etc.). In an embodiment, the width can be a predetermined amount, such as specified by a user and desirably selected so that it is greater than the maximum expected deformation towards the substantially insoluble resist. In an embodiment, the width is selected from the range of 5nm to 300nm, the range of 5nm to 200nm, the range of 5nm to 100nm, the range of 5nm to 50nm, the range of 10nm to 40nm or the range of 5nm to 30nm or the range of 5nm to 20nm. In this embodiment, the width can not be changed as a part of the calibration routine. Or, in an embodiment, the predetermined width can be a starting point, and the width can change as a parameter of viscosity or a parameter associated with viscosity, etc.

[0160] In an embodiment, the width may be a range (such as the range just described above) which may act as a constraint. Thus, in such an embodiment, a certain width may be a starting point, and the width may vary with parameters such as viscosity or associated with viscosity, but is constrained within a given range.

[0161] In an embodiment, the width may vary at different locations along the boundary liquid layer. For example, the width may be different in different developed or open areas or at different portions along a developed or open area. As another example, the width may be different at the top surface of the resist (e.g., in a top view) but not in the sidewalls of the developed area (e.g., in a side view). In an embodiment, a relationship (e.g., a ratio) between the sidewall width and the top surface width may be specified to constrain the difference in width.

[0162] Thus, in embodiments, when short-range interaction of forces is desired, a region of liquid material having a finite (and relatively small, such as 30 nm or less) width may be confined to the boundary of the developed or open region and the resist, the region at the boundary having a smaller width than the resist at the boundary. On the opposite side of the developed or open region, the model may have another material, such as a solid, that does not deform, or deforms significantly less than the material of the region having a finite width (e.g., has a significantly higher viscosity than the material of the region having a finite width), or has boundary conditions equivalent to specifying low or no deformation at that location. Thus, the velocity field at the opposite side may be zero or close to zero, resulting in little or no change in deformation at that location (and at locations outward therefrom until another region of finite width is encountered).

[0163] Fig.11 The invention depicts the use of, for example, pattern information (e.g., contours extracted from aerial images, as described earlier with respect to Figure 3C Specifically, Fig.11 Depicted are (i) an initial profile Mi (e.g., input profiles Mi1, Mi2, and Mi3 extracted from AI), (ii) a deformed profile M1o (including profiles 111, 1112, and 1113) obtained from a full fluid dynamics model (i.e., where the entire resist is considered as a liquid rather than a thin layer of resist), and (iii) another deformed profile M2o (including profiles 1121, 1122, and 1123) obtained from a resist deformation model according to the present disclosure.

[0164] exist Fig.11 , the deformation profiles 1111 and 1113 obtained from the full fluid model (which are part of the output M1o) predict that (e.g., M1o) the profiles deform outward into the resist 1010 relative to the corresponding input profiles Mi1 and Mi2, respectively, and the deformation profile 1112 deforms inward relative to the corresponding input profile Mi3. Such deformation profiles 1111, 1112, and 1113 are due to the interaction between the deformations of the individual profiles Mi1, Mi2, and Mi3, since the entire resist 1010 is assumed to be fluid. In other words, the deformation of one feature (or its profile) may result in the deformation of an adjacent feature (or its profile).

[0165] The deformation profiles 1121 and 1123 obtained from the thin film fluid model (of the present disclosure), which are part of the output M2o, predict that (e.g., M2o) the profiles are deformed inwardly relative to the corresponding input profiles Mi1 and Mi2, respectively, and that the deformation profile 1122 is not deformed or is slightly deformed relative to the corresponding input profile Mi3. In such a model, the deformation of one feature is independent of another adjacent feature, because it is assumed that only the thin layer around the feature will deform due to the fluid flow, while the resist 1010 area away from the thin layer is not deformed or is only negligibly deformed. It can be seen that there is a relatively slight but significant deformation at the curved portion of the profile such as 1121 relative to the input profile Mi1, and it is very different from the deformation of the profile 1111 at the similar curved portion of the input profile Mi1.

[0166] In an embodiment, the resist deformation model includes formulas related to the thin film model, such as (i) a momentum equation simplified according to the lubrication approximation, (ii) conservation of mass and volume, and (iii) boundary conditions. The following description provides example formulas for the resist deformation model to determine the deformation caused by the horizontal flow component and the vertical flow component of the fluid flow of the boundary liquid layer.

[0167] In an embodiment, assuming that the viscosity of the boundary liquid layer is constant, the horizontal flow component of the fluid flow of the boundary layer (e.g., 930) can be determined based on the pressure-driven lubrication flow. The horizontal flow is determined while conserving the volume of the feature and the volume of the boundary liquid layer. In the horizontal flow, the curvature of the feature profile imposes constraints on the way deformation occurs. In an embodiment, the deformation calculation is performed on multiple vertices along the feature profile, each vertex is a location on the feature profile. The calculation also involves: determining the Jacobian of the time derivative of the deformation based on a finite difference method.

[0168] Further, when the concave portion of the contour moves, a plurality of vertices may overlap, which is undesirable. Thus, redistribution of the plurality of vertices is performed to adjust the spacing between the vertices.

[0169] In an embodiment, the pressure at the apex may be determined using the following equation.

[0170]

[0171] In the above equation, i is the i-th vertex among multiple vertices; The line between the nearest vertices perpendicular to the i-th vertex; r ci is the average distance to the nearest neighbor of the i-th vertex, where if x is the position of the vertex, then r ci Depend on given; and is the force at the ith vertex. In an embodiment, the force may be determined using, for example, a Stokes-based model, without using a full fluid model, but only using the previously discussed procedure to determine the force.

[0172] According to the present disclosure, the fluid flow rate depends on the viscosity μ, the boundary liquid layer thickness δ, and the pressure gradient in the horizontal plane

[0173] To calculate the flow rate, the velocity u is integrated over the layer thickness δ. According to the present model, the viscosity does not depend on the shear rate or shear strain rate of the strain, i.e., the boundary liquid is a Newtonian liquid. The viscosity may depend on the height H of the resist. For a simple explanation of these concepts, it is assumed that the viscosity is constant throughout the entire layer thickness. However, the viscosity may vary.

[0174] The velocity can be determined based on the momentum equation in the lubrication approximation and the boundary conditions as described by the following system of equations:

[0175]

[0176] u=0 at y=0…(14)

[0177]

[0178] In the above system of equations, y is the coordinate in the normal direction, i.e., the distance from the solid surface (i.e., the resist) surrounding the feature, and x is the tangent coordinate. In an embodiment, the velocity gradient is set to The dynamic boundary condition of is zero at the free surface (e.g., the outer edge represents the surface of the boundary layer at the contour of the feature). Further, the kinematic boundary condition is set so that the velocity u is zero at the solid (e.g., the inner edge represents the surface of the boundary layer at the side opposite to the contour of the feature).

[0179] As the pressure changes, the above equation may cause the layer thickness to take zero or negative values, which may make the simulation unstable. Therefore, the layer thickness is determined halfway between the points i-th vertex and i+1-th vertex as follows: δ = min(δ i ,δ i+1 )

[0180] Furthermore, the pressure gradient Located between the points i-th vertex and i+1-th vertex (e.g. midway).

[0181] The speed can then be calculated using the following equation:

[0182]

[0183] Further, the integral of the velocity gives the flow rate Q of the boundary layer in the horizontal plane (eg, equation below). This formula gives the flow rate Q from one vertex to the next, depending on the pressure gradient between these vertices and the layer thickness.

[0184]

[0185] In addition, the time derivatives of the state variables including the layer thickness are determined at each time step. For example, d t δ=(Q i+1 / 2 -Q i-1 / 2 ) / r ci In an embodiment, the time derivative of the thickness may be integrated to determine the thickness at any given point.

[0186] Time derivative calculation involves the calculation of the Jacobian determinant of the time derivative for efficient calculation. The time derivative of a point (e.g., the i-th vertex) depends on the position of its nearest neighbor and the next nearest neighbor. This Jacobian determinant is a five-diagonal matrix, and is therefore very sparse. The Jacobian matrix can be calculated in a variety of different ways. In an embodiment, the five diagonals refer to the main diagonal with non-zero elements (i.e., the longest diagonal of the matrix, such as through (1,1) to (n,n) in the case of a square matrix) and the two diagonals with non-zero elements adjacent to the main diagonal, and the remaining elements of the Jacobian matrix are zero. At first glance, automatic differentiation (e.g., via a predefined ODE solver) can be used. However, this may make the code more difficult to read and less portable. For these reasons, automatic differentiation is not required. In another embodiment, the Jacobian matrix can be calculated analytically. However, analytical calculation requires configuration and reconfiguration of control equations and their derivatives, which is very time-consuming and error-prone. On the other hand, according to the present model, the resulting Jacobian is sparse, which allows the use of finite differences. The finite difference method provides faster computation time and less cost, since it only requires, for example, 6 evaluations of the derivative when a vertex and its four neighboring vertices are used to calculate the deformation.

