CN112363371A - Narrow-band level set calculation method for light source mask collaborative optimization semi-implicit discretization - Google Patents

Narrow-band level set calculation method for light source mask collaborative optimization semi-implicit discretization Download PDF

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CN112363371A
CN112363371A CN202011180414.9A CN202011180414A CN112363371A CN 112363371 A CN112363371 A CN 112363371A CN 202011180414 A CN202011180414 A CN 202011180414A CN 112363371 A CN112363371 A CN 112363371A
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level set
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CN112363371B (en
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沈逸江
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Guangdong University of Technology
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    • G—PHYSICS
    • G03—PHOTOGRAPHY; CINEMATOGRAPHY; ANALOGOUS TECHNIQUES USING WAVES OTHER THAN OPTICAL WAVES; ELECTROGRAPHY; HOLOGRAPHY
    • G03F—PHOTOMECHANICAL PRODUCTION OF TEXTURED OR PATTERNED SURFACES, e.g. FOR PRINTING, FOR PROCESSING OF SEMICONDUCTOR DEVICES; MATERIALS THEREFOR; ORIGINALS THEREFOR; APPARATUS SPECIALLY ADAPTED THEREFOR
    • G03F7/00—Photomechanical, e.g. photolithographic, production of textured or patterned surfaces, e.g. printing surfaces; Materials therefor, e.g. comprising photoresists; Apparatus specially adapted therefor
    • G03F7/70—Microphotolithographic exposure; Apparatus therefor
    • G03F7/70483—Information management; Active and passive control; Testing; Wafer monitoring, e.g. pattern monitoring
    • G03F7/70491—Information management, e.g. software; Active and passive control, e.g. details of controlling exposure processes or exposure tool monitoring processes
    • G—PHYSICS
    • G03—PHOTOGRAPHY; CINEMATOGRAPHY; ANALOGOUS TECHNIQUES USING WAVES OTHER THAN OPTICAL WAVES; ELECTROGRAPHY; HOLOGRAPHY
    • G03F—PHOTOMECHANICAL PRODUCTION OF TEXTURED OR PATTERNED SURFACES, e.g. FOR PRINTING, FOR PROCESSING OF SEMICONDUCTOR DEVICES; MATERIALS THEREFOR; ORIGINALS THEREFOR; APPARATUS SPECIALLY ADAPTED THEREFOR
    • G03F7/00—Photomechanical, e.g. photolithographic, production of textured or patterned surfaces, e.g. printing surfaces; Materials therefor, e.g. comprising photoresists; Apparatus specially adapted therefor
    • G03F7/70—Microphotolithographic exposure; Apparatus therefor
    • G03F7/70483—Information management; Active and passive control; Testing; Wafer monitoring, e.g. pattern monitoring
    • G03F7/70491—Information management, e.g. software; Active and passive control, e.g. details of controlling exposure processes or exposure tool monitoring processes
    • G03F7/705—Modelling or simulating from physical phenomena up to complete wafer processes or whole workflow in wafer productions

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Abstract

The invention discloses a light source mask collaborative optimization semi-implicit discretization narrow-band level set calculation method, which comprises the steps of taking discretization circuit board graph variables and discretization light source variables as observation matrixes, taking corresponding imaging matrixes and discretization matrixes of target patterns as constraint conditions, and constructing a Lagrangian equation comprising distance regularization level set terms and the constraint conditions to obtain a time-varying differential equation; then respectively and implicitly and explicitly discretizing a diffusion term and a non-diffusion term in the three-diagonal linear equation set, constructing a three-diagonal linear equation set in the horizontal direction and the vertical direction by using additive operator splitting, and efficiently solving by using a Thomas method. The invention can improve the optimization convergence efficiency while reducing the optimization dimension.

