Method and system for solving Fraunhofer diffraction field of mask pattern

Abstract

The invention belongs to the related technical field of integrated circuits, and discloses a method and a system for solving a Fraunhofer diffraction field of a mask pattern. The method comprises the following steps of: performing topological analysis on a mask pattern to extract a boundary of the mask pattern, and fitting the extracted boundary to obtain a mask contour formed by fitting a plurality of closed rings; converting a two-dimensional diffraction field integral of the mask pattern into a one-dimensional line integral along a contour curve; the diffraction field of each profile curve in each closed ring in the mask profile is calculated, the sum of the diffraction fields of all the profile curves in all the closed rings is the one-dimensional line integral diffraction field of the mask pattern along the profile curve, and therefore the two-dimensional diffraction field surface integral is obtained, namely the Fraunhofer diffraction field of the needed mask pattern. According to the invention, the problem of low calculation precision and efficiency in the diffraction field calculation process is solved.

Description

TECHNICAL FIELD

[0001] The present application belongs to the technical field of integrated circuits, and more particularly, to a method and system for solving a Fraunhofer diffraction field of a mask pattern. BACKGROUND

[0002] A photolithography system can be considered as a low-pass filter system. How to accurately and quickly evaluate the degree to which a mask pattern is affected by imaging of the photolithography system has always been a problem in calculating lithography invariants. For Fraunhofer far-field imaging simulation, the prior art directly uses fast FFT to simulate and calculate a mask pattern.

[0003] A traditional method discretizes a mask into a pixel matrix (Pixelated Mask) and uses fast Fourier transform (FFT) to calculate a frequency spectrum. The calculation complexity is , where is a grid resolution. As a process node is scaled down, the grid resolution must be exponentially increased to ensure accuracy, which results in a sharp increase in calculation amount. The photolithography system is essentially a low-pass filter, and only central low-frequency components participate in imaging (usually only a very small part of the frequency spectrum). After the FFT calculates all frequency domain data, most high-frequency components are truncated and discarded, which causes a great waste of computing power. Pixelization representation introduces an unavoidable staircase effect when processing a curvilinear mask or a diagonal line, which leads to simulation distortion at a sub-pixel level. In inverse lithography technology (ILT) and other gradient-based optimization tasks, a binary pixel mask is not differentiable. The prior art usually introduces a Sigmoid function for continuous approximation, which not only increases the calculation overhead but also causes a physical deviation between the simulation model and the actual mask (hard truncation).

[0004] Therefore, there is a need for a diffraction field calculation method for a mask pattern that can solve the above problems. SUMMARY

[0005] In view of the above defects or improvement needs of the prior art, the present application provides a method and system for solving a Fraunhofer diffraction field of a mask pattern, which solves the problems of low calculation accuracy and efficiency in the diffraction field calculation process.

[0006] To achieve the above object, according to one aspect of the present application, a method for solving a Fraunhofer diffraction field of a mask pattern is provided, which comprises the following steps: topologically analyzing a mask pattern to extract a boundary of the mask pattern, fitting the extracted boundary to obtain a mask profile formed by a plurality of closed loops, converting a two-dimensional area integral of the Fraunhofer diffraction field of the mask pattern into a one-dimensional line integral along a profile curve. The diffraction field of each profile curve in each closed loop in the mask profile is calculated respectively, and the sum of the diffraction fields of all profile curves in all closed loops is the one-dimensional line integral diffraction field of the mask pattern along the profile curve, thereby obtaining the Fraunhofer diffraction field of the mask pattern.

[0007] Further preferably, the fitting of the extracted boundary employs a piecewise Bezier curve.

[0008] Further preferably, the dimension reduction employs Green's formula.

[0009] Further preferably, the formula for reducing the two-dimensional area integral of the Fraunhofer diffraction field of the mask pattern to a one-dimensional line integral along the profile curve is as follows:

[0010] wherein, is the two-dimensional area integral of the Fraunhofer diffraction field of the mask pattern, , is the spatial frequency coordinate of the obtained diffraction field, i is the imaginary unit, e is the natural logarithm, represents the closed profile integral, is all curve profiles of the mask, is the spatial coordinate of the input mask, is the differential with respect to x, and is the differential with respect to y, respectively, is the input mask field.