[0187] A further consideration in determining deformations due to horizontal flow involves concave corners or rounded edges. In a concave corner, vertices will approach each other. Eventually, they will meet, i.e., intersect. When this happens, the thin film approximation may become invalid because the radius of curvature should be large relative to the layer thickness. When the vertices intersect, the radius of curvature is zero. Therefore, when the vertices coincide, the normal direction is not defined, resulting in simulation instability. To alleviate this problem, multiple vertices are continuously redistributed so that they cannot coincide. The present disclosure is not limited to continuously redistributing vertices, i.e., at each time step, and those skilled in the art may also perform the redistribution at discrete time intervals.

[0188] In an embodiment, the redistribution of vertices may be defined on a time scale that is at least approximately the magnitude of the time scale of the layer thickness evolution. Using continuity, the reduction in layer thickness per unit time may be determined as the derivative of the flow rate with respect to the tangent coordinate. The time scale thus obtained may be different for different vertices.

[0189] Further, it is determined how to redistribute the vertices, and how to change the layer thickness for each vertex. The example redistribution is based on the following three specifications. First, the redistribution should conserve the total volume of the feature. This is achieved by moving any vertex only in a direction parallel to the line between the nearest neighbor vertices. Second, the redistribution should converge, that is, it should not have any growth pattern. This can be achieved by using diffusion of the vertex spacing as the redistribution, where the velocity of each vertex is proportional to its distance to the line halfway between the nearest neighbor vertices. The proportionality constant determines the time scale of the redistribution. Third, the volume of the boundary liquid layer should be conserved. This can be achieved by evaluating the geometric representation of the boundary liquid layer.

[0190] Reference Fig. 12A , the boundary liquid layer is represented by layer thickness and vertex position. The inner edge BLi between the boundary liquid layer and the solid part (i.e., resist) is drawn with a dotted line. And the outer edge BLo is drawn with a solid line, and dots 1208, 1209, 1210, 1211, and 1212 are sample vertices. For the middle vertex 1210 (e.g., the i-th vertex), the center line, tangent 1230, and the direction of motion are shown. In each vertex on the outer edge (i.e., at the contour of the feature), position and layer thickness are state variables. Using redistribution, normal vectors and tangent vectors are obtained from the instantaneous positions of the outer edge BLo vertices 1208, 1209, 1210, 1211, and 1212.

[0191] Redistribution moves external vertices (i.e., vertices on the outer edge BLo of the boundary layer) in the direction of the line between adjacent external edge vertices. If the normal direction does not change, the internal edge vertices (i.e., vertices along the internal edge BLi) (not shown) will move in the direction of the line between adjacent internal edge vertices. In order to take into account the changes in the normal direction, an additional change in the layer thickness must be applied. When an external vertex moves in the tangential direction, this will cause the corresponding internal vertex and its nearest neighbors to move in the tangential direction of the corresponding external vertex. The change in layer thickness must be such that their direction of movement is in the tangential direction of these internal vertices. In an embodiment, redistribution is achieved based on the following time scale equation below, where Δx is the distance between consecutive or coherent vertices, and λ is the spatial period of 2Δx.

[0192]

[0193] In an embodiment, the velocity in the interior normal direction should be zero so that the volume of the boundary liquid layer is conserved, which yields the following equation:

[0194]

[0195] In the above equation, is the external tangent velocity of the internal vertex; refers to the external tangent direction; is the internal normal direction; and = refers to the external normal direction. Equation 19 above gives the time rate of change of the layer thickness in the case of redistribution of vertices, which conserves the volume of the boundary liquid layer. This time rate of change of the layer thickness conserves the volume when the external vertex moves in the external tangent direction. This time rate of change of the layer thickness should be added to the time rate of change of the layer thickness due to the flow, which causes the external vertex to move in the external normal direction.

[0196] Furthermore, when the neighboring outer vertices move, this changes the normal direction. The time derivative of the normal vector depends on the velocity of the neighboring vertices and on the distance between the neighboring vertices. This is because the neighboring vertices determine the tangent direction. Since the normal vector can change during the simulation, it should be considered in the changes due to the flow. The flow of the material creates a velocity component of the outer vertices in the outer normal direction. This changes the normal direction of the neighboring vertices, which changes the inner tangent velocity.

[0197] The end of the redistribution process results in a first displacement component due to the horizontal flow of the boundary layer. Further, the first displacement component is adjusted to include a second displacement due to the vertical flow of the fluid flow.

[0198] In an embodiment, the vertical direct current component of the fluid flow of the boundary layer (e.g., 930) can be determined as follows. The calculation of the vertical direct current is assumed to be independent of the horizontal flow for modeling purposes. Thus, in an embodiment, deformations due only to horizontal flow, only to vertical direct current, or a combination thereof can be determined. In an embodiment, the vertical direct current component causes an additional component of the time derivative of the layer thickness, which is used to determine the deformation of the boundary layer. The deformation is calculated in the lubrication approximation, where vertical shear stresses are ignored. If the deformation only changes gradually in the horizontal direction over a distance much larger than the layer thickness, the vertical shear stress can be ignored.

[0199] In an embodiment, the vertical flow is determined based on a resist shrinkage model. Fig. 12B An example nomenclature for defining the vertical flow of the resist deformation model is illustrated, and the shrinkage model is defined as follows:

[0200]

[0201] In the above equation and referring to Fig. 12B , y is the coordinate in the normal direction, that is, the distance from the solid surface of the feature; x is the tangent coordinate; is the time derivative of the layer thickness due to horizontal flow, δ is the layer thickness, κ is the volume contraction; and h is the height of the feature.

[0202] To calculate the vertical flow, the shape of the side surface (e.g., Fig. 10B ), and a model of the resist deformation due to shrinkage is applied based on a lubrication approximation. Further, it is assumed that the deformation is incompressible. In an embodiment, the deformation is a two-dimensional deformation, which would be a good approximation of the deformation of a line feature, for example. However, this formula is not limited to a specific shape of a feature and can be extended for any feature shape.

[0203] The resist deformation model in vertical flow determination is based on a formula where the stress τ i,j The divergence of vanishes in the material of the feature because there are no body forces acting on the material. All forces act on the boundary. Therefore, elastic formulas describing the displacements in the boundary liquid layer can be employed, where stresses and strains are eliminated from the formulas. For example, the linear elastic formula defined by the Navier-Cauchy equations can be used with the above assumptions to determine how the boundary liquid layer deforms due to the vertical flow.

[0204] In an embodiment, the Navier equations are used to obtain the governing equations for horizontal displacement only (in the x-direction), given as follows:

[0205]

[0206] In addition to the above equations, in an embodiment, the boundary condition at the top y = δ is that the stress should vanish because there is no material beyond the feature edge to pull. Also, the boundary condition at the bottom y = 0 is that the displacement should vanish. Fig.13 An example of a deformed shape of a cross section of the resist in a vertical plane is shown in FIG. Fig.13 In FIG. 1 , edges 1301 , 1303 and 1305 are away from the resist and deformed due to the vertical flow discussed above. It can be seen that according to the boundary conditions, no deformation occurs at the edges at the solid part or the resist.

[0207] Fig.14A14 is a flow chart of an example method 1400 for determining a deformation of a pattern to be formed in a resist during a patterning process based on a thin film-based fluid model simplified according to a lubrication approximation. In an embodiment, a first deformation of the feature is simulated based on a fluid flow in a first plane (e.g., a horizontal flow). Further, the first deformation is further deformed (referred to as a second deformation) based on a fluid flow in a second plane (e.g., a vertical plane) to account for the vertical flow discussed earlier. The result is a pattern deformation that takes into account the deformation due to the horizontal flow and the vertical flow.

[0208] In process P141, method 1400 involves: inputting pattern information 1401 about a pattern to be formed into a resist deformation model, the model being configured to simulate deformation of a portion of the resist, the portion including a boundary liquid layer (e.g., a layer of liquid at a boundary between a developed region in the resist and a region of the resist surrounding the developed region) Fig. 10C 930 in the table).

[0209] In an embodiment, the pattern information 1401 is in the form of an image including (multiple) contours of (multiple) features as input. In an embodiment, the contours are from above relative to Figure 3B and 3C The extracted contours (eg, in 320) are extracted from the discussed aerial image (eg, in 310). The extracted contours may be a grayscale image or a binary image. In an embodiment, the extracted contours refer to points where forces due to surface tension will be applied.

[0210] Process P143 involves determining a first deformation 1410 based on the horizontal flow discussed earlier. In an embodiment, determining the deformation further comprises defining a plurality of vertices (e.g., at Fig. 12A and 15 Determine the capillary force at a given vertex along the contour and redistribute the vertices so that (i) the volume of the developed regions in the resist is conserved, (ii) the redistribution should converge, and (iii) the volume of the boundary liquid layer is conserved. An example redistribution process is described earlier, as with horizontal flow and Fig. 12A Related.

[0211] For example, determining the deformation due to horizontal flow involves: determining the force at each vertex (contour of the feature) of the boundary layer The velocity and flow rate at each vertex of the profile and redistribute the vertices to obtain a first displacement of the vertices at the edge (at the resist) opposite to the developed area on the boundary layer of the developed area (ie, the profile of the feature). Fig. 14B is an example flow chart of a method for determining deformation 1410 based on horizontal flow.

