Electron beam lithography layout grading correction method based on genetic algorithm

Abstract

The invention relates to an electron beam lithography layout correction method based on a genetic algorithm, and aims to improve the quality of an exposed pattern by optimizing layout design. The method comprises the following steps: reading a to-be-corrected layout, dividing the to-be-corrected layout into a plurality of rectangular areas, and mapping the rectangular areas into a layout matrix according to a preset pixel size; then calculating an energy distribution matrix, and adjusting the size of each rectangular region to carry out global correction until convergence; and finally, discretized filling is carried out on the adjustment region, and optimal pixel distribution is selected through iterative optimization by using a genetic algorithm and taking pixels as population individuals, so that fine correction of the local region is realized. Meanwhile, up-down sampling is adopted in a solution space to reduce a search range, and a dynamic fitness function is adopted to reduce the number of iterations. According to the method, hierarchical optimization of the layout is realized by utilizing simple correction and accurate correction, and the method has the characteristics of high correction precision and high calculation efficiency.

Description

Technical Field

[0001] The present invention relates to the field of computational lithography, and more particularly to a genetic algorithm-based method for electron beam lithography layout correction. This method effectively reduces the negative impact of proximity effects on pattern quality during electron beam lithography by optimizing layout design. The method is applicable to high-precision manufacturing applications such as microelectromechanical systems (MEMS), photolithography masks, and optoelectronic devices, and can achieve nanometer-scale and even subnanometer-scale pattern correction. Background Art

[0002] Electron beam lithography (EBL) is a high-resolution, maskless lithography technology widely used in the manufacture of microelectromechanical systems (MEMS), photolithography masks, and optoelectronic devices, capable of achieving nanometer-scale and even subnanometer-scale pattern accuracy. This technology exposes the photoresist using a focused electron beam, followed by development to remove the exposed or unexposed areas, thereby forming the target pattern. However, as process node sizes continue to shrink, the proximity effect (PE) generated during the exposure process has become increasingly significant in affecting the final developed pattern, becoming a key factor limiting pattern accuracy. The proximity effect is primarily caused by forward and backscattering of electrons in the photoresist and substrate, resulting in energy deposition in unexposed areas near the pattern, which in turn causes pattern distortion.

[0003] Currently, the correction methods for the proximity effect mainly include dose correction and shape correction. Among them, dose correction requires the electron beam exposure equipment to have the ability to quickly adjust the exposure dose, which places high demands on hardware performance; while shape correction improves the proximity effect by optimizing the layout design and has low dependence on equipment. Brian D. Cook and Soo-Young Lee proposed a hierarchical correction method called "PYRAMID", which reduces the polygon size through global correction and adjusts the local area in combination with the energy deposition calculation of local key points. Although this method can alleviate the proximity effect to a certain extent, its correction accuracy is still difficult to meet the strict requirements of advanced nodes.

[0004] Therefore, it is urgent to develop a more accurate shape correction method. Summary of the Invention

[0005] In order to overcome the deficiencies of the above-mentioned prior art, the present invention proposes a hierarchical electron beam lithography layout correction method based on a genetic algorithm. The method first reduces the layout polygon size through a simple global correction, extracts the difference area between the global optimized layout and the target graphic as the solution space, and uses up and down sampling technology to narrow the search range, significantly improving the computational efficiency. In fine local correction, a genetic algorithm is used to iteratively optimize the downsampled pixel points, and the fitness is evaluated by combining the intersection-over-union ratio, and the optimal solution is gradually approached through elite selection, crossover and mutation operations. After the optimization is completed, the high-resolution layout is restored by upsampling to ensure the correction accuracy. The purpose of the present invention is to use a genetic algorithm to optimize the results of electron beam layout shape correction, reduce the negative impact of the proximity effect on the graphic quality without adjusting the dose distribution, and improve the correction accuracy.

[0006] The solution of the present invention is as follows:

[0007] A method for correcting electron beam lithography patterns based on a genetic algorithm, the method comprising:

[0008] Step 1. Preprocessing of layout data. Discretize the layout and obtain the coordinate list of each polygon vertex of the input layout V = {V1, V2, ..., V n}, where V i ={(x1,y1),(x2,y2),...,(x m ,y m )}. Set the pixel sizes in the x and y directions to dx and dy respectively, and map the vertex coordinates according to the pixel sizes:

[0009]

[0010] And update the vertex list V and create the edge list S.