[0011] Further preferably, the formula for calculating the diffraction field of each profile curve in each closed loop is as follows:

[0012] wherein, is the one-dimensional line integral of a certain profile curve in the mask pattern at the diffraction field coordinate , i is the imaginary unit, e is the natural logarithm, represents the line integral symbol, , is the spatial frequency coordinate of the obtained diffraction field, s is the curve parameter, and ds is the differential with respect to s, a and b are the minimum value and the maximum value of the range of s, respectively, is the input mask field.

[0013] Further preferably, the formula for calculating the one-dimensional line integral diffraction field of the mask pattern along the profile curve is as follows:

[0014] wherein, , ) is the coordinate of the obtained diffraction field frequency, is one-dimensional line integral of the mask pattern along the profile curve at the diffraction field coordinate , M represents that the mask pattern is composed of M closed regions, and represents the first closed region is composed of segment curve, represents the transmittance of the first closed region is , represents the summation symbol, represents the integral symbol, s is the curve parameter, , b is the minimum value and the maximum value of the s value range respectively, represents the coordinates of the points on the profile curve segment, and is the derivative of the coordinate with respect to the parameter , is the differential of s.

[0015] Further preferably, the fitting of the extracted boundary adopts non-uniform rational B-spline, B-spline curve, Fourier descriptor, implicit function representation, high-order polynomial fitting or Hermite spline; the dimension reduction adopts Gauss divergence theorem, Stokes formula, complex function line integral or boundary element method.

[0016] According to another aspect of the present application, a system for solving the Fraunhofer diffraction field of a mask pattern is provided, which comprises an executor for executing the above-mentioned method for solving the Fraunhofer diffraction field of a mask pattern.

[0017] According to still another aspect of the present application, a computer storage medium having a computer program stored thereon is provided, the computer program being used for implementing the above-mentioned method for solving the Fraunhofer diffraction field of a mask pattern.

[0018] Overall, compared with the prior art, the above technical solutions conceived by the present application have the following beneficial effects: 1. The present application abandons the traditional pixel grid representation, and adopts parameterized curve (such as Bezier curve) to accurately describe the mask profile; converts the two-dimensional area integral of the mask transmittance function into one-dimensional line integral along the profile, and then deduces and solves the diffraction field; the analytical calculation based on the parameterized curve eliminates the "step effect", and can provide theoretically accurate solution regardless of how the simulation grid is subdivided, thereby greatly improving the calculation accuracy and efficiency.

[0019] 2. The present application uses Bezier curve to change the numerous sampling points of mask Fresnel imaging problem to the edge integral of mask, compared with the traditional FFT (Fast Fourier Transform) based method, without high resolution gridding processing to mask, completely eliminates the aliasing effect, especially in processing super large scale mask or non-orthogonal special-shaped pattern, the method ensures the calculation accuracy, significantly reduces the memory overhead and greatly improves the calculation efficiency.

[0020] 3. The present application converts the two-dimensional area integral of mask transmission function into one-dimensional line integral along the boundary Bezier curve by Green formula. The calculation complexity is reduced from the "area pixel number" to the "contour control point number". When processing lithography mask with sparse spectrum characteristics, the operation speed can be improved by several orders of magnitude, and naturally supports sparse sampling.

[0021] 4. The one-dimensional line integral of diffraction field along the contour curve establishes a direct function relationship between the spectrum and the control point coordinates. This relationship comes from the analytical representation of the mask contour, so it is smooth and continuously derivable. When performing mask optimization, the control point coordinates can be directly differentiated without any artificial approximation, greatly improving the convergence speed and solution quality of the inverse lithography algorithm.

[0022] 5. In order to pursue accuracy, the traditional FFT method requires very high sampling frequency, resulting in huge memory space for storing mask matrix. The present application stores mask information through a series of Bezier control points, rather than storing high-resolution bitmaps. The data compression rate is very high, which significantly reduces the memory peak occupancy when performing full-chip simulation or large-scale multi-level optimization, making it possible to complete real-time simulation of super complex masks on ordinary computing hardware. BRIEF DESCRIPTION OF DRAWINGS

[0023] Figure 1 is a schematic diagram of fitting the boundary contour with Bezier curve segments according to the preferred embodiment of the present application.