[0212] exist Fig. 14B In the process of using pattern deformation 1401 to determine the first deformation is an iterative process involving processes P1431, P1433, P1435 and P1437. Process P1431 involves: using, for example, equations 3 to 5 discussed earlier to determine the forces at the vertices of the plurality of vertices Force at the vertex It is also used to calculate the pressure at the vertex using, for example, Equation 12 discussed above. Further, process P1433 involves: determining velocity and flow rate based on pressure gradient and layer thickness. The pressure gradient at the vertex is determined by the pressure in equation 12, and is also used to determine the velocity discussed relative to equations 13 to 16. Based on the velocity, the flow rate from one vertex to another vertex can be determined using equation 17, as discussed above. Process P1435 involves: redistributing multiple vertices based on a time scale, such as defined by equation 18, and three criteria, the three criteria being: (i) the total volume of the feature is conserved, (ii) the redistribution should converge, and (iii) the volume of the boundary liquid layer should be conserved, as discussed above. The redistribution involves: determining the time scale based on the horizontal component of the flow rate of the boundary liquid layer, and moving the vertices based on the time scale.

[0213] In addition, process P1437 involves determining the first deformation 1410 using, for example, a time derivative of the flow rate (eg, Equation 17). In an embodiment, process P143 also involves determining a rate of change of the thickness of the boundary liquid layer (eg, d ) based on the flow rate of the boundary layer. t δ=(Q i+1 / 2 -Q i-1 / 2 ) / r ci )); and based on the redistribution of vertices (e.g. Fig.15 1521 in ), adjusting the other multiple vertices along the free surface (e.g., Fig.15 1522 in the above).

[0214] In process P145, the deformation 1410 due to horizontal flow is adjusted based on the vertical flow. Process P145 involves determining a second deformation 1420 based on the vertical flow (e.g., using equations 20 and 21), as discussed earlier. The calculation of the vertical flow is independent of the horizontal flow. In an embodiment, the second deformation is determined using a contraction model that includes the time derivative of the layer thickness due to the horizontal flow (e.g., 1410) and the height of the feature relative to the substrate. The governing equations cause the divergence of stress to be zero. Further, a lubrication approximation is adopted in which vertical shear stresses are ignored. These assumptions lead to a simplified version of the Navier equations (e.g., equation 21), based on which the velocity and deformation 1420 of the boundary liquid layer and the features associated therewith can be determined.

[0215] Fig.15 An example of deformation of an input profile 1501 (e.g. extracted from an aerial image, as discussed earlier) based on a resist deformation model (e.g. thin layer based using a lubrication approximation) is shown. Fig.15 , the input profile 1501 is also surrounded by another profile 1502 of width δ, thereby forming a boundary liquid layer 1510 having a width δ. Therefore, with respect to employing the resist deformation model discussed above, the profile 1501 is used as a profile at the feature, and the other profile 1502 is used as a profile away from the feature (i.e., at the resist assumed to be solid). Further, a plurality of vertices are defined on the profiles 1501 and 1502, and the deformation of the boundary layer is determined relative to the plurality of vertices. In an embodiment, the plurality of vertices on the profile 1501 move according to the velocity and fluid flow equations discussed above, and the movement of the other plurality of vertices of the other profile 1502 is also determined based on the pressure gradient across the boundary layer and the width of the boundary layer.

[0216] In an embodiment, based on the horizontal flow formula, the deformation of the vertices along the profile 1501 is determined, as discussed above, for example, in method 1400 (particularly in process P143). Applying the formula discussed in process P143 results in first deformations 1521 and 1521 of the profiles 1501 and 1502, respectively. Thus, a deformed boundary layer 1520 due to the horizontal flow is obtained.

[0217] Further, the deformation of the vertices on the deformed profiles 1521 and 1522 is determined based on the vertical flow formula, for example as discussed above in process P145. Applying the vertical flow formula results in deformed profiles 1531 and 1532, as shown. Thus, a final deformed boundary layer 1530 is obtained that takes into account the deformation caused by the vertical flow.

[0218] In an embodiment, a method for determining deformation of a resist during patterning is provided. The method includes: obtaining a resist deformation model of a resist having a pattern, the resist deformation model being configured to simulate fluid flow of the resist due to capillary forces acting on a profile of at least one feature of the pattern; and determining, via a processor and the resist deformation model, deformation of a resist pattern to be developed based on an input pattern of the resist deformation model.

[0219] In an embodiment, the resist deformation model is based on the linearized Navier-Stokes flow equations.

[0220] In an embodiment, the fluid flow is characterized by Stokes flow and / or Hull-Shaw flow.

[0221] In an embodiment, the fluid flow is characterized by a fundamental solution that decays at a rate of at least 1 / R with distance to a specific location in the resist and is either regular in the resist or singular only at the specific location in the resist.

[0222] In an embodiment, determining deformation involves: defining multiple vertices along a profile of at least one feature; determining capillary forces at a given vertex along the profile of at least one feature; and determining a velocity flow field of the fluid flow due to capillary forces based on a superposition of velocity responses at one or more of the multiple vertices along the profile of at least one feature and boundary conditions, wherein the velocity response is due to capillary forces acting on vertices adjacent to the given vertex.

[0223] In an embodiment, defining the plurality of vertices includes distributing the plurality of vertices to be evenly spaced while conserving an area or volume of a contour of the at least one feature.

[0224] In an embodiment, the capillary force acting on a given vertex is the sum of the tension forces on either side of the given vertex. In an embodiment, the velocity response is characterized by Stokes sub-fields, where Stokes sub-fields are velocity fields due to point forces in Stokes flow.

[0225] In an embodiment, determining the velocity flow field further involves: decomposing the velocity field into a squeeze flow and a higher-order velocity flow; applying a boundary condition corresponding to the squeeze flow; and removing the squeeze flow from the velocity field based on the boundary condition.

[0226] In an embodiment, squeeze flow is resist flow due to a net inward flux or a net outward flux through a resist domain boundary, resulting in large-scale migration of features.

[0227] In an embodiment, applying the boundary condition involves: setting a flow rate through a boundary of the resist domain to zero; and / or setting a velocity on a boundary of the resist domain to a no flux condition.

[0228] In an embodiment, by providing rotators of appropriate strength at one or more corners of the boundary of the resist domain, the total inward flow rate through the resist domain boundary is set to zero.

[0229] In an embodiment, rotators of equal magnitude and alternating sign are placed at the four corners of the rectangular resist domain when crossing the domain boundary. In an embodiment, determining the velocity flow field further involves: obtaining a velocity at a given vertex along the contour of the at least one feature based on the velocities of all other vertices due to capillary forces at each of the other vertices by multiplication with the Stokes rotators.

[0230] In an embodiment, the contribution of the velocity flow field due to features far away from a given vertex is negligible.

[0231] In an embodiment, the method further comprises: determining a force at a local area within the resist; and obtaining a deformation of the entire area of ​​the resist based on simulating a resist deformation model using the force at the local area.

[0232] In an embodiment, the input pattern is a grayscale image and / or a binary image. In an embodiment, the input pattern is a design pattern or an aerial image.

[0233] In an embodiment, the method further comprises generating a binary image, the generating involving: obtaining a pattern forming device pattern corresponding to the input pattern; generating an aerial image based on the pattern forming device pattern through simulation of a process model; and extracting a boundary of the pattern in the aerial image to generate the binary image.

[0234] In an embodiment, the method further involves: using a resist deformation model, calculating a critical dimension between a pair of locations, the pair of locations being disposed on a boundary of at least one feature in a developed resist pattern; and calculating an error between the calculated critical dimension and a measured critical dimension of an actual developed resist pattern.

[0235] In an embodiment, calculating the error includes comparing using a cross-correlation matrix between the printed wafer data and the simulated image.

[0236] In an embodiment, the deformation is determined at a plurality of locations, each location corresponding to a point located on a boundary of a developed portion of the developed resist pattern for the input pattern.

[0237] In an embodiment, the resist is a chemically or non-chemically amplified negative resist or a positive resist.

[0238] In an embodiment, a method for determining a parameter of a patterning process is provided, the method involving: obtaining (i) a patterning process model, including a resist deformation model of a resist having a pattern, the resist deformation model being configured to simulate a fluid flow of the resist due to capillary forces acting on a profile of at least one feature of the pattern, and (ii) a target pattern; determining, via a processor, the resist pattern based on a simulation of the patterning process model, wherein the target pattern is used as an input to the patterning process model, wherein there is a difference between the resist pattern and the target pattern; and determining, via the processor, a value of a parameter of the patterning process based on the simulation of the patterning process, the value of the parameter being determined such that the difference between the resist pattern and the target pattern is reduced.

[0239] In an embodiment, the parameters of the patterning process include at least one of: dose, focus, and optical proximity correction.

[0240] In an embodiment, the method further involves applying, during the patterning process, values ​​of parameters of the patterning process to the lithographic apparatus.