Description

Narrow-band level set calculation method for light source mask collaborative optimization semi-implicit discretization
Technical Field
The invention relates to the technical field of optical proximity correction in the photoetching resolution enhancement technology, in particular to a narrow-band level set calculation method for light source mask collaborative optimization semi-implicit discretization.
Background
Projection lithography systems are a core device for fabricating very large scale integrated circuits with line widths in the micrometer and nanometer range. The projection photoetching system mainly comprises an illumination system, a mask, a projection objective, a pupil, a silicon wafer coated with photoresist and the like, wherein a light wave emitted by a light source irradiates and penetrates through the mask to generate a mask near field, and a mask pattern is transferred onto the silicon wafer through projection of the projection objective, low-pass filtering of the pupil and etching of the photoresist, but the shape on the silicon wafer is distorted due to information loss in the image transmission process, and the distortion becomes more obvious along with the reduction of the wavelength, the increase of the numerical aperture and the improvement of the complexity of the technological process.
As semiconductor integration strength continues to increase, lithography technology nodes enter 22nm nodes, and images printed on silicon wafers using 193nm deep ultraviolet projection lithography systems must be improved in resolution and fidelity by Resolution Enhancement Techniques (RETs) and Optical Proximity Correction (OPC). The SMO technology is a common RETs technology, is used for improving the imaging performance of a key area in an integrated circuit, is an important component of OPC as an inverse lithography technology (ILT for short), and puts higher demands on optimization and image processing technologies, and puts forward a new calculation strategy to improve the calculation efficiency of a pixelated OPC technology. The existing pixelation SMO technology regards a light source and a mask as a pixel map, improves the performance of photoetching imaging by optimizing the intensity values of all light source pixels and mask pixels and adjusting the incidence angle of the light source, but the large number of optimization variables and the sensitivity of an ILT algorithm to the optimization step greatly influence the synthesis efficiency of the light source and the mask. Meanwhile, the computational cost of the pixelated SMO technology is further increased by the continuously improved integration density of the circuit layout, the perception of the process manufacturability and the reduction of high precision. Therefore, the computational efficiency and convergence efficiency of the existing pixelized SMO techniques are to be further improved.
The related literature (Optics Express, 2017, 25 (18): 21775) proposes a light source mask collaborative optimization algorithm based on level set evolution. The method characterizes the light source and mask pattern profiles as a zero level set for a level set function, thereby realizing light source mask collaborative optimization through the evolution of the level set function according to the normal direction speed. In addition, the method uses a Couerant-Friedrichs-Lewy (CFL for short) condition to restrict the iteration step length and ensure the stability of the level set evolution.
However, the method has the following two disadvantages:
firstly, the method optimizes and updates all observation points on the light source and the mask layout in the iterative process, the gradients of the aerial image and the cost function on the observation points of the light source and the mask are required to be calculated, the calculation efficiency is low, and the method is not beneficial to large-scale collaborative optimization simulation of the light source and the mask.
Second, the above method uses a simple and widely used explicit discretization method, and the iteration step is suppressed to generate excessive iterations due to the limitation of the koron-friedrichs-column dimension (CFL), resulting in slow convergence; the corresponding implicit discretization method overcomes the stability constraint, can use a large enough iteration step, but needs to solve a linear equation set with a considerable scale, has high computational complexity, and is difficult to apply to an actual OPC technology.
In summary, the conventional SMO method needs to be further improved and enhanced in terms of optimizing mask frame, calculating efficiency, and converging efficiency.
Disclosure of Invention
The invention aims to overcome the defects of the prior art, and provides a light source mask co-optimization (SMO for short) semi-implicit (SI for short) discretization narrow-band level set (NL) calculation method which can reduce the optimization dimension and improve the optimization convergence efficiency. In addition, instead of optimizing all mask pixels, local optimization is selected for observation points (Monitoring pixels) in narrow bands adjacent to the zero level set of the level set function, so as to reduce the computational complexity.