[0011] Step 2. Split the polygon. Using the scanline algorithm, scan the horizontal edges in the edge list S. If the currently scanned edge is not the top or bottom edge of the polygon, perform the following splitting operation on it: For each horizontal edge in the polygon, add a small offset Δ to the vertex coordinates at both ends. For each offset vertex, determine whether it is inside the polygon. If an endpoint is inside the polygon, extend the horizontal edge to the nearest vertical edge in the polygon and add a new vertex at the intersection. Simultaneously update the vertex list and the edge list V S. Finally, calculate the midpoints of each edge of the split polygon and extract the midpoint list O.

[0012] Step 3. Calculate the exposure energy deposition distribution. First, set an initial dose d0 for the exposure area and 0 for the non-exposure area, generating a two-dimensional dose matrix D(x,y). Then calculate the layout energy deposition according to the following formula:

[0013]

[0014] Where, For convolution operations, a two-dimensional Fourier transform can also be used to accelerate the convolution operation. E(x, y) is a two-dimensional energy deposition matrix, and f(r) is the energy distribution point spread function of a single electron beam, which describes the influence of the proximity effect. The present invention adopts the form of a double Gaussian function, as follows:

[0015]

[0016] Where r is the distance from the current pixel to the exposure point; α is the forward scattering coefficient, which is the characteristic half-width of the forward scattering range; β is the backscattering coefficient, which is the characteristic half-width of the backscattering range; η is the ratio of the amount of electrons consumed by forward scattering to backscattering.

[0017] Step 4. Perform a simple correction on the layout. Dynamically adjust the length of each rectangle's horizontal and vertical sides based on the relationship between the midpoint energy deposition and the development threshold. For the horizontal side, adjust the length according to the following formula:

[0018]

[0019] Where, L h is the current horizontal side length, E h is the energy deposition value at the midpoint of the edge, T is the development threshold, and T0 is the allowable error. For vertical edges, the edge length is adjusted according to the following formula:

[0020]

[0021] Where, L w is the current vertical side length, E w is the energy deposition value of the midpoint of the edge. After all rectangles are adjusted, the two-dimensional energy deposition distribution is calculated according to step 3, and the final graphic matrix H is obtained through the development model. The threshold development model used in the present invention is expressed as:

[0022]

[0023] Then repeat steps 3 and 4 until the side lengths of all rectangles no longer need to be adjusted.

[0024] Step 5. Confirm the optimal solution space and calculate the initial population. The search range of the optimal solution is the pixel area between the simply corrected graphic and the target graphic. The coordinates of all pixels in the area are extracted to form the basic solution space (each pixel point uses binary encoding: 0 / 1 represents unselected / selected state). Then downsample the coordinates of the solution space to further narrow the search range. There are two main downsampling strategies adopted in the present invention: neighborhood downsampling and sparse downsampling. Neighborhood downsampling is to synthesize k pixels in the pixel neighborhood into a new pixel unit, which can better retain structural features. Sparse downsampling is to select k pixels at a certain interval to synthesize into a new pixel unit, which has better robustness. After the search range is downsampled, the space complexity is reduced from O(N 2 ) is reduced to O((N / k) 2 ), improving computational efficiency. Each solution in the population contains the selected state of each pixel unit, expressed as where θ i The value is 0 or 1.

[0025] Step 6. Calculate the population fitness. The solution in the population is upsampled and restored, and superimposed with the optimized layout in step 4. The energy deposition distribution and the developed pattern matrix are calculated using equations (2) and (5). The main criteria for measuring pattern quality are the intersection of union (IoU) e1 and the mean square error (MSE) e2 between the developed pattern H and the target pattern P, which are expressed as follows:

[0026]

[0027] The fitness function used in this invention is a weighted superposition of IoU and MSE, and the specific expression is as follows:

[0028] Fitness=s1(t)(1-e1)+s2(t)e2 (15)

[0029] Where s1,s2∈[0,1] are the weights of the two errors. The weights are dynamically adjusted in the subsequent iteration process to accelerate convergence. The dynamic adjustment strategy used in the present invention is as follows:

[0030]

[0031] s2(t)=1-s1(t) (17)

[0032] Where t is the current iteration number, t0 is the iteration number of the transition point, and k is the coefficient of change.