[0241] In an embodiment, a method for determining deformation of a pattern to be formed in a patterning process is provided. The method involves: inputting pattern information about the pattern to be formed into a resist deformation model, the model being configured to simulate deformation of a portion of the resist, the portion including a boundary liquid layer at a boundary between a developed region in the resist and a region of the resist surrounding the developed region, wherein the model is configured to determine a first deformation component of the boundary liquid layer caused by a fluid flow of the boundary liquid layer and a second deformation component of the boundary liquid layer caused by the fluid flow of the boundary liquid layer; and determining, via a processor, a deformation of the pattern to be formed in the resist based on the input pattern information, wherein the deformation includes a combination of the first deformation component and the second deformation component of the boundary liquid layer.

[0242] In an embodiment, the thickness of the boundary liquid layer is less than the length of the developed area in the resist at the boundary.

[0243] In an embodiment, the first deformation component is determined in the horizontal plane based on a horizontal component of the flow rate of the boundary liquid layer, and the second deformation is determined in the horizontal plane based on a vertical component of the flow rate of the boundary liquid layer.

[0244] In an embodiment, a first boundary condition is defined in the model, the first boundary condition comprising a velocity gradient approximately equal to zero at a free surface of the boundary liquid layer, the free surface being opposite to the developed area or the open area.

[0245] In an embodiment, a second boundary condition is defined in the model, the second boundary condition comprising a velocity approximately equal to zero at the surface at the developed area or the open area.

[0246] In an embodiment, the model defines a velocity according to the thickness of the boundary liquid layer and a pressure gradient across the thickness of the boundary liquid layer.

[0247] In an embodiment, the model defines a flow rate of the boundary liquid layer as an integral of the velocity over the thickness of the boundary liquid layer.

[0248] In an embodiment, a deformation according to the flow rate of the boundary liquid layer is defined in the model.

[0249] In an embodiment, determining the deformation involves: determining a first deformation component in a first plane caused by a horizontal component of the flow rate of the boundary liquid layer; and determining a final deformation in the first plane by adjusting the first deformation component based on a vertical component of the flow rate of the boundary liquid layer in a second plane, the second plane being perpendicular to the first plane.

[0250] In an embodiment, the first plane is a horizontal plane.

[0251] In an embodiment, determining the deformation further involves: defining multiple vertices along the contour of a developed area or an open area in the resist; determining the capillary force at a given vertex along the contour; and redistributing the vertices so that (i) the volume of the developed area in the resist is conserved, and (ii) the volume of the boundary liquid layer is conserved.

[0252] In an embodiment, redistributing the vertices involves: determining a time scale based on a horizontal component of a flow rate of the boundary liquid layer; and moving the plurality of vertices based on the time scale and the flow rate of the boundary layer.

[0253] In an embodiment, the method further involves: determining a rate of change of a thickness of the boundary liquid layer based on a flow rate of the boundary layer; and adjusting another plurality of vertices along the free surface based on the redistributed vertices based on the rate of change of the thickness of the boundary liquid layer.

[0254] In an embodiment, the model is a thin film model simplified based on a lubrication approximation, wherein the lubrication approximation includes a vertical shear stress value of zero.

[0255] In addition, a method for determining the deformation of a pattern to be formed in a patterning process is provided. The method includes: inputting pattern information about the pattern to be formed into a resist deformation model, the model being configured to simulate the deformation of a portion of the resist, the portion including a boundary liquid layer at a boundary between a developed area in the resist and an area of ​​the resist surrounding the developed area, wherein the model is configured to determine the deformation of the boundary liquid layer caused by a horizontal fluid flow of the boundary liquid layer; and determining (e.g., in process P143) the deformation of the pattern to be formed in the resist by simulating the resist deformation model based on the input pattern information. The thickness of the boundary liquid layer is less than the length of the developed area in the resist at the boundary.

[0256] In an embodiment, a non-transitory computer program product including machine-readable instructions is provided, the machine-readable instructions being used to cause a processor to implement the execution of the aforementioned method.

[0257] Fig.16 1 is a block diagram illustrating a computer system 100 that performs one or more aspects of the methods and processes disclosed herein. The computer system 100 includes a bus 102 or other communication mechanism for transmitting information and a processor 104 (or multiple processors 104 and 105) coupled to the bus 102 for processing information. The computer system 100 also includes a main memory 106, such as a random access memory (RAM) or other dynamic storage device, which is coupled to the bus 102 for storing information and instructions to be executed by the processor 104. The main memory 106 can also be used to store temporary variables or other intermediate information during the execution of instructions to be executed by the processor 104. The computer system 100 also includes a read-only memory (ROM) 108 or other static storage device, which is coupled to the bus 102 for storing static information and instructions for the processor 104. A storage device 110, such as a magnetic disk or optical disk, is provided and coupled to the bus 102 for storing information and instructions.

[0258] The computer system 100 may be coupled to a display 112, such as a cathode ray tube (CRT) or a flat panel or touch panel display, via the bus 102 for displaying information to a computer user. An input device 114 including alphanumeric and other keys is coupled to the bus 102 for communicating information and command selections to the processor 104. Another type of user input device is a cursor control 116, such as a mouse, trackball, or cursor direction keys, for communicating direction information and command selections to the processor 104 and controlling cursor movement on the display 112. The input device typically has two degrees of freedom on two axes, a first axis (e.g., x) and a second axis (e.g., y), which allows the device to specify a position in a plane. A touch panel (screen) display may also be used as an input device.

[0259] According to one embodiment, portions of the processes described herein may be performed by computer system 100 in response to processor 104 executing one or more sequences of one or more instructions contained in main memory 106. Such instructions may be read into main memory 106 from another computer-readable medium such as storage device 110. Execution of the sequences of instructions contained in main memory 106 causes processor 104 to perform the process steps described herein. One or more processors in a multi-processing arrangement may also be employed to execute the sequences of instructions contained in main memory 106. In alternative embodiments, hardwired circuitry may be used in place of software instructions, or in combination with software instructions. Therefore, the description herein is not limited to any specific combination of hardware circuitry and software.

[0260] The term "computer-readable medium" as used herein refers to any medium that participates in providing instructions to processor 104 for execution. Such media can take many forms, including but not limited to non-volatile media, volatile media, and transmission media. Non-volatile media include, for example, optical or magnetic disks, such as storage device 110. Volatile media include dynamic memory, such as main memory 106. Transmission media include coaxial cables, copper wires, and optical fibers, including the wires that make up bus 102. Transmission media can also take the form of sound waves or light waves, such as those generated during radio frequency (RF) and infrared (IR) data communications. Common forms of computer-readable media include, for example, floppy disks, foldable disks, hard disks, magnetic tape, any other magnetic media, CD-ROMs, DVDs, any other optical media, punch cards, paper tape, any other physical media with a pattern of holes, RAM, PROMs and EPROMs, flash EPROMs, any other memory chips or memory cartridges, carrier waves described below, or any other media from which a computer can read.

[0261] Various forms of computer-readable media may be involved in carrying one or more sequences of one or more instructions to processor 104 for execution. For example, the instructions may initially be carried on a disk of a remote computer. The remote computer can load the instructions into its dynamic memory and send the instructions over a telephone line using a modem. A modem local to computer system 100 can receive the data on the telephone line and use an infrared transmitter to convert the data into an infrared signal. An infrared detector coupled to bus 102 can receive the data carried in the infrared signal and place the data on bus 102. Bus 102 carries the data to main memory 106, from which processor 104 retrieves and executes the instructions. The instructions received by main memory 106 may optionally be stored on storage device 110 before or after execution by processor 104.

[0262] Preferably, the computer system 100 also includes a communication interface 118 coupled to the bus 102. The communication interface 118 provides a two-way data communication coupled to a network link 120, which is connected to a local network 122. For example, the communication interface 118 can be an integrated services digital network (ISDN) card or a modem to provide a data communication connection with a corresponding type of telephone line. As another example, the communication interface 118 can be a local area network (LAN) card to provide a data communication connection with a compatible LAN. A wireless link can also be implemented. In any such implementation, the communication interface 118 sends and receives electrical signals, electromagnetic signals, or optical signals that carry digital data streams representing various types of information.

[0263] The network link 120 typically provides data communication through one or more networks to other data devices. For example, the network link 120 may provide a connection through a local network 122 to a host computer 124 or data equipment operated by an Internet service provider (ISP) 126. The ISP 126, in turn, provides data communication services through a global packet data communication network, now commonly referred to as the "Internet" 128. Both the local network 122 and the Internet 128 use electrical, electromagnetic, or optical signals that carry digital data streams. The signals through the various networks and the signals on the network link 120 and through the communication interface 118, which carry the digital data to and from the computer system 100, are exemplary forms of carrier waves that transport the information.