In order to achieve the purpose, the technical scheme provided by the invention is as follows:
a narrow-band level set calculation method for light source mask collaborative optimization semi-implicit discretization comprises the following steps:
s1, initializing the light source to Ns×NsLight source pattern J, patterns M and I for rasterizing mask pattern and target into NxN0；
S2, selecting a level set function philJ or M, and the profile of the light source and mask pattern is regarded as a level set function philZero level set of
Figure BDA0002749961390000031
In the above formula, r represents a space coordinate (x, y), lintAnd lextPredefined negative and positive numbers;
s3, constructing a vector imaging model of wafer imaging:
I＝Γ(J，M)＝sig(Ia).
in the above formula, I is wafer imaging, IaFor aerial image imaging, gamma (-) is a wafer imaging model, sig (-) is an S-shaped activation function for approximating the exposure development process, and the expression is
Figure BDA0002749961390000037
S4, constructing a light source mask collaborative optimization problem as the following energy formula:
Figure BDA0002749961390000032
in the above formula, μ is a constant,
Figure BDA0002749961390000033
for the distance regularization level set term, defined as:
Figure BDA0002749961390000034
Eext(φ) is an external energy term used to minimize mask pattern distortion, defined as:
Figure BDA0002749961390000035
in the above formula, the first and second carbon atoms are,
Figure BDA0002749961390000036
for the region boundary of the level set function,
Figure BDA0002749961390000038
is a gradient operator;
s5, constructing a narrow band region adjacent to a zero level set of a level set function, and obtaining a stable time model:
Figure BDA0002749961390000041
in the above formula, Δ is Laplace operator, t is artificial time, v (r, t) is normal direction velocity of level set function evolution, BbA narrow band region containing the adjacent phi zero level set, and b is a designated narrow band width;
s6, performing semi-implicit discretization on the partial differential equation in the step S5, and decomposing the solved problem into two linear equation sets in the coordinate axis direction by using Additive Operator Splitting (AOS);
Figure BDA0002749961390000042
in the above formula, t is discretized into tkK τ, k 0, 1, 2, …, τ being the iteration step,
Figure BDA0002749961390000043
(mask) or
Figure BDA0002749961390000044
(light source) is to stack phi into a vector according to the dictionary order, namely, the column vectors, and construct a non-diffusion term g (r, t) -v (r, t) -mu delta omega;
s7, rapidly solving the three-diagonal linear equation set by using a Thomas method and updating the light source and the mask;
s8, repeating the steps S5-S7 until the pattern error is less than the specified value or the update times reaches the upper limit.
Further, the step S5 is specifically configured to construct a narrowband region adjacent to the zero level set of the level set function as follows:
S5-1when k is 0, a narrow band is constructed
Figure BDA0002749961390000045
Where Z is the set of all observation points spanning the zero level set,
Figure BDA0002749961390000046
a narrow-band region having a width r around an observation point (x, y);
constructing a stable time model:
Figure BDA0002749961390000047
in the above formula, Δ is Laplace operator, t is artificial time, v (r, t) is normal direction velocity of level set function evolution, BbA narrow band region containing the adjacent phi zero level set, and b is a designated narrow band width;
s5-2, executing steps S6 and S7 to update the mask and the light source;
s5-3, calculating Z in the k iterationk+1Is composed of
Figure BDA0002749961390000048
A set of observation points spanning a zero level set;
s5-4, constructing a new narrow band
Figure BDA0002749961390000051
Further, the concrete steps of performing semi-implicit discretization on the partial differential equation in the step S5, and decomposing the solved problem into two linear equation sets in the coordinate axis direction by using additive operator splitting are as follows:
s6-1, use
