Lithography simulation method and optical proximity effect correction method

Abstract

To provide a lithography simulation method and optical proximity effect correction method capable of simulating a mask shape as a curved shape.SOLUTION: The lithography simulation method according to this embodiment includes obtaining a mask shape that is transferred from a mask substrate to a wafer substrate using a projection exposure apparatus. Furthermore, this lithography simulation method comprises obtaining control points on the contour shapes included in the mask shape, the functional form of polynomial parametric representation, and the degree of the functional form. Additionally, this lithography simulation method predicts the resist pattern by performing a Fourier transformation of the contour shapes using polynomial parametric representation based on the control points, functional form, and degree.SELECTED DRAWING: Figure 6

Description

Technical Field

[0001] The present embodiment relates to a lithography simulation method and a proximity effect correction method.

Background Art

[0002] For the purpose of increasing the capacity, the circuit structure of semiconductor memories has changed from a planar structure to a three-dimensional structure, and high stacking has been progressing. However, in order to reduce costs, reduction in the planar direction is also continuing. Although reduction in the planar direction is possible by using extreme ultraviolet light, the cost is very high, so it is necessary to extend the life of immersion lithography using an argon fluoride laser as a light source. As a typical technique for reduction in the planar direction using immersion lithography, there is a technique called inverse lithography in which a mask shape for obtaining a resist shape on a desired wafer is obtained by solving an inverse problem. Generally, the mask shape obtained by inverse lithography has a complicated curved shape. However, since it is difficult to simulate a mask shape represented by a curve, a complicated curve is approximated by a straight line and simulated. Therefore, in order to reduce the error when approximating a curve by a straight line, it is necessary to finely divide the straight line to reduce the difference from the curve, and there has been a problem that the data amount becomes too large.

Prior Art Documents

Patent Documents

[0003]

Patent Document 1

Summary of the Invention

Problems to be Solved by the Invention

[0004] Provided are a lithography simulation method and a proximity effect correction method capable of simulating a mask shape as a curved shape.

Means for Solving the Problems

[0005] The lithography simulation method according to this embodiment includes obtaining a mask shape transferred from a mask substrate to a wafer substrate using a projection exposure apparatus. Further, this lithography simulation method includes obtaining control points on a contour figure included in the mask shape, a functional form of a polynomial parametric representation, and the degree of the functional form. Further, this lithography simulation method includes predicting a resist pattern by performing a Fourier transform on the contour figure using a polynomial parametric representation based on the control points, the functional form, and the degree.

Brief Description of the Drawings

[0006]

Figure 1

Figure 2

Figure 3A

Figure 3B

Figure 4

Figure 5

Figure 6

Figure 7

Figure 8

Embodiments for Carrying Out the Invention

[0007] Hereinafter, embodiments of the present invention will be described with reference to the drawings. These embodiments do not limit the present invention. The drawings are schematic or conceptual, and the ratios of the respective parts are not necessarily the same as those in reality. In the specification and the drawings, the same reference numerals are given to the same elements as those described above with respect to the previous drawings, and detailed descriptions are appropriately omitted.

[0008] (First Embodiment) In the semiconductor lithography process, even if the designed pattern arrangement and shape are directly created as the mask arrangement and shape, the resist shape on the wafer transferred through the exposure apparatus does not become the desired shape due to the optical proximity effect. Therefore, a correction called optical proximity effect correction is performed so that the resist shape on the wafer becomes the desired shape. In recent years, with the progress of miniaturization, in addition to obtaining the desired shape, it is required for optical proximity effect correction to arrange sub-resolution assist features (SRAFs) that cannot be resolved on the wafer in order to make it less likely to vary against the variations in the wafer process. Furthermore, with the progress of miniaturization, in order to further enhance the above robustness, it is required to represent the mask shape not by a simple rectangle but by an arbitrary curve.

[0009] FIG. 1 is a diagram showing an example of a method for manufacturing a semiconductor device according to the first embodiment. The central part of FIG. 1 shows a flowchart for obtaining a mask shape from a design pattern.

[0010] First, using the designed data as input, SRAF is generated in the design data (S10). The SRAF may be generated by an inverse lithography technique or may be generated according to a predetermined rule.

[0011] Next, optical proximity effect correction is performed (S20). Optical proximity effect correction is a process in which when the lithography simulation result of transfer to the wafer substrate deviates from the desired shape, the pattern is corrected and lithography simulation is performed again. The lithography simulation method is included in the optical image calculation described in FIG. 1.

[0012] Next, verification is performed (S30). By inputting the layout data combining the design pattern with the proximity effect correction and the SRAF pattern as mask-shaped data into a lithography simulator (described as optical image calculation in FIG. 1), the shape of the resist pattern on the wafer is obtained.