[0264] The computer system 100 can send messages and receive data, including program code, through the network(s), network link 120, and communication interface 118. In the Internet example, the server 130 can transmit the requested code for the application program through the Internet 128, ISP 126, local network 122, and communication interface 118. For example, one such download application can provide the processes described herein. The received code can be executed by the processor 104 as it is received, and / or stored in the storage device 110 or other non-volatile storage device for later execution. In this way, the computer system 100 can obtain the application code in the form of a carrier wave.

[0265] Fig.17 An exemplary lithographic projection apparatus for use with the methods described herein is schematically depicted. The apparatus comprises:

[0266] - an illumination system IL to condition the radiation beam B. In this particular case, the illumination system further comprises a radiation source SO;

[0267] a first stage (e.g. mask table) MT provided with a patterning device holder for holding a patterning device MA (e.g. reticle) and connected to a first positioner for accurately positioning the patterning device relative to the object PS;

[0268] a second object table (substrate table) WT provided with a substrate holder for holding a substrate W (e.g. a silicon wafer coated with resist) and connected to a second positioner for accurately positioning the substrate relative to the item PS;

[0269] - a projection system ("lens") PS (eg a refractive, reflective or catadioptric optical system) to image the illuminated portion of the patterning device MA onto a target portion C of the substrate W (eg comprising one or more dies).

[0270] As depicted herein, the device is transmissive (e.g., with a transmissive mask). However, in general, it may also be reflective, e.g., with a reflective mask). Alternatively, the device may employ another class of patterning devices as an alternative to using a classical mask; examples include a programmable mirror array or an LCD matrix.

[0271] A source SO (e.g. a mercury lamp or an excimer laser) generates a radiation beam. This beam is fed into an illumination system (illuminator) IL, for example directly or after passing through an adjustment component such as a beam expander Ex. The illuminator IL may include an adjustment component AD for setting the outer and / or inner radial extent (commonly referred to as σ outer and σ inner, respectively) of the intensity distribution in the beam. In addition, it will typically include various other components, such as an integrator IN and a condenser CO. In this way, the beam B impinging on the patterning device MA has a desired uniformity and intensity distribution in its cross section.

[0272] about Fig.17 It should be noted that the source SO can be within the housing of the lithographic projection apparatus (for example, this is usually the case when the source SO is a mercury lamp), but it can also be remote from the lithographic projection apparatus, the radiation beam it generates being introduced into the apparatus (for example, with the aid of suitable guide mirrors); this latter scenario is usually the case when the source SO is an excimer laser (for example, based on KrF, ArF or F2 laser irradiation).

[0273] The light beam PB then intercepts the patterning device MA, which is held on the patterning device table MT. After passing the patterning device MA, the light beam PB passes through a lens PL, which focuses the light beam B onto a target portion C of the substrate W. With the aid of the second positioning component (and the interferometry component IF), the substrate table WT can be accurately moved, for example to position a different target portion C in the path of the light beam PB. Similarly, the first positioning component can be used to accurately position the patterning device MA relative to the path of the light beam B, for example after mechanically retrieving the patterning device MA from the patterning device library or during a scan. Typically, movement of the stage MT, WT will be realized with the aid of a long-stroke module (coarse positioning) and a short-stroke module (fine positioning), which are used in the embodiment of the present invention. Fig.17 However, in the case of a wafer stepper (as opposed to a step-and-scan tool), the patterning device table MT may be connected to a short-stroke actuator only, or may be fixed.

[0274] The depicted tool can be used in two different modes:

[0275] - in step mode, the patterning device table MT is kept essentially stationary and the entire patterning device image is projected once (i.e. a single "flash") onto a target portion C. The substrate table WT is then shifted in the x and / or y direction so that a different target portion C can be illuminated by the beam PB;

[0276] - In scan mode, essentially the same scenario applies, except that a given target portion C is not exposed in a single "flash". Instead, the patterning device table MT is movable in a given direction (the so-called "scanning direction", e.g. the y-direction) at a speed v, so that the projection beam B is used to scan over the patterning device image; concurrently, the substrate table WT is simultaneously moved in the same or opposite direction at a speed V=Mv, where M is the magnification of the lens PL (typically M=1 / 4 or 1 / 5). In this way, a relatively large target portion C can be exposed without having to sacrifice resolution.

[0277] Fig.18 Another exemplary lithographic projection apparatus 1000 that may be used for the methods described herein is schematically depicted.

[0278] The lithographic projection apparatus 1000 comprises:

[0279] - Source collector module SO;

[0280] - an illumination system (illuminator) IL configured to condition a radiation beam B (e.g. EUV radiation);

[0281] a support structure (e.g. mask table) MT configured to support a patterning device (e.g. mask or reticle) MA and connected to a first positioner PM configured to accurately position the patterning device;

[0282] a substrate table (eg wafer stage) WT configured to hold a substrate (eg resist coated wafer) W and connected to a second positioner PW configured to accurately position the substrate; and

[0283] - a projection system (eg a reflective projection system) PS configured to project the pattern imparted to the radiation beam B by the patterning device MA onto a target portion C of the substrate W (eg comprising one or more dies).

[0284] As depicted here, the device 1000 is reflective (e.g., employing a reflective mask). Note that because most materials are absorptive in the EUV wavelength range, the mask can have a multilayer reflector comprising multiple stacks of, for example, molybdenum and silicon. In one example, the multilayer reflector has 40 layer pairs of molybdenum and silicon, where each layer is a quarter wavelength thick. Even smaller wavelengths can be produced with X-ray lithography. Since most materials are absorptive at both EUV and X-ray wavelengths, a thin sheet of patterned absorbing material on the patterning device topography (e.g., a TaN absorber on top of a multilayer reflector) defines where features will be printed (positive resist) or not (negative resist).

[0285] Reference Fig.18 , the illuminator IL receives a beam of extreme ultraviolet radiation from a source collector module SO. Methods of generating EUV radiation include, but are not necessarily limited to, converting a material into a plasma state having at least one element (e.g., xenon, lithium, or tin) having one or more emission lines in the EUV range. In one such method, generally referred to as laser produced plasma ("LPP"), a plasma may be generated by irradiating a fuel (such as a droplet, stream, or cluster of a material having a line emitting element) with a laser beam. The source collector module SO may be part of an EUV radiation system that includes a device not provided in the EUV radiation system. Fig.18 The laser shown in Figure 1 is used to provide a laser beam for exciting the fuel. The resulting plasma emits output radiation, such as EUV radiation, which is collected using a radiation collector disposed in a source collector module. The laser and source collector module may be separate entities, such as when a CO2 laser is used to provide a laser beam for fuel excitation.

[0286] In this case, the laser is not considered to form part of the lithographic apparatus, and the radiation beam is delivered from the laser to the source collector module with the aid of a beam delivery system including, for example, suitable guide mirrors and / or a beam expander. In other cases, the source may be an integral part of the source collector module, for example when the source is a discharge produced plasma EUV generator, often referred to as a DPP source.

[0287] The illuminator IL may include an adjuster for adjusting the angular intensity distribution of the radiation beam. Typically, at least the outer and / or inner radial extent (generally 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 a faceted field and a pupil mirror device. The illuminator may be used to adjust the radiation beam to have a desired uniformity and intensity distribution in its cross section.

[0288] A radiation beam B is incident on and patterned by a patterning device (e.g. a mask) MA, which is held on a support structure (e.g. a mask table) MT. After reflection from the patterning device (e.g. a mask) MA, the radiation beam B passes through a projection system PS which focuses the beam onto a target portion C of a substrate W. With the aid of a second positioner PW and a position sensor PS2 (e.g. an interferometric device, a linear encoder or a capacitive sensor), the substrate table WT can be accurately moved, for example to position different target portions C in the path of the radiation beam B. Similarly, a first positioner PM and a further position sensor PS1 can be used to accurately position the patterning device (e.g. a mask) MA relative to the path of the radiation beam B. The patterning device (e.g. a mask) MA and the substrate W can be aligned using the patterning device alignment marks M1, M2 and the substrate alignment marks P1, P2.

[0289] The depicted device 1000 may be used in at least one of the following modes:

[0290] 1. In step mode, the support structure (e.g. mask table) MT and the substrate table WT remain substantially stationary while the entire pattern imparted to the radiation beam is projected at one time onto a target portion C (i.e. a single static exposure). The substrate table WT is then shifted in the X and / or Y direction so that a different target portion C can be exposed.

[0291] 2. In scan mode, the support structure (e.g. mask table) MT and substrate table WT are scanned synchronously as a pattern imparted to the radiation beam is projected onto a target portion C (i.e. a single dynamic exposure). The speed and direction of the substrate table WT relative to the support structure (e.g. mask table) MT may be determined by the (de-)magnification and image reversal characteristics of the projection system PS.

[0292] 3. In another mode, the support structure (e.g. mask table) MT remains essentially stationary, thereby holding the programmable patterning device, and the substrate table WT is moved or scanned, while a pattern imparted to the radiation beam is projected onto a target portion C. In this mode, typically a pulsed radiation source is employed, and the programmable patterning device is updated as required after each movement of the substrate table WT or between successive radiation pulses during a scan. This mode of operation can readily be applied to maskless lithography, which utilizes a programmable patterning device, such as a programmable mirror array of the type cited above.