[0033] Step 7. Generate a new population through the selection, crossover, and mutation operations of the genetic algorithm. The selection operation refers to retaining the individuals closest to the ideal solution from the current population to the new population. The method used in the present invention is the optimal retention method. First, all individuals are substituted into formula (8) to calculate the fitness, and then sorted in ascending order. The individuals with the highest ranking are selected and copied to the new population to ensure that the best individuals are retained in each iteration. The crossover operation refers to randomly selecting two individuals from the population, and through the exchange combination of two chromosomes, the excellent characteristics of the parent individuals p1 and p2 are inherited to the offspring individuals c1 and c2, thereby generating excellent individuals. The crossover method used in the present invention is the uniform crossover method, which exchanges all chromosomes in the two parent generations with a probability q0, as shown in the following formula:

[0034]

[0035]

[0036] Where, Represents the i-th chromosome in the parent and offspring, r i is a random number between 0 and 1. After the crossover, the offspring individuals are added to the next generation population. The mutation operation refers to modifying the chromosomes of some individuals to prevent falling into a local optimal solution during the iteration process. The mutation method used in this invention is the global uniform position flipping method, which flips each chromosome in the individual with probability q1. The specific expression is as follows:

[0037]

[0038] Where θ i Represents the i-th chromosome in the individual. After the genetic algorithm selection, crossover and mutation operations are performed on the original population, new individuals will be generated to form a new population for the next round of iterative calculation.

[0039] Step 8. Fine-tune the layout until the genetic algorithm terminates and the optimal solution is obtained. When the termination condition is met, the algorithm stops iterating and outputs the current optimal solution. Termination conditions include: the current number of iterations has reached the set maximum number of iterations, the iteration error has converged, and the iteration error has met the current accuracy requirement, that is, the fitness is less than the set error value δ.

[0040] Compared with the existing technology, the genetic algorithm-based electron beam lithography layout hierarchical correction method of the present invention significantly improves the correction accuracy and computational efficiency of the proximity effect (PE) in electron beam lithography (EBL) by combining global correction and local fine optimization. Its technical effects are mainly reflected in the following aspects: the electron beam energy deposition distribution is accurately modeled by a double Gaussian function, the influence of forward scattering and backscattering is quantified, and the corrected layout is closer to the target pattern after exposure and development. Correction is achieved only by optimizing the layout shape (rather than adjusting the electron beam dose), which is applicable to all types of electron beam lithography equipment without the need for hardware upgrades. Through elite retention, uniform crossover and global mutation operations, the problem of traditional optimization methods easily falling into local optimality is avoided, ensuring the global optimal solution. BRIEF DESCRIPTION OF THE DRAWINGS

[0041] Figure 1 It is a schematic diagram of the overall process of the present invention.

[0042] Figure 2 It is a layout graphic, the white area is the exposure area, and the black area is the non-exposure area.

[0043] Figure 3 Schematic diagrams of the search range of the layout graph and genetic algorithm after simple correction in the present invention, wherein the red area represents the search range.

[0044] Figure 4 It is the developed pattern of the uncorrected layout.

[0045] Figure 5 These are the layout patterns and development patterns obtained by the present invention. DETAILED DESCRIPTION

[0046] In order to more clearly illustrate the present invention, the following will be described in detail with reference to specific examples. It should be noted that these examples are only used to illustrate the embodiments of the present invention and do not constitute a limitation on the scope of the present invention.

[0047] Figure 1The overall process of correcting a layout in the present invention is demonstrated. The method first imports two-dimensional layout data and maps the coordinates in the layout to a pixel grid. The polygon is divided into rectangular regions, and a vertex list, midpoint list, and edge list for all rectangles are generated. The discrete layout matrix is convolved with the single-electron beam energy distribution function to obtain the layout energy distribution. The energy deposition at the midpoints of all rectangular edges is calculated, and the size of each rectangle is adjusted through simple corrections, and the energy deposition is calculated until the size of all rectangles requires no adjustment. The region of simple correction is set as the solution space, and the search range is narrowed through up- and down-sampling to set the initial population. The fitness of the population is calculated, and the genetic algorithm selection, crossover, and mutation operations are performed on the individuals in the population to generate a new population. When the conditions for iterative termination are met, the iteration is terminated and the optimal solution is output, completing the layout correction.