Figure BDA0002749961390000052
Represents tkTime mesh node riElement ω (r) of (C)i，tk) And will diffuse the term
Figure BDA0002749961390000053
Implicit discretization and explicit discretization of "balloon force" g (r, t) ═ v (r, t) - μ Δ ω, a semi-implicit discretization scheme is defined as
Figure BDA0002749961390000054
In the above formula, the first and second carbon atoms are,
Figure BDA0002749961390000055
4 adjacent nodes which are the node i, and g (i, k) is a discretization value of g (r, t);
s6-2, converting the semi-implicit discretization formula into a matrix vector form:
Figure BDA0002749961390000056
in the above formula, ArAn interaction matrix representing the r direction, the element a of whichijrIs defined as:
Figure BDA0002749961390000057
s6-3, selection in the x-axis direction
Figure BDA0002749961390000058
The observation point in (1);
s6-4, solving the linear equation set (I-2 tau A)x(ωk))uk+1＝ωk+τgkObtaining uk+1；
S6-5, selection in y-axis direction
Figure BDA0002749961390000059
The observation point in (1);
s6-6, solving a linear equation set (I-2 tau A)y(ωk))vk+1＝ωk+τgkObtaining vk+1；
S6-7, average omegak+1＝0.5(uk+1+vk+1)。
Compared with the prior art, the principle and the advantages of the scheme are as follows:
first, the light source mask collaborative optimization framework involved in the present scheme includes regularization of the distance level set of the objective function. Compared with the traditional level set method, the scheme not only ensures the stability of level set evolution and the accuracy of calculation without repeated level set initialization, but also provides a diffusion item required in semi-implicit discretization.
Secondly, the light source mask collaborative optimization algorithm related to the scheme only optimizes observation points in narrow bands adjacent to the zero level set of the level set function, so that the optimization dimensionality is reduced, and the calculation efficiency is improved.
Thirdly, the light source mask collaborative optimization algorithm related to the scheme adopts semi-implicit discretization, overcomes the stability requirement of step length constraint of explicit discretization numerical calculation, enables a large enough iteration step length to be possible, and improves the convergence efficiency.
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In order to more clearly illustrate the embodiments of the present invention or the technical solutions in the prior art, the services required for the embodiments or the technical solutions in the prior art will be briefly described below, it is obvious that the drawings in the following description are only some embodiments of the present invention, and for those skilled in the art, other drawings can be obtained according to these drawings without creative efforts.
FIG. 1 is a flowchart of a narrow-band level set calculation method for light source mask collaborative optimization semi-implicit discretization according to the present invention;
FIG. 2 is a schematic illustration of an optimized light source, mask pattern and its imaging in a photoresist at a nominal exposure using a conventional level set method;
FIG. 3 shows the SMO method according to the present invention, the light source iteration step τs0.08 and 0.1, respectively, mask iteration step τmOptimized light source, mask pattern and photoresist obtained at 1.0, 1.2 and 1.5 respectivelyAn imaging schematic;
FIG. 4 is a schematic diagram showing a comparison of convergence efficiency between SMO based on semi-implicit discretization and the SMO method involved in the present invention using a conventional level set method, a narrowband level set method;
FIG. 5 is a schematic diagram showing comparison of computational efficiencies based on semi-implicit discretization SMO and the SMO method involved in the present invention using a conventional level set method, a narrowband level set method;
FIG. 6 shows the light source iteration step τ in the SMO method according to the present inventions0.08 and 0.1, respectively, mask iteration step τmThe convergence efficiency at 1.0, 1.2 and 1.5 respectively is shown in comparison.