[0013] As described above, with the shape of the design pattern as it is, it is impossible to obtain a desired shape on the wafer due to the optical proximity effect. Therefore, the shape of the design pattern is deformed so that the resist shape on the wafer becomes the desired shape, and the deformation of the design pattern and the lithography simulation are repeated until the resist shape on the wafer becomes the desired shape. After confirming that the lithography simulation is performed with the combination of the shape due to the final deformation and the SRAF and the shape is the desired one, it is sent to the mask manufacturing process. At this time, the lithography simulation is performed on a mask shape represented by an arbitrary curve, but being a curve may cause a problem in the lithography simulation. This problem will be described later.

[0014] Next, the optical image calculation flow (lithography simulation method) included in step S20 will be described.

[0015] FIG. 2 is a diagram showing an example of a lithography simulation method according to the first embodiment.

[0016] Regarding the conditions related to exposure in the exposure process (light source shape S(ξ), pupil function P(f)) and the mask shape data m(x) as inputs, the optical image I(x) is taken as the output after calculation.

[0017] First, the Fourier transform of the mask shape data m(x) is performed (S110). The light source shape S(ξ) is discretized and regarded as an aggregate of point light sources. The optical image for one of the point light sources is obtained.

[0018] Next, the optical image calculation is performed for each point light source (ξ) (S120).

[0019] Next, weighting and integration are performed on the light source intensity distribution S (S130). That is, integration is executed to obtain the sum of the optical images at each point light source.

[0020] Next, the Fourier transform included in step S110 and the mask shape having a curved shape will be described.

[0021] FIG. 3A shows an example of a figure surrounded by a curve. FIG. 3B shows an example of a figure obtained by linearly dividing the figure shown in FIG. 3A.

[0022] As shown in FIG. 3B, when divided by eight straight lines, the deviation from the original curve is large, and it does not become a good approximation unless the number of divisions is increased. Increasing the number of divisions increases the data amount, which becomes a problem. This is one of the problems when performing lithography simulation on a mask shape represented by an arbitrary curve. Therefore, the figure surrounded by the curve in FIG. 3A can be represented by eight control points using, for example, a cubic Bezier curve, and an increase in the data amount can be suppressed.

[0023] When performing lithography simulation on a mask shape represented by a Bezier curve or the like, usually, only a figure surrounded by straight lines as shown in FIG. 3B could be handled. Therefore, in the present embodiment, the curved figure is processed as it is without being linearly divided. Specifically, the curve is parametrically represented as a Bezier curve or a Spline curve and Fourier-transformed.

[0024] Next, the Fourier transform using parametric representation will be described.

[0025] Let the region in the x-y plane surrounded by a curve or a straight line be D. D gives the mask shape. To perform Fourier transform on this means to calculate (Equation 1).

Equation

[0026] FIG. 4 is a diagram showing an example of a region D surrounded by a straight line according to a comparative example and coordinates.

[0027] As a comparative example, the mask shape represented by the region D surrounded by a straight line as shown in FIG. 4 is Fourier-transformed. Let the coordinates of the endpoints of the straight line be (x j , y j )(i = 0, 1, ···, N - 1), and a j and S j,j+1 defined by (Equation 2) and (Equation 3) are used to obtain (Equation 1), which becomes (Equation 4).

Number

Number

Number

[0028] Next, a method of treating the curve as it is in the present embodiment will be described.

[0029] FIG. 5 is a diagram showing an example of a region D surrounded by a curve and a closed curve C of a boundary according to the first embodiment.

[0030] To perform the double integral in the region D of (Equation 1), a closed curve C that gives the boundary of the region D as shown in FIG. 5 is considered. Assuming that P(x, y) and Q(x, y) are continuous and differentiable within the region D, (Equation 5) holds by Green's theorem.

Number

[0031] FIG. 6 is a diagram showing an example of a parametric display of a region D surrounded by a curve according to the first embodiment.

[0032] Taking the region D enclosed by a curve and the points j (j = 0, 1, ···, N - 1) on the curve in Fig. 6, a parametric representation such as a Bezier curve or a Spline curve is performed. The points j on the curve are called control points and their coordinates are (x j , y j ). The curve between point j and point (j + 1) can be expressed as in (Equation 6) and (Equation 7). Here, m is the degree of the curve, which is usually a natural number up to 3.

Equation

Equation

[0033] The Fourier transform of region D is obtained from this parametric representation. Here, the coefficients a jp , b jp of the parametric representation are as shown in Table 1 and Table 2. Table 1 shows the coefficients of x j in the parametric representation. Table 2 shows the coefficients of y j in the parametric representation.