[0024] Figure 2 is a schematic diagram of mask contour extraction according to the preferred embodiment of the present application, wherein (a) is the input mask pattern, and (b) is the Bezier curve extracted mask contour. DETAILED DESCRIPTION

[0025] In order to make the purpose, technical scheme and advantages of the present application clearer, the present application will be further described in detail below in combination with the drawings and examples. It should be understood that the specific examples described herein are only used to explain the present application and do not limit the present application. In addition, the technical features involved in each embodiment of the present application described below can be combined with each other as long as they do not conflict with each other.

[0026] A solving method of a Fraunhofer diffraction field of a mask pattern, the specific steps are as follows: (1) Vectorization coding of the mask pattern Topological analysis is performed on the input mask pattern, and the closed boundary is extracted. The boundary is fitted by using a piecewise Bezier curve, and the profile of the mask is obtained by fitting. The order of the Bezier curve is automatically selected according to the curvature change of the profile (such as linear, quadratic or cubic Bezier curve), as shown in formula (1).

[0027]

[0028] Wherein, s is a parameter variable, the value is usually between . P(s) represents the point on the Bezier curve corresponding to the parameter s, is the i-th control point, and n is the order of the Bezier curve. For the n-order Bezier curve, it contains control points, P(s) represents the sum.

[0029] Therefore, the mask pattern is parameterized as a sequence of ordered control points. This representation is continuous in mathematics, and completely eliminates the quantization error caused by pixelization.

[0030] As shown in Figure 1 , the blue line from P0 to P3 is the mask surface profile, and four points P0-P3 are obtained by fitting the piecewise Bezier curve.

[0031] For the mask shown in the left figure below, the profile of the curve and the Bezier curve control points are extracted by Bezier curve fitting coding. As shown in Figure 2 , the red and blue points are the control points of the series of Bezier curve segments obtained by fitting coding.

[0032] Finally, the mask pattern is coded as a set composed of multiple closed loops, and each loop is represented by a sequence of ordered Bezier control points. This representation not only retains the sub-pixel level accuracy of the mask edge, but also lays the foundation for subsequent analytical Fourier transform.

[0033] In practical applications, the following curves can also be used for boundary fitting based on the Bezier curve (Bezier Curve) in the present application, specifically: (a) Non-uniform rational B-spline (NURBS): Compared with the ordinary Bezier curve, NURBS can more accurately represent circular arcs and other quadratic curves, and has better local control, which can directly replace the Bezier control point for spectral analysis and derivation.

[0034] (b) B-splines: connecting control points by piecewise polynomial basis functions, which are more complex in form but have consistent derivative and integral properties with Bezier curves.

[0035] (c) Fourier Descriptors: representing the contour edge as Fourier series expansion of azimuthal angle, and directly calculating the corresponding mask spectrum by using the harmonic property in frequency domain.

[0036] (d) implicit function representation (e.g. Level Sets): although Level Sets are usually used for pixelated processing, under certain conditions, by extracting the contour equation of Level Sets, it is also possible to realize non-pixelated simulation by using integral transform.

[0037] (e) high-order polynomial fitting or Hermite Splines: using edge point position and derivative information to construct an analytical curve, which is still essentially a parametric polynomial integral.

[0038] Meanwhile, it should be emphasized that even if the Bezier curve is degenerated into a high-density polygonal line segment (Piecewise Linear), as long as it is applied to the frequency domain analysis conversion by using Green's formula or similar methods, it should still be within the protection scope of the present application.

[0039] (2) reducing the two-dimensional area integral of the mask transmission function to one-dimensional line integral along the contour Taking deep ultraviolet lithography scalar imaging as an example, the mask near field can be expressed as where is the transmittance of the mask, and is 0 or a constant for a binary mask. The Fraunhofer diffraction field of a binary mask (i.e. the Fourier transform of the mask near field) is:

[0040] where is the two-dimensional area integral of the Fraunhofer diffraction field of the mask pattern, represents the two-dimensional area integral, represents the spatial region enclosed by the mask contour, i is the imaginary unit, e is the natural logarithm, is the spatial coordinate of the input mask, and are the differentials with respect to x and y, respectively, is the mask field of the input mask pattern, and usually the transmittance in each closed region of the mask is constant. ) is the spatial frequency of the diffraction field. In the simulation of lithography imaging, the diffraction field needs to be sampled.