Detailed Description
The invention will be further illustrated with reference to specific examples:
the principle of the invention is as follows: in order to improve the operation efficiency and the convergence efficiency, the invention constructs the light source mask collaborative optimization problem as an energy formula, namely:
Figure BDA0002749961390000071
wherein the first term is a distance regularization level set term to ensure a level set function sign distance characteristic
Figure BDA0002749961390000072
The second term is an external energy term used for minimizing the distortion degree of the mask pattern and forcing optimization to be carried out in the direction of reducing the pattern error PE, namely, the value of the optimized aerial image on an observation point is gradually close to the value of the target circuit board diagram;
in one aspect, the invention involves using narrowband computations that optimize only a narrowband region B containing a neighborhood of the zero level set of phibB is a specified narrow bandwidth. Due to the fact that the optimization dimension is reduced, the calculation efficiency can be effectively improved;
on the other hand, the SMO method of the AOS-SI technology used in the invention uses a Level-set semi-implicit discretization method to convert the time model into a semi-implicit equation, which overcomes the stability requirement of the explicit discretization method on iteration step constraint, so that a large enough iteration step becomes possible, and the convergence efficiency is effectively improved;
as shown in fig. 1, the method for calculating a narrow-band level set by using a light source mask to cooperatively optimize semi-implicit discretization according to the embodiment of the present invention includes the following specific steps:
s1, initializing the light source to Ns×NsLight source pattern J, patterns M and I for rasterizing mask pattern and target into NxN0；
S2, selecting a level set function philJ or M, and the profile of the light source and mask pattern is regarded as a level set function philZero level set of
Figure BDA0002749961390000073
In the above formula, r represents a space coordinate (x, y), lintAnd lextPredefined negative and positive numbers; s3, constructing a vector imaging model of wafer imaging:
I＝Γ(J，M)＝sig(Ia)；
in the above formula, I is wafer imaging, IaFor aerial image imaging, gamma (-) is a wafer imaging model, sig (-) is an S-shaped activation function for approximating the exposure development process, and the expression is
Figure BDA0002749961390000081
S4, constructing a light source mask collaborative optimization problem as the following energy formula:
Figure BDA0002749961390000082
in the above formula, μ is a constant,
Figure BDA0002749961390000083
for the distance regularization level set term, defined as:
Figure BDA0002749961390000084
Eext(φ) is an external energy term used to minimize mask pattern distortion, defined as:
Figure BDA0002749961390000085
in the above formula, the first and second carbon atoms are,
Figure BDA0002749961390000086
for the region boundary of the level set function,
Figure BDA0002749961390000087
is a gradient operator;
s5, constructing a narrow band region adjacent to a zero level set of a level set function, and obtaining a stable time model:
when S5-1 and k equals 0, a narrow band is constructed
Figure BDA0002749961390000088
Where Z is the set of all observation points spanning the zero level set,
Figure BDA0002749961390000089
a narrow band region having a width b with an observation point (x, y) as a center;
constructing a stable time model:
Figure BDA00027499613900000810
in the above formula, Δ is Laplace operator, t is artificial time, v (r, t) is normal direction velocity of level set function evolution, BbA narrow band region containing the adjacent phi zero level set, and b is a designated narrow band width;
s5-2, executing steps S6 and S7 to update the mask and the light source;
s5-3, calculating Z in the k iterationk+1Is composed of
Figure BDA00027499613900000811
A set of observation points spanning a zero level set;
s5-4, constructing a new narrow band
Figure BDA00027499613900000812
S6, performing semi-implicit discretization on the partial differential equation in the step S5, and decomposing the solved problem into two linear equation sets in the coordinate axis direction by using Additive Operator Splitting (AOS):
s6-1, use
Figure BDA0002749961390000091
Represents tkTime mesh node riElement ω (r) of (C)i，tk) And will diffuse the term
Figure BDA0002749961390000092
Implicit discretization and explicit discretization of "balloon force" g (r, t) ═ v (r, t) - μ Δ ω, a semi-implicit discretization scheme is defined as
Figure BDA0002749961390000093
In the above formula, the first and second carbon atoms are,
Figure BDA0002749961390000094
4 adjacent nodes which are the node i, and g (i, k) is a discretization value of g (r, t);
discretizing t into tkK τ, k 0, 1, 2, …, where τ is the iteration step;
stacking the column vectors into a vector structure in dictionary order
Figure BDA0002749961390000095
(mask) or
Figure BDA0002749961390000096
(light source);
constructing a non-diffusion term g (r, t) ═ -v (r, t) - μ Δ ω;
s6-2, converting the semi-implicit discretization formula into a matrix vector form:
Figure BDA0002749961390000097
in the above formula, ArAn interaction matrix representing the r direction, the element a of whichijrIs defined as:
Figure BDA0002749961390000098
s6-3, selection in the x-axis direction
Figure BDA0002749961390000099
The observation point in (1);
s6-4, solving the linear equation set (I-2 tau A)x(ωk))uk+1＝ωk+τgkObtaining uk+1；
S6-5, selection in y-axis direction
Figure BDA00027499613900000910
The observation point in (1);
s6-6, solving a linear equation set (I-2 tau A)y(ωk))vk+1＝ωk+τgkObtaining vk+1；
S6-7, average omegak+1＝0.5(uk+1+vk+1)；
S7, rapidly solving the three-diagonal linear equation set by using a Thomas method and updating the light source and the mask;
s8, repeating the steps S5-S7 until the Pattern Error (PE) is less than the specified value or the number of updates reaches the upper limit.