[0293] Fig.19 The apparatus 1000 is shown in more detail and includes a source collector module SO, an illumination system IL, and a projection system PS. The source collector module SO is constructed and arranged so that a vacuum environment can be maintained in a closed structure 220 of the source collector module SO. The EUV radiation emitting plasma 210 can be formed by a plasma source generated by a discharge. EUV radiation can be generated by a gas or vapor, such as xenon, lithium vapor, or tin vapor, in which a very hot plasma 210 is created to emit radiation in the EUV range of the electromagnetic spectrum. The very hot plasma 210 is created by, for example, a discharge that results in at least partially ionized plasma. In order to effectively generate radiation, a partial pressure of, for example, 10Pa of Xe, Li, Sn vapor, or any other suitable gas or vapor may be required. In an embodiment, a plasma that excites tin (Sn) is provided to generate EUV radiation.

[0294] Radiation emitted by the hot plasma 210 enters the collector chamber 212 from the source chamber 211 via an optional gas barrier or contaminant trap 230 (also referred to as a contaminant barrier or fin trap in some cases) positioned in or behind an opening in the source chamber 211. The contaminant trap 230 may include a channel structure. The contaminant trap 230 may also include a gas barrier or a combination of a gas barrier and a channel structure. As is known in the art, the contaminant trap or contaminant barrier 230 further indicated herein includes at least a channel structure.

[0295] The collector chamber 211 may include a radiation collector CO, which may be a so-called grazing incidence collector. The radiation collector CO has an upstream radiation collector side 251 and a downstream radiation collector side 252. Radiation passing through the collector CO may be reflected from the grating spectral filter 240 to be focused in a virtual source point IF along the optical axis indicated by the dotted line 'O'. The virtual source point IF is generally referred to as an intermediate focus, and the source collector module is arranged so that the intermediate focus IF is located at or near an opening 221 in the enclosure 220. The virtual source point IF is an image of the radiation emitting plasma 210.

[0296] The radiation then passes through the illumination system IL, which may include a faceted field mirror device 22 and a faceted pupil mirror device 24, which are arranged to provide a desired angular distribution of the radiation beam 21 at the patterning device MA and a desired uniformity of radiation intensity at the patterning device MA. Upon reflection of the radiation beam 21 at the patterning device MA held by the support structure MT, a patterned beam 26 is formed, and the patterned beam 26 is imaged by the projection system PS via reflective elements 28, 30 onto a substrate W held by a substrate table WT.

[0297] More elements than shown may typically be present in the illumination optics unit IL and the projection system PS. Depending on the type of lithographic apparatus, a grating spectral filter 240 may optionally be present. Further, there may be more mirrors than shown in the figures, for example with Fig.19 Compared to that shown, there may be between 1 and 6 additional reflective elements in the projection system PS.

[0298] like Fig.19 As illustrated, the collector optics CO is depicted as a nested collector with grazing incidence reflectors 253, 254 and 255, merely as an example of a collector (or collector mirror). The grazing incidence reflectors 253, 254 and 255 are arranged axially symmetrically around the optical axis O, and preferably, this type of collector optics CO may be used in combination with a discharge produced plasma source, generally referred to as a DPP source.

[0299] Alternatively, the source collector module SO may be Fig. 20 A portion of an LPP radiation system is shown. The laser LA is arranged to deposit laser energy into a fuel such as xenon (Xe), tin (Sn) or lithium (Li), thereby creating a highly ionized plasma 210 with an electron temperature of tens of eV. Energetic radiation generated during the deexcitation and recombination of these ions is emitted from the plasma, collected by a collector optics CO at near normal incidence and focused onto an opening 221 in a closed structure 220.

[0300] The embodiments may also be described using the following aspects:

[0301] 1. A method for determining deformation of a resist during a patterning process, the method comprising:

[0302] obtaining a resist deformation model of a resist having a pattern, the resist deformation model being configured to simulate fluid flow of the resist due to capillary forces acting on a profile of at least one feature of the pattern; and

[0303] A deformation of a resist pattern to be developed is determined based on an input pattern of the resist deformation model via a processor and the resist deformation model.

[0304] 2. The method of clause 1, wherein the resist deformation model is based on linearized Navier-Stokes flow equations.

[0305] 3. The method according to aspect 1 or aspect 2, wherein the fluid flow is characterized by Stokes flow and / or Hull-Shaw flow.

[0306] 4. A method according to aspect 1 or aspect 2, wherein the fluid flow is characterized by a fundamental solution that decays at a rate at least as fast as 1 / R with the distance to a specific location in the resist and is regular in the resist or is singular only at the specific location in the resist.

[0307] 5. According to the method of any one of aspects 1 to 4, determining the deformation comprises:

[0308] defining a plurality of vertices along a profile of the at least one feature;

[0309] determining a capillary force at a given vertex along a profile of the at least one feature; and

[0310] determining a velocity flow field of the fluid flow due to capillary forces based on a superposition of velocity responses at one or more of a plurality of vertices along a profile of the at least one feature, and boundary conditions,

[0311] The velocity response is due to capillary forces acting at vertices adjacent to a given vertex.

[0312] 6. The method according to any one of clauses 5, wherein defining the plurality of vertices comprises distributing the plurality of vertices so that they are evenly spaced apart while conserving an area or volume of a profile of at least one feature.

[0313] 7. The method of any one of aspects 1 to 6, wherein the capillary force acting on a given vertex is the sum of the tension forces on either side of the given vertex.

[0314] 8. The method of clause 5, wherein the velocity response is characterized by Stokes particles, wherein Stokes particles are velocity fields due to point forces in Stokes flow.

[0315] 9. The method according to aspect 5, wherein determining the velocity flow field further comprises:

[0316] Decomposing the velocity field into a squeeze flow and a high-order velocity flow;

[0317] applying boundary conditions corresponding to the extrusion flow; and

[0318] The squeeze flow is removed from the velocity field based on the boundary conditions.

[0319] 10. The method of clause 9, wherein the squeeze flow is resist flow due to a net inward flux or a net outward flux across a resist domain boundary, resulting in large-scale migration of features.

[0320] 11. The method according to any one of aspects 7 to 10, wherein applying the boundary condition comprises:

[0321] setting the flow rate across the border of the resist domain to zero; and / or

[0322] The velocity across the boundary of the resist domain was set to a no-flux condition.

[0323] 12. The method of clause 9, wherein the total inward flow rate across the resist domain boundary is set to zero by providing rotators of appropriate strength at one or more corners of the resist domain boundary.

[0324] 13. The method of clause 10, wherein rotators of equal magnitude and alternating sign are placed at the four corners of the rectangular resist domain when traversing the domain boundary.

[0325] 14. According to the method described in any one of aspects 5 to 13, determining the velocity flow field further comprises: obtaining the velocity at a given vertex along the profile of the at least one feature based on the velocities of all other vertices due to capillary forces at each of the other vertices multiplied by Stokes subtraction.

[0326] 15. The method of clause 5, wherein contributions to the velocity flow field due to features far from a given vertex are negligible.

[0327] 16. The method according to any one of aspects 5 to 13, further comprising:

[0328] determining a force at a localized region within the resist; and

[0329] The deformation of the entire area of ​​the resist is obtained based on a simulation of the resist deformation model using the force at the local area.

[0330] 17. A method according to any one of clauses 1 to 16, wherein the input pattern is a grayscale image and / or a binary image.

[0331] 18. The method according to any one of clauses 1 to 17, wherein the input pattern is a design pattern, a resist image, a mask pattern and / or an aerial image.

[0332] 19. The method of clause 18, further comprising generating the binary image, the generating comprising:

[0333] obtaining a patterning device pattern corresponding to an input pattern;

[0334] generating an aerial image based on the patterning device pattern via simulation of a process model; and

[0335] The boundary of the pattern is extracted in the aerial image to generate the binary image.

[0336] 20. The method according to any one of aspects 1 to 19, further comprising:

[0337] calculating a critical dimension between a pair of locations disposed on a boundary of the at least one feature in the developed resist pattern using the resist deformation model; and

[0338] The error between the calculated critical dimension and the measured critical dimension of the actual developed resist pattern is calculated.

[0339] 21. The method of clause 20, wherein calculating the error comprises comparing using a cross-correlation matrix between printed wafer data and simulated images.

[0340] 22. A method according to any one of clauses 1 to 21, wherein the deformation is determined at a plurality of locations, each location corresponding to a point located on a boundary of a developed portion of the developed resist pattern for an input pattern.

[0341] 23. The method according to any one of aspects 1 to 22, wherein the resist is a chemically amplified or non-chemically amplified negative resist or a positive resist.

[0342] 24. A method for determining parameters of a patterning process, the method comprising:

[0343] obtaining (i) a patterning process model including a resist deformation model of a resist having a pattern, the resist deformation model being configured to simulate fluid flow of the resist due to capillary forces acting on a profile of at least one feature of the pattern, and (ii) a target pattern;

[0344] determining, via a processor, a resist pattern based on a simulation of a patterning process model, wherein a target pattern is used as an input to the patterning process model, wherein a difference exists between the resist pattern and the target pattern; and

[0345] A value of a parameter of the patterning process is determined, via a processor, based on the simulation of the patterning process, the value of the parameter being determined such that a difference between the resist pattern and the target pattern is reduced.