[0041] Using Green's formula (3), the area integral in (2) can be converted into a one-dimensional line integral of the diffraction field, i.e. formula (11).

[0042]

[0043] Q, P are functions, and the region is a closed region, represents the edge, is the spatial coordinate of the input mask, is the partial derivative with respect to x, y respectively, is all the curve profiles of the mask, represents the spatial region enclosed by the mask profile, represents the closed profile integral represents a two-dimensional area integral.

[0044] The specific derivation process is as follows: For formula (3), it is only necessary to find a set of specific functions P, Q that satisfy , so that the area integral can be converted into the edge integral.

[0045] It is not difficult to find that when and satisfy such conditions, for the region , here , that is, .

[0046] Calculate ,

[0047]

[0048]

[0049] where is the spatial coordinate of the input mask, , is the partial derivative with respect to x, y respectively, , is the frequency coordinate of the diffraction field, is the input mask field, is the natural logarithm, and i is the imaginary unit.

[0050] Therefore, the integral term on the left side of Green's formula

[0051] where , are partial derivatives with respect to x, y, respectively, P, Q are functions, are spatial coordinates of the input mask, , are frequency coordinates of the resulting diffracted field, is the input mask field, is the natural logarithm, i is the imaginary unit.

[0052] i.e.

[0053] where are spatial coordinates of the input mask, , are frequency coordinates of the resulting diffracted field, is the input mask field, is the natural logarithm, i is the imaginary unit, are partial derivatives with respect to x, y, respectively.

[0054] Substitute the result into the Green formula

[0055] where , are frequency coordinates of the resulting diffracted field, are spatial coordinates of the input mask, are partial derivatives with respect to x, y, respectively, is the input mask field, is the natural logarithm, i is the imaginary unit, denotes a two-dimensional area integral, denotes a closed contour integral, is the set of all curve contours of the mask, denotes the spatial region enclosed by the mask contours.

[0056] i.e.

[0057] where , are frequency coordinates of the resulting diffracted field, are spatial coordinates of the input mask, are partial derivatives with respect to x, y, respectively, is the input mask field, is the natural logarithm, i is the imaginary unit, denotes a two-dimensional area integral, denotes a closed contour integral, all the curve profiles of the mask, represents the space region enclosed by the mask profile.

[0058] For , equals 0, a meaningless case occurs. From the physical meaning of Fraunhofer diffraction field, the value of , corresponds to the area of the mask region, so it is necessary to find , again that satisfy the condition. Here let Then:

[0059] where is the spatial coordinate of the input mask, , are the partial derivatives with respect to x and y respectively, and P' and Q' are functions. At this time, the Green formula is

[0060] where, is the spatial coordinate of the input mask, are the differentials with respect to x and y respectively, represents the closed profile integral, represents the two-dimensional area integral, is all the curve profiles of the mask, represents the space region enclosed by the mask profile, A is the area of the region, and dA represents the differential of the area.

[0061] From the above, formula (11) is derived

[0062] where, is the two-dimensional area integral of the Fraunhofer diffraction field of the mask pattern. , are the spatial frequency coordinates of the obtained diffraction field, i is the imaginary unit, e is the natural logarithm, represents the closed profile integral, is all the curve profiles of the mask, is the spatial coordinate of the input mask, are the differentials with respect to x and y respectively, is the input mask field. Generally, each closed region of the mask has a constant transmittance, and generally , (0 < T < 1) or 0, respectively, represent the transmissive region and the non-transmissive region.

[0063] According to the definition of Fourier transform, the mask spectrum is the integral of the mask transmission function over a two-dimensional region. According to Green's theorem, if the function is constant in a closed region, the two-dimensional area integral of the region can be accurately converted into a line integral along the edge profile of the region.