Table 1

Table 2

[0034] As an example, when a quadratic Spline curve is adopted as the parametric representation, using the variable defined in (Equation 8), the Fourier transform of region D can be expressed by the imaginary error function (erfi(u)) as in (Equation 9).

Equation

Equation

[0035] As yet another example, when a cubic Bézier curve is adopted as a parametric representation, the Fourier transform of region D becomes (Equation 11) and (Equation 12) using the variables defined in (Equation 10). Here, m in (Equation 12) k and x k are the weight in the Gauss-Legendre quadrature method and the k-th solution where the n-th Legendre polynomial P n (x) = 1. As an example, Table 3 shows the weights m k and the k-th solution x k for n = 20. Note that the value of n can be arbitrarily set, and it is known that appropriate values of m k and x k exist as a table depending on the value of n.

Number

Number

Number

Table 3

[0036] Figure 7 is a graph showing an example of the result of the Fourier transform of a circle with radius a. The dots in Figure 7 show the result of performing a Fourier transform on a circle with radius a using a cubic Bézier curve as a parametric representation and eight control points. The solid line in Figure 7 shows the result using the Bessel function.

[0037] It is known that the exact Fourier transform of a circle with radius a can be expressed by a Bessel function as in (Equation 13) by another method. As shown in Figure 7, the result using the Bessel function almost coincides with the result using the cubic Bézier curve.

Number

[0038] Also, the integration of the function defined by (Equation 11) is performed by the Gauss-Legendre quadrature method. However, higher accuracy can be achieved by adopting n according to the phase fluctuation in the integration interval defined by (Equation 14). If n in the Gauss-Legendre quadrature method is set to be about three times or more of M represented by (Equation 14), calculations can be performed with very high accuracy. That is, the degree (n) of the Legendre polynomial in the Gauss-Legendre quadrature method is set according to the phase change of the integrand within the integration interval. Note that M is calculated by substituting the value of (Equation 11) into (Equation 14).

Number

[0039] Therefore, when performing the Fourier transform of a figure surrounded by a curve, the curve is parametrically represented, the double integral is converted into a single integral by Green's theorem, and even for a function that cannot be integrated, fast and accurate calculations can be performed by using the Gauss-Legendre quadrature method.

[0040] As described above, according to the first embodiment, first, a mask shape transferred from a mask substrate to a wafer substrate is acquired using a projection exposure apparatus. Next, control points on the contour figure included in the mask shape, the functional form of the polynomial parametric representation, and the degree of the functional form are acquired. Hereinafter, unless otherwise specified, "degree" refers to the degree of the functional form. Next, based on the control points, the functional form, and the degree, the contour figure is Fourier-transformed using the polynomial parametric representation to predict (simulate) the resist pattern. As a result, the mask shape can be represented by an interpolation curve so as to suppress the data amount, and the curve can be directly Fourier-transformed. As a result, the mask shape can be simulated in the form of a curve shape.

[0041] Also, the degree is 2 or higher. Thereby, a curve can be represented by the polynomial parametric representation. The higher the degree, the more complex the curve that can be represented. However, since higher computational power is required as the degree increases, the degree may be 2 or higher and 3 or lower.

[0042] When the degree is quadratic, the Fourier transform of the contour shape is performed by using the error function, the imaginary error function, the complementary error function, or the complementary imaginary error function.

[0043] When the degree is cubic, the Fourier transform of the contour shape is performed by numerical integration. This is because it is necessary to calculate the indefinite integral. Note that it is not limited to the Gauss-Legendre quadrature method described above, and other numerical integrations such as the simplex method may be used. However, when using the Gauss-Legendre quadrature method, the computational load is less, and numerical integration can be performed with higher accuracy.

[0044] Also, the control points are the minimum coordinates for representing the curve and are input to the simulator by the user. (Equation 6) and (Equation 7) are functions for interpolating between the control points. The function form of the function for interpolating between the control points and the degree of the function form are input to the simulator by the user. The simulator includes an arithmetic unit such as a CPU (Central Processing Unit). The arithmetic unit acquires the design pattern, acquires the input control points, function form, and degree, and performs the Fourier transform.

[0045] Also, before acquiring the control points, function form, and degree, the user may be able to select a first mode in which the contour shape is Fourier-transformed using polynomial parametric representation. Note that in addition to the first mode, a method similar to the comparative example described in FIG. 4 may also be selectable. The simulator may further include a display control unit that performs display on a display unit such as a display. The display control unit causes the display unit to display a UI (User Interface) that allows the user to select the first mode.

[0046] FIG. 8 is a diagram showing an example of the display on the display unit.