[0346] 25. The method of clause 24, wherein parameters of the patterning process include at least one of: dose, focus, and optical proximity correction.

[0347] 26. The method according to any one of aspects 24 to 25, further comprising:

[0348] During the patterning process, values ​​of parameters of the patterning process are applied to the lithographic apparatus.

[0349] 27. A method for determining a deformation of a pattern to be formed in a patterning process, the method comprising:

[0350] inputting pattern information about a pattern to be formed into a resist deformation model configured to simulate deformation of a portion of the resist including a boundary liquid layer at a boundary between a developed region in the resist and a region of the resist surrounding the developed region,

[0351] wherein the model is configured to determine a first deformation component of the boundary liquid layer caused by a fluid flow in the boundary liquid layer and a second deformation component of the boundary liquid layer caused by the fluid flow in the boundary liquid layer; and

[0352] A deformation of a pattern to be formed in the resist is determined based on the input pattern information via a processor, wherein the deformation includes a combination of a first deformation component and a second deformation component of the boundary liquid layer.

[0353] 28. The method of clause 27, wherein the thickness of the boundary liquid layer is less than the length of the developed area in the resist at the boundary.

[0354] 29. A method according to any one of aspects 27 to 28, wherein the first deformation component is determined in a horizontal plane based on a horizontal component of the flow rate of the boundary liquid layer, and the second deformation is determined in a horizontal plane based on a vertical component of the flow rate of the boundary liquid layer.

[0355] 30. A method according to any one of aspects 27 to 29, wherein a first boundary condition is defined in the model, the first boundary condition comprising a velocity gradient approximately equal to zero at a free surface of the boundary liquid layer, the free surface being opposite to the developed area or the open area.

[0356] 31. The method of any one of aspects 27 to 30, wherein a second boundary condition is defined in the model, the second boundary condition comprising a velocity approximately equal to zero at the surface at the developed or open area.

[0357] 32. A method according to aspect 31, wherein the model defines a velocity according to the thickness of the boundary liquid layer and a pressure gradient across the thickness of the boundary liquid layer.

[0358] 33. A method according to clause 32, wherein the model defines a flow rate of the boundary liquid layer as an integral of the velocity over the thickness of the boundary liquid layer.

[0359] 34. The method of clause 29, wherein a deformation according to a flow rate of a boundary liquid layer is defined in the model.

[0360] 35. The method according to any one of aspects 27 to 34, wherein determining the deformation comprises:

[0361] determining a first deformation component in a first plane caused by a horizontal component of the flow rate of the boundary liquid layer; and

[0362] A final deformation in the first plane is determined by adjusting the first deformation component based on a vertical component of the flow rate of the boundary liquid layer in a second plane, the second plane being perpendicular to the first plane.

[0363] 36. A method according to clause 35, wherein the first plane is a horizontal plane.

[0364] 37. The method according to any one of aspects 27 to 36, wherein determining the deformation further comprises:

[0365] defining a plurality of vertices along the outline of the developed or open areas in the resist;

[0366] determining the capillary force at a given vertex along the profile; and

[0367] The vertices are redistributed such that (i) the volume of the developed regions in the resist is conserved; and (ii) the volume of the boundary liquid layer is conserved.

[0368] 38. The method of clause 37, wherein redistributing vertices comprises:

[0369] determining a time scale based on a horizontal component of the flow rate of the boundary liquid layer; and

[0370] Based on the time scale and the flow rate of the boundary layer, multiple vertices are moved.

[0371] 39. The method according to aspect 38, further comprising:

[0372] determining a rate of change of a thickness of a boundary liquid layer based on a flow rate of the boundary layer; and

[0373] Another plurality of vertices along the free surface are adjusted based on the redistributed vertices based on the rate of change of the thickness of the boundary liquid layer.

[0374] 40. The method of any one of aspects 27 to 39, wherein the model is a thin film model simplified based on a lubrication approximation, wherein the lubrication approximation includes a vertical shear stress value of zero.

[0375] 41. A method for determining a deformation of a pattern to be formed in a patterning process, the method comprising:

[0376] inputting pattern information about a pattern to be formed into a resist deformation model, the model being configured to simulate deformation of a portion of the resist, the portion including a boundary liquid layer at a boundary between a developed region in the resist and a region of the resist surrounding the developed region, wherein the model is configured to determine deformation of the boundary liquid layer caused by a horizontal fluid flow of the boundary liquid layer; and

[0377] A deformation of a pattern to be formed in the resist is determined, via a processor, by simulating a resist deformation model based on input pattern information.

[0378] 42. The method of clause 41, wherein the thickness of the boundary liquid layer is less than the length of the developed area in the resist at the boundary.

[0379] 43. A non-transitory computer program product comprising machine-readable instructions for causing a processor to implement the method of any one of aspects 1 to 42.

[0380] Although specific reference may be made herein to the manufacture of devices such as ICs, it should be clearly understood that the description herein has many other possible applications. For example, it may be used to manufacture integrated optical systems, guidance and detection patterns for magnetic domain memories, liquid crystal display panels, thin film magnetic heads, etc. Those skilled in the art will appreciate that in the context of such alternative applications, any use of the terms "reticle", "wafer" or "die" herein should be considered interchangeable with the more general terms "mask", "substrate" and "target portion", respectively.

[0381] It is noted that the terms "mask", "reticle" and "patterning device" are used interchangeably herein. Moreover, those skilled in the art will recognize that, particularly in the context of lithography simulation / optimization, the terms "mask" / "patterning device" and "design layout" are used interchangeably, because in lithography simulation / optimization, a physical patterning device is not necessarily used, but a design layout can be used to represent a physical patterning device.

[0382] In this document, the terms "radiation" and "beam" are used to cover all types of electromagnetic radiation, including ultraviolet radiation (e.g., having a wavelength of 365, 248, 193, 157 or 126 nm) and extreme ultraviolet radiation (EUV), e.g., having a wavelength in the range of 5 to 20 nm.

[0383] As used herein, the terms "optimizing" and "optimization" refer to adjusting a lithography projection apparatus and / or a patterning process so that the result and process of a patterning process (such as lithography) have more desirable characteristics, such as a higher projection accuracy of a design layout on a substrate, a larger process window, etc. The terms "optimizing" and "optimization" do not necessarily require that the result and / or process of lithography have the most desirable characteristics, such as the highest projection accuracy of a design layout on a substrate, the largest process window, etc.

[0384] The pattern forming device cited above includes or can form a design layout. The design layout can be generated using a CAD (computer-aided design) program, which is generally referred to as EDA (electronic design automation). Most CAD programs follow a predetermined set of design rules to create a functional design layout / pattern forming device. These rules are set by processing and design constraints. For example, the design rules define the spatial tolerances between circuit devices (such as gates, capacitors, etc.) or interconnects to ensure that the circuit devices or lines do not interact with each other in an undesirable manner. Design rule restrictions are generally referred to as "critical dimensions" (CD). The critical dimension of a circuit can be defined as the minimum width of a line or hole or the minimum space between two lines or two holes. Therefore, CD determines the total size and density of the designed circuit. Of course, one of the goals in integrated circuit manufacturing is to faithfully reproduce the original circuit design (via a pattern forming device) on a substrate.

[0385] The terms "mask" or "patterning device" as used herein may be broadly interpreted as referring to a general purpose patterning device that can be used to impart a patterned cross-section to an incoming radiation beam, corresponding to the pattern to be created in a target portion of the substrate; in this context, the term "light valve" may also be used. In addition to classical masks (transmissive or reflective; binary, phase-shifting, hybrid, etc.), other examples of such patterning devices include programmable mirror arrays and programmable LCD arrays.

[0386] The concepts disclosed herein can simulate or mathematically model any patterning process and may be particularly useful for imaging techniques that are capable of producing shorter and shorter wavelengths. Examples of such imaging techniques that have been used include EUV (extreme ultraviolet), DUV lithography, which is capable of producing wavelengths of 193 nm using ArF lasers and / or 157 nm using fluorine lasers. Moreover, EUV lithography is capable of producing wavelengths in the range of about 5 nm to about 20 nm by using, for example, a synchrotron or by hitting a material (solid or plasma) with high-energy electrons to produce photons in this range.

[0387] Although the concepts disclosed herein may be used in patterning processes involving imaging on substrates such as silicon wafers, it should be understood that the disclosed concepts may be used with any type of lithography system, such as those used for imaging on substrates other than silicon wafers.