[0064] Although the above embodiment uses Green's theorem to realize the conversion from two-dimensional spatial domain area integral to one-dimensional edge line integral, in actual application, the following methods can also be used for dimension reduction processing: (a) 2D form of Gauss's divergence theorem: by defining a specific vector field (whose divergence is equal to the complex transmission function of the mask), the flux integral in the region is converted into the flux integral on the closed curve by using the divergence theorem.

[0065] (b) Stokes' theorem: in the perspective of complex plane or three-dimensional manifold simplification, the integral of the mask function is converted into the edge curl integral by the exterior differential form.

[0066] (c) Cauchy integral formula and its corollaries: if the mask edge is characterized as an analytic curve on the complex plane, the properties of analytic functions can be used to convert the area integral into an edge path integral based on complex variables.

[0067] (d) BEM idea conversion: use the Green function as the weight function to convert the control equation (such as the source term integral of the wave equation) into an integral equation containing only boundary variables.

[0068] The above methods are consistent with the Bessel-Green framework used in the present application in terms of physical mechanism, and the convergence and accuracy gain of the calculation results are both derived from the core idea of "converting from surface to line and analytical characterization" proposed in the present application.

[0069] (3) Bessel profile analytical integral: The above profile integral is decomposed into several Bessel curve paths. Bessel curve has an analytical expression in the form of a polynomial, and its Fourier integral term in the frequency domain component ( ) can be derived to obtain an analytical solution or an efficient numerical approximation solution. To obtain the mapping relationship from "Bessel curve control points" to "Fraunhofer diffraction field", take a Bessel curve as an example for derivation, and denote the obtained one-dimensional diffraction field line integral as .

[0070] Taking a third-order Bessel curve as an example, the coordinates of each point on the curve are , and according to formula (1), the coordinates of the points on the curve profile are obtained:

[0071] where, on the curve segment denotes the coordinates of a point on the profile curve segment, and are the coordinates and coordinates of the control points of the Bezier curve, i.e. the coordinates of the control points , , denotes the summation symbol, which is 0 to 3 in this example of a third-order Bezier curve, and is obtained from equation (12) the derivative of the coordinates with respect to the parameter and :

[0072] where, and denote the derivative of the coordinates with respect to the parameter s, and are the coordinates and coordinates of the control points of the Bezier curve, i.e. the coordinates of the control points , , denotes the summation symbol, which is 0 to 3 in this example of a third-order Bezier curve.

[0073] By substitution, we obtain , and substituting into equation (5) gives: The mapping relationship of a profile line in a closed region:

[0074] where denotes the mask integral of the curve segment in the profile, , are the spatial frequency coordinates of the obtained diffraction field, is the natural logarithm, i is the imaginary unit, and s is the curve parameter, which is taken as , denotes the integral symbol, denotes the coordinates of a point on the profile curve segment, denotes the derivative value of the point with respect to , i.e. the tangent, is the input mask field, is the differential of s.

[0075] As can be seen from the transformation relationship of formula (5), formula (8) is the mapping relationship from the "Bézier curve control point" to the "François diffraction field".

[0076] For a closed contour, it can be divided into several contour segments. These contour segments correspond to a series of control points. As long as the first control point and the last control point coincide, the closed contour can be obtained.

[0077] Let the mask pattern consist of M closed regions, where the first region is... A closed region is composed of Segment curve Composition, and the transmittance of the closed region is That is, within this closed area Therefore, we have:

[0078] in,( , () represents the frequency coordinates of the obtained diffraction field. In the diffraction field coordinates The one-dimensional line integral of the mask pattern along the contour curve is given by M, where M represents the mask pattern consisting of M closed regions. and Indicates the first A closed region is composed of Composed of segmented curves, Indicates the first The transmittance of a closed region is , The summation symbol is used to represent the summation symbol. This indicates the integral sign, and s is the curve parameter, which here takes the value of , This represents the coordinates of a point on the contour curve segment. and for coordinates for parameters The derivative of It is the differential of s.

[0079] (4) Full-frequency domain analytical calculation: For each spatial frequency sampling point, the pre-derived Bessel segment integral formula is directly applied, and the total spectrum of the mask is obtained by accumulating the contributions of all boundary segments. During the calculation, the filtering effect of spatial resolution in lithographic imaging can be directly considered, and only the center segment is calculated. , Frequency points within the specified frequency range can avoid wasting computing resources.