To demonstrate the effectiveness and superiority of the embodiments of the invention, the following simulations were performed:
FIG. 2 is a schematic view of an unoptimized light source and mask imaged through a projection lithography system. 201 is a light source pattern before optimization, 202 is a mask pattern and a target pattern, 203 is a schematic diagram of imaging in the photoresist at an optimal focal plane under a rated exposure dose, a pattern error PE is 3372, and the pattern error is defined as the square of the Euler distance between the imaging of the photoresist and the target pattern. Where black represents a non-light emitting region, i.e., light intensity is 0, and white represents a light emitting region, i.e., light intensity is 1. 204 is the light source pattern optimized by the level set method, 205 is the mask pattern optimized by the level set, 206 is the image in the photoresist at the best focal plane under the optimized rated exposure dose, and the pattern error PE is 371. In this system the illumination system wavelength is 193nm, the annular illumination source incident angle is between 0.6 and 0.9, the system numerical aperture is 1.35, the size of each grid of the image is 4nm, and the steepness and threshold of the photoresist function are 85 and 0.25, respectively.
Fig. 3 is a schematic view of an image of a light source and a mask through a projection lithography system optimized by the SMO algorithm according to the present invention. τ for mask iteration stepmRepresenting the light source iteration step by τsAnd (4) showing. The first action takes τm＝1.0，τsThe optimized light source, the mask pattern and the imaging schematic diagram in the photoresist at the optimal focal plane under the optimized rated exposure dose are 0.08, and the pattern error PE is 501; second row with τm＝1.2，τs0.08, its pattern error PE is 424; third row with τm＝1.5，τs0.08, the pattern error PE is 367; fourth row by τm＝1.0，τs0.1, the pattern error PE is 498; the fifth element adopts taum＝1.2，τs0.1, the pattern error PE is 419; line six employs τm＝1.5，τsThe pattern error PE is 367 for 0.1.
As shown in FIG. 4, the SMO calculation method based on semi-implicit discretization SMO and the invention adopts a level set algorithm and a narrow-band level set algorithmThe convergence efficiency of the method is compared with the schematic diagram. As can be seen in fig. 4, the level set algorithm converges to the pattern error PE 371 using 50 iterations; the narrow-band level set algorithm uses 40 iterations to converge to a pattern error PE 365; based on semi-implicit discretization SMO, 25 iterations are used, and step length tau is adoptedm＝1.0，τsWhen the value is 0.08, the pattern error PE 536 converges, and the step τ is adoptedm＝1.5，τsWhen the value is 0.1, the pattern error PE 383 converges; the SMO algorithm uses 25 iterations and adopts step length taum＝1.0，τsWhen the value is 0.08, the pattern error PE 501 converges, and the step τ is usedm＝1.5，τsWhen the value is 0.1, the pattern error PE 367 converges.