[0388] In the block diagrams, the components illustrated are depicted as discrete functional blocks, but the embodiments are not limited to systems in which the functionality described herein is organized as illustrated. The functionality provided by each of the components may be provided by software or hardware modules organized differently than currently depicted, such as such software or hardware may be mixed, combined, replicated, decomposed, distributed (e.g., within a data center or geographically), or organized differently. The functionality described herein may be provided by one or more processors of one or more computers that execute code stored on a tangible, non-transitory, machine-readable medium. In some cases, a third-party content delivery network may host some or all of the information communicated over the network, in which case, to the extent that the information (e.g., content) is considered to be supplied or otherwise provided, the information may be provided by sending instructions to retrieve the information from the content delivery network.

[0389] Unless otherwise specifically stated, as will be apparent from the discussion, it will be understood that throughout this specification, discussions using terms such as "processing," "computing," "calculating," "determining," etc. refer to the actions or processes of a specific apparatus, such as a special purpose computer or similar special purpose electronic processing / computing device.

[0390] The reader should understand that this application describes multiple inventions. These inventions are grouped into a single document rather than dividing these inventions into multiple isolated patent applications because their related subject matter helps save costs in the application process. However, the different advantages and aspects of such inventions should not be combined. In some cases, embodiments solve all the defects mentioned herein, but it should be understood that the present invention is independently useful, and some embodiments only solve a subset of such problems or provide other unmentioned benefits that are obvious to those skilled in the art who review this disclosure. Due to cost constraints, some inventions disclosed herein may not be claimed at present, and may be claimed in a later filing, such as a continuation application or by amending the current claims. Similarly, due to space constraints, the Abstract and Summary of the Invention sections of this document should not be regarded as a comprehensive list containing all such inventions or all aspects of such inventions.

[0391] It should be understood that the description and drawings are not intended to limit the disclosure to the particular forms disclosed, but on the contrary, the intention is to cover all modifications, equivalents, and alternatives falling within the spirit and scope of the invention, as defined by the appended claims.

[0392] In view of this description, modifications and alternative embodiments of various aspects of the present invention will be apparent to those skilled in the art. Therefore, the description and drawings are to be interpreted as illustrative only, and are intended to teach those skilled in the art the general manner of performing the present invention. It is to be understood that the forms of the present invention shown and described herein will be considered as examples of embodiments. Elements and materials may replace those illustrated and described herein, parts and processes may be reversed or omitted, certain features may be used independently, and embodiments or features of embodiments may be combined, all of which will be apparent to those skilled in the art after having benefited from this description. Changes may be made to the elements described herein without departing from the spirit and scope of the present invention described in the following claims. The headings used herein are for organizational purposes only and are not meant to be used to limit the scope of the description.

[0393] As used in this application, the word "may" is used in a permissive sense (i.e., meaning having the potential to), rather than in a mandatory sense (i.e., meaning must). The words "include," "including," and "includes," etc. mean including, but not limited to. As used in this application, the singular forms "a," "an," and "the" include plural referents unless the context clearly indicates otherwise. Thus, for example, reference to "an" element or "an" element includes combinations of two or more elements, even though other terms and phrases, such as "one or more," are used for one or more elements. Unless otherwise indicated, the term "or" is non-exclusive, i.e., covers both "and" and "or." Terms describing conditional relationships (e.g., "in response to X, Y," "at X, Y," "if X, Y," "when X, Y," etc.) encompass causal relationships where the antecedent is a necessary causal condition, the antecedent is a sufficient causal condition, or the antecedent is a contributing causal condition of the effect, e.g., "when condition Y is obtained, state X occurs" is common to "X occurs only when Y" and "X occurs when Y and Z." Such conditional relationships are not limited to the effect obtained immediately following the antecedent, as some effects may be delayed, and in the conditional statement, the antecedent is connected to its effect, e.g., the antecedent is related to the likelihood of the effect occurring. Unless otherwise indicated, a statement where multiple attributes or functions are mapped to multiple objects (e.g., one or more processors perform steps A, B, C, and D) encompasses all such attributes or functions mapped to all such objects and subsets of attributes or functions mapped to subsets of attributes or functions (e.g., all processors perform steps A to D, respectively, and processor 1 performs step A, processor 2 performs step B and part of step C, and processor 3 performs part of step C and step D). Further, unless otherwise indicated, a statement that a value or action is "based on" another condition or value encompasses instances where the condition or value is the only factor as well as instances where the condition or value is one factor among multiple factors. Unless otherwise indicated, a statement that "each" instance of a set has a certain property should not be read to exclude that certain other identical or similar members of the larger set do not have that property, i.e., each does not necessarily mean every one. References to range selections include the endpoints of the range.

[0394] In the above description, any process, description or box in the flowchart should be understood to represent a module, segment or portion of code, which includes one or more executable instructions for implementing specific logical functions or steps in the process, and alternative implementations are included within the scope of the exemplary embodiments of the present invention, in which functions may be performed out of order as shown or discussed, depending on the functionality involved, including substantially concurrently or in reverse order, as understood by those skilled in the art.

[0395] Although certain embodiments have been described, these embodiments are presented by way of example only and are not intended to limit the scope of the present disclosure. Indeed, the novel methods, devices, and systems described herein may be implemented in a variety of other forms; furthermore, various omissions, substitutions, and changes in the form of the methods, devices, and systems described herein may be made without departing from the spirit of the present disclosure. The accompanying claims and their equivalents are intended to cover such forms or modifications that fall within the scope and spirit of the present disclosure.

Claims

1. A method for determining deformation of a resist during a patterning process, the method comprising: obtaining a resist deformation model of the resist having a pattern, the resist deformation model being configured to simulate fluid flow of the resist due to capillary forces acting on a profile of at least one feature of the pattern, wherein the pattern extends in a plane and the capillary forces have at least a force component in or parallel to the plane; as well as A deformation of a resist pattern to be developed is determined based on an input pattern of the resist deformation model via a processor and the resist deformation model.

2. A method according to claim 1, wherein the resist deformation model is based on the Navier-Stokes flow equations, and wherein the fluid flow is characterized by Stokes flow and / or Hull-Shaw flow, and wherein the fluid flow is characterized by a fundamental solution that decays at a rate at least as fast as 1 / R with a distance, referred to as R, to a specific location in the resist and is either regular in the resist or singular only at the specific location in the resist.

3. The method of claim 1 , wherein determining the deformation comprises: defining a plurality of vertices along a profile of the at least one feature; determining a capillary force at a given vertex along a profile of the at least one feature; and A velocity flow field of the fluid flow due to capillary forces is determined based on a superposition of velocity responses at one or more of a plurality of vertices along a profile of the at least one feature and based on boundary conditions, wherein the velocity response is due to capillary forces acting at vertices adjacent to a given vertex.

4. The method of claim 3, wherein defining the plurality of vertices comprises: The plurality of vertices are distributed so that they are evenly spaced apart while conserving area or volume of a profile of at least one feature, wherein the capillary force acting on a given vertex is the sum of the tension forces on either side of the given vertex.

5. The method of claim 3, wherein the velocity response is characterized by Stokes quanta, wherein the Stokes quanta represent the velocity field due to point forces in the Stokes flow.

6. The method according to claim 3, wherein determining the velocity flow field further comprises: Decomposing the velocity flow field into a squeeze flow and a high-order velocity flow; applying a first boundary condition corresponding to the extrusion flow; and Based on the first boundary condition, the squeeze flow is removed from the velocity flow field.

7. The method of claim 6, wherein the squeeze flow is resist flow due to a net inward flux or a net outward flux across a resist domain boundary, thereby causing large-scale migration of features in the resist pattern.

8. The method of claim 6, wherein applying the first boundary condition comprises: The flow rate across the border of the resist domain is set to zero; and / or The velocity across the boundary of the resist domain was set to a no-flux condition.

9. The method of claim 7, wherein the total inward flow rate across the resist domain boundary is set to zero by providing rotators of selected strength at one or more corners of the resist domain boundary.

10. The method of claim 3, wherein determining the velocity flow field further comprises: The velocity at a given vertex along the profile of the at least one feature is obtained based on the velocities of all other vertices due to capillary forces at each of the other vertices by multiplication with the Stokes sub-force, and wherein contributions to the velocity flow field due to features far from the given vertex are neglected.

11. The method according to claim 3, further comprising: determining a force at a localized region within the resist; as well as The deformation of the entire area of ​​the resist is obtained based on a simulation of the resist deformation model using the force at the local area.

12. The method according to claim 1, wherein the input pattern is a design pattern, a resist image, a mask pattern and / or an aerial image, wherein the resist is a chemically amplified or non-chemically amplified negative resist or positive resist, and wherein further the input pattern is a grayscale image and / or a binary image.

13. The method according to claim 1, further comprising: calculating a critical dimension between a pair of locations disposed on a boundary of the at least one feature in the developed resist pattern using the resist deformation model; as well as calculating the error between the calculated critical dimension and the measured critical dimension of the actual developed resist pattern, Wherein calculating the error comprises using a cross-correlation matrix between the printed wafer data and the simulated image for comparison.

14. The method of claim 1, wherein the deformation is determined at a plurality of locations, each location corresponding to a point located on a boundary of a developed portion of a developed resist pattern for an input pattern.

15. A non-transitory computer program product comprising machine-readable instructions for causing a processor to implement the method according to claim 1.

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