[0080] Those skilled in the art can easily understand that the above description is only the preferred embodiment of the present application, and is not intended to limit the present application. Any modification, equivalent replacement and improvement made within the spirit and principle of the present application shall be included in the protection scope of the present application.

Claims

1. A method for solving the Fraunhofer diffraction field of a mask pattern, characterized in that, The method includes the following steps: Topological analysis is performed on the mask pattern to extract its boundaries, and the extracted boundaries are fitted to obtain a mask contour formed by fitting multiple closed loops. The two-dimensional surface integral of the Fraunhofer diffraction field of the mask pattern is converted into a one-dimensional line integral along the contour curve. The diffraction field of each contour curve in each closed loop of the mask contour is calculated separately. The sum of the diffraction fields of all contour curves in all closed loops is the one-dimensional line integral diffraction field of the mask pattern along the contour curve, thereby obtaining the Fraunhofer diffraction field of the desired mask pattern.

2. The method for solving the Fraunhofer diffraction field of a mask pattern as described in claim 1, characterized in that, The extracted boundaries are fitted using piecewise Bézier curves.

3. The method for solving the Fraunhofer diffraction field of a mask pattern as described in claim 2, characterized in that, The dimensionality reduction is achieved using Green's formula.

4. The method for solving the Fraunhofer diffraction field of a mask pattern as described in claim 3, characterized in that, The formula for reducing the two-dimensional area integral of the Fraunhofer diffraction field of the mask pattern to a one-dimensional line integral along the contour curve is as follows: in, For the two-dimensional surface integral of the Fraunhofer diffraction field of the mask pattern, ( , (i) represents the spatial frequency coordinates of the obtained diffraction field, where i is the imaginary unit and e is the natural logarithm. Represents the integral of a closed profile. For all the curve contours of the mask, Input the spatial coordinates of the mask, Let be the differentials with respect to x and y, respectively. The input mask field.

5. The method for solving the Fraunhofer diffraction field of a mask pattern as described in claim 4, characterized in that, The formula for calculating the diffraction field of each profile curve in each closed loop is as follows: in, In the diffraction field coordinates The one-dimensional line integral of a contour curve in a mask pattern, where i is the imaginary unit. It is the natural logarithm. Indicates the line integral symbol, ( , Let ) represent the spatial frequency coordinates of the obtained diffraction field, s be the curve parameters, and ds be the derivative with respect to s. b represents the minimum and maximum values ​​of s, respectively. The input mask field.

6. The method for solving the Fraunhofer diffraction field of a mask pattern as described in claim 5, characterized in that, The formula for calculating the one-dimensional line integral diffraction field of the mask pattern along the contour curve is as follows: in,( , () represents the frequency coordinates of the obtained diffraction field. In the diffraction field coordinates The one-dimensional line integral of the mask pattern along the contour curve is given by M, where M represents the mask pattern consisting of M closed regions. and Indicates the first A closed region is composed of Composed of segmented curves, Indicates the first The transmittance of a closed region is ,, The summation symbol is used to represent the summation symbol. Indicates the integral sign, where s is the curve parameter. b represents the minimum and maximum values ​​of s, respectively. This represents the coordinates of a point on the contour curve segment. and for Coordinates for parameters The derivative of It is the differential of s.

7. The method for solving the Fraunhofer diffraction field of a mask pattern as described in claim 1, characterized in that, The extraction boundary is fitted using non-uniform rational B-splines, B-spline curves, Fourier descriptors, implicit function representations, higher-order polynomial fitting, or Hermitian splines; the dimensionality reduction is achieved using Gaussian divergence theorem, Stokes' theorem, line integrals of complex functions, or the boundary element method.

8. A system for solving the Fraunhofer diffraction field of a mask pattern, characterized in that, The system includes an actuator for performing a method for solving the Fraunhofer diffraction field of a mask pattern as described in any one of claims 1-7.

9. A computer storage medium having a computer program stored thereon, characterized in that, The computer program is used to implement the method for solving the Fraunhofer diffraction field of a mask pattern as described in any one of claims 1-7.

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