[0048] Figure 2 A layout diagram is shown, where the white area is the exposure area and the black area is the non-exposure area. Figure 2 The layout is used to perform proximity effect correction. The specific steps are as follows:

[0049] Step 1. Layout data preprocessing. Read the layout data and discretize the layout into a two-dimensional binary matrix, where 0 represents the non-exposure area and 1 represents the exposure area. Get the coordinate list of each polygon of the input layout V = {V1, V2, ..., V n}, where V i ={(x1,y1),(x2,y2),...,(x m ,y m )}. Set the pixel sizes in the x and y directions to dx and dy to 0.1 nm respectively, and map the vertex coordinates according to the pixel size:

[0050]

[0051] And update the vertex list V and create the edge list S.

[0052] Step 2. Split the polygon. Using the scanline algorithm, scan the horizontal edges in the edge list S. If the currently scanned edge is not the top or bottom edge of the polygon, perform the following splitting operation on it: For each horizontal edge in the polygon, add a small offset Δ to the vertex coordinates at both ends. For each offset vertex, determine whether it is inside the polygon. If an endpoint is inside the polygon, extend the horizontal edge to the nearest vertical edge in the polygon and add a new vertex at the intersection. Simultaneously update the vertex list and the edge list V S. Finally, calculate the midpoints of each edge of the split polygon and extract the midpoint list O.

[0053] Step 3. Calculate the exposure energy deposition distribution. First, set an initial dose d0 for the exposure area and 0 for the non-exposure area, generating a two-dimensional dose matrix D(x,y). Then calculate the layout energy deposition according to the following formula:

[0054]

[0055] Where, For convolution operations, a two-dimensional Fourier transform can also be used to accelerate the convolution operation. E(x, y) is a two-dimensional energy deposition matrix, and f(r) is the energy distribution point spread function of a single electron beam, which describes the influence of the proximity effect. The present invention adopts the form of a double Gaussian function, as follows:

[0056]

[0057] Where r is the distance from the current pixel to the exposure point; α is the forward scattering coefficient, which is the characteristic half-width of the forward scattering range; β is the backscattering coefficient, which is the characteristic half-width of the backscattering range; and η is the ratio of the electrons consumed by forward scattering to backscattering. The parameters of the point spread function here are set to: α = 11.194, β = 472.462, and η = 1.156.

[0058] Step 4. Perform a simple correction on the layout. Dynamically adjust the length of each rectangle's horizontal and vertical sides based on the relationship between the midpoint energy deposition and the development threshold. For the horizontal side, adjust the length according to the following formula:

[0059]

[0060] Where, L h is the current horizontal side length, E h is the energy deposition value at the midpoint of the edge, T is the development threshold, which is set to 0.5 times the maximum energy deposition max(E(x,y)), and T0 is the allowable error, which is set to 0.1 times max(E(x,y)). For vertical edges, the edge length is adjusted according to the following formula:

[0061]

[0062] Where, L w is the current vertical side length, E w is the energy deposition value of the midpoint of the edge. After all rectangles are adjusted, the two-dimensional energy deposition distribution is calculated according to step 3, and the final graphic matrix H is obtained through the development model. The threshold development model used in the present invention is expressed as:

[0063]

[0064] Then repeat steps 3 and 4 until the side lengths of all rectangles no longer need to be adjusted.

[0065] Step 5. Confirm the optimal solution space and calculate the initial population. The search range of the optimal solution is the pixel area between the simple corrected graph and the target graph, such as Figure 3 As shown, all pixel coordinates in the area are extracted to form the basic solution space (each pixel point is encoded in binary value: 0 / 1 represents unselected / selected state). Then the coordinates of the solution space are downsampled to further narrow the search range. The downsampling strategy adopted in this embodiment is neighborhood downsampling. Neighborhood downsampling is to