Fig. 5 is a schematic diagram showing comparison of computational efficiency of SMO based on semi-implicit discretization based on level set algorithm, narrow-band level set algorithm and SMO algorithm according to the present invention. Wherein the iteration step sizes of the level set algorithm and the narrow-band level set algorithm are constrained by CFL conditions, and the iteration step size based on the semi-implicit discretization SMO is the same as that of the SMO algorithm related by the invention. As can be seen from fig. 4, the level set algorithm uses 50 iterations, the total duration is 200.4 minutes, and the average time per iteration is 4.01 minutes; the narrow-band level set algorithm uses 40 iterations, the total time length is 41.0 minutes, and the average time of each iteration is 1.02 minutes; step length tau is adopted based on semi-implicit discretization SMOm＝1.0，τsAt 0.08, 25 iterations are used, with a total duration of 105.5 minutes, with each iteration averaging 4.22 minutes, using a step τm＝1.5，τsWhen the time is 0.1, 25 iterations are used, the total time length is 107.5 minutes, and the average time of each iteration is 4.30 minutes; the SMO algorithm related by the invention adopts step length taum＝1.0，τsAt 0.08, 25 iterations are used, with a total duration of 49.1 minutes, with each iteration averaging 1.96 minutes, using a step τm＝1.5，τsWhen equal to 0.1, 25 iterations were used, with a total duration of 49.9 minutes, with an average time of 1.99 minutes per iteration.
Fig. 6 is a schematic diagram illustrating the convergence efficiency of the SMO algorithm according to the present invention when different iteration steps are used. The SMO algorithm of the present invention uses 25 iterationsBy using τm＝1.0，τsWhen the value is 0.08, the pattern error PE 501 converges; using τm＝1.2，τsWhen equal to 0.08, the pattern error PE 424 converges; using τm＝1.5，τsWhen the value is 0.08, the pattern error PE 367 converges; using τm＝1.0，τsWhen the value is 0.1, the pattern error PE 498 is converged; using τm＝1.2，τsWhen the value is 0.1, the pattern error PE 419 converges; using τm＝1.5，τsWhen the value is 0.1, the pattern error PE 367 converges.
As can be seen from comparison of fig. 2, 3, 4, 5, and 6, compared with the existing SMO algorithm, the SMO algorithm of the present invention can use a larger iteration step size than the narrow band level set algorithm, thereby improving the convergence efficiency; compared with the SMO algorithm based on semi-implicit type, the optimization dimension is reduced, and therefore the operation efficiency is improved.
The above-mentioned embodiments are merely preferred embodiments of the present invention, and the scope of the present invention is not limited thereto, so that variations based on the shape and principle of the present invention should be covered within the scope of the present invention.

Claims (3)

1. A narrow-band level set calculation method for light source mask collaborative optimization semi-implicit discretization is characterized by comprising the following steps:
s1, initializing the light source to NS×NSLight source pattern J, patterns M and I for rasterizing mask pattern and target into NxN0；
S2, selecting a level set function philJ or M, and the profile of the light source and mask pattern is regarded as a level set function philZero level set of
Figure FDA0002749961380000011
In the above formula, r represents a space coordinate (x, y), lintAnd lextPredefined negative and positive numbers;
s3, constructing a vector imaging model of wafer imaging:
I＝Γ(J,M)＝sig(Ia)；
in the above formula, I is wafer imaging, IaFor aerial image imaging, gamma (-) is a wafer imaging model, sig (-) is an S-shaped activation function for approximating the exposure development process, and the expression is
Figure FDA0002749961380000012
S4, constructing a light source mask collaborative optimization problem as the following energy formula:
Figure FDA0002749961380000013
in the above formula, μ is a constant,
Figure FDA0002749961380000014
for the distance regularization level set term, defined as:
Figure FDA0002749961380000015
Eext(φ) is an external energy term used to minimize mask pattern distortion, defined as:
Figure FDA0002749961380000016
in the above formula, the first and second carbon atoms are,
Figure FDA0002749961380000017
for the region boundary of the level set function,
Figure FDA0002749961380000018
is a gradient operator;
s5, constructing a narrow band region adjacent to a zero level set of a level set function, and obtaining a stable time model:
Figure FDA0002749961380000021
in the above formula, Δ is Laplace operator, t is artificial time, v (r, t) is normal direction velocity of level set function evolution, BbA narrow band region containing the adjacent phi zero level set, and b is a designated narrow band width;
s6, performing semi-implicit discretization on the partial differential equation in the step S5, and decomposing the solved problem into two linear equation sets in the coordinate axis direction by using additive operator splitting;
Figure FDA0002749961380000022
in the above formula, t is discretized into tkK τ, k 0, 1, 2, …, τ being the iteration step,
Figure FDA0002749961380000023
or
Figure FDA0002749961380000024
Stacking phi into a vector according to the dictionary sequence, namely stacking column vectors, and constructing a non-diffusion term g (r, t) -v (r, t) -mu delta omega;
s7, rapidly solving the three-diagonal linear equation set by using a Thomas method and updating the light source and the mask;
s8, repeating the steps S5-S7 until the pattern error is less than the specified value or the update times reaches the upper limit.
2. The method for calculating the narrow-band level set of the light source mask collaborative optimization semi-implicit discretization according to claim 1, wherein the step S5 of constructing the narrow-band region adjacent to the zero level set of the level set function specifically comprises the steps of:
when S5-1 and k equals 0, a narrow band is constructed
Figure FDA0002749961380000025
Where Z is the set of all observation points spanning the zero level set,
Figure FDA0002749961380000026
a narrow band region having a width b with an observation point (x, y) as a center;
constructing a stable time model:
Figure FDA0002749961380000027
in the above formula, Δ is Laplace operator, t is artificial time, v (r, t) is normal direction velocity of level set function evolution, BbA narrow band region containing the adjacent phi zero level set, and b is a designated narrow band width;
s5-2, executing steps S6 and S7 to update the mask and the light source;
s5-3, calculating Z in the k iterationk+1Is composed of
Figure FDA0002749961380000028
A set of observation points spanning a zero level set;
s5-4, constructing a new narrow band
Figure FDA0002749961380000031
3. The narrow-band level set calculation method for light source mask collaborative optimization semi-implicit discretization according to claim 1, wherein the semi-implicit discretization is performed on partial differential equations in step S5, and the solving of the solution problem into two linear equation sets in coordinate axis direction by using additive operator splitting comprises the following specific steps:
s6-1, use
Figure FDA0002749961380000032
Represents tkTime mesh node riElement ω (r) of (C)i，tk) And will diffuse the term
Figure FDA0002749961380000033
Implicit discretization and explicit discretization of "balloon force" g (r, t) ═ v (r, t) - μ Δ ω, a semi-implicit discretization scheme is defined as
Figure FDA0002749961380000034
In the above formula, the first and second carbon atoms are,
Figure FDA0002749961380000035
4 adjacent nodes which are the node i, and g (i, k) is a discretization value of g (r, t);
s6-2, converting the semi-implicit discretization formula into a matrix vector form:
Figure FDA0002749961380000036
in the above formula, ArAn interaction matrix representing the r direction;
s6-3, selection in the x-axis direction
Figure FDA0002749961380000037
The observation point in (1);
s6-4, solving the linear equation set (I-2 tau A)x(ωk))uk+1＝ωk+τgkObtaining uk+1；
S6-5, selection in y-axis direction
Figure FDA0002749961380000038
The observation point in (1);
s6-6, solving a linear equation set (I-2 tau A)y(ωk))vk+1＝ωk+1＝ωk+τgkObtaining vk+1；
S6-7, average omegak+1＝0.5(uk+1+vk+1)。
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