[0047] When a spline curve is selected as the parametric display, the user presses the "SPLINE" button. When a Bezier curve is selected as the parametric display, the user presses the "BEZIER" button. When a second-degree is selected, the user presses the "second-degree" button. When a third-degree is selected, the user presses the "third-degree" button. When a degree of four or higher is selected, the user presses the "n-degree (n>=4)" button and enters the degree in the "n=?" input field.

[0048] Also, when the first mode is selected, the control points, function form, and degree may be obtained. For example, after the arithmetic unit obtains an input (information) indicating that the first mode has been selected, it obtains the control points, function form, and degree.

[0049] In addition, the first mode may be made selectable by the user before the Fourier transform is performed.

[0050] Also, the lithography simulation method according to the present embodiment may be included in the process of an optical proximity effect correction method or a semiconductor device manufacturing method, or may be an independent process (for example, step S30 shown in FIG. 1). In the optical proximity effect correction method, after predicting the resist pattern, the pattern of the mask substrate is corrected and the resist pattern is predicted repeatedly until the predicted resist pattern becomes the desired shape.

[0051] At least a part of the lithography simulation method and the optical proximity effect correction method according to this embodiment may be configured by hardware or software. When configured by software, a program that realizes at least a part of the functions of the lithography simulation method and the optical proximity effect correction method may be stored in a recording medium such as a flexible disk or a CD-ROM, and read into a computer and executed. The recording medium is not limited to removable ones such as magnetic disks and optical disks, and may be a fixed recording medium such as a hard disk device or a memory. Further, a program that realizes at least a part of the functions of the lithography simulation method and the optical proximity effect correction method may be distributed via a communication line (including wireless communication) such as the Internet. Furthermore, the program may be encrypted, modulated, compressed, and distributed via a wired or wireless line such as the Internet, or stored in a recording medium and distributed.

[0052] Although some embodiments of the present invention have been described, these embodiments are presented as examples and are not intended to limit the scope of the invention. These embodiments can be implemented in various other forms, and various omissions, replacements, and changes can be made without departing from the gist of the invention. These embodiments and their modifications are included in the scope and gist of the invention, and are also included in the invention described in the claims and its equivalent scope.

Explanation of Reference Numerals

[0053] C Closed curve, D Region, x j Coordinate, y j Coordinate

Claims

1. Obtaining a mask shape to be transferred from a mask substrate to a wafer substrate using a projection exposure apparatus, obtaining control points on a contour graphic included in the mask shape, a functional form of a polynomial parametric representation, and a degree of the functional form, predicting a resist pattern by performing Fourier transform on the contour graphic using the polynomial parametric representation based on the control points, the functional form, and the degree, A lithography simulation method comprising the above.

2. The lithography simulation method according to claim 1, wherein the functional form is a parametric spline curve or a Bézier curve.

3. The lithography simulation method according to claim 1 or claim 2, wherein the degree is 2 or higher.

4. The lithography simulation method according to claim 3, wherein the degree is 2 or higher and 3 or lower.

5. The lithography simulation method according to claim 1, wherein performing Fourier transform on the contour graphic includes performing double integral to single integral conversion based on Green's theorem to perform Fourier transform on the contour graphic.

6. The functional form is a parametric spline curve or a Bézier curve, the degree is 2, The lithography simulation method according to claim 1, wherein performing Fourier transform on the contour graphic includes performing Fourier transform on the contour graphic by using an error function, an imaginary error function, a complementary error function, or a complementary imaginary error function.

7. The functional form is a parametric spline curve or a Bézier curve, the degree is 3, The lithography simulation method according to claim 1, wherein performing Fourier transform on the contour graphic includes performing Fourier transform on the contour graphic by numerical integration using the Gauss-Legendre quadrature method.

8. The lithography simulation method according to claim 7, wherein the degree of the Legendre polynomial in the Gauss-Legendre quadrature method is set according to the phase change of the integrand within the integration interval.

9. The lithography simulation method according to claim 1, further comprising enabling selection of a first mode for performing Fourier transform on the contour graphic using the polynomial parametric representation before obtaining the control points, the functional form, and the degree.

10. The obtaining of the control points, the function form, and the degree includes obtaining the control points, the function form, and the degree when the first mode is selected, according to the lithography simulation method of claim 9.

11. Obtaining a mask shape transferred from a mask substrate to a wafer substrate using a projection exposure apparatus, obtaining control points on a contour graphic included in the mask shape, a function form of polynomial parametric representation, and the degree of the function form, predicting a resist pattern by Fourier-transforming the contour graphic using the polynomial parametric representation based on the control points, the function form, and the degree, repeating correction of the pattern of the mask substrate and prediction of the resist pattern until the predicted resist pattern becomes a desired shape, which comprises an optical proximity effect correction method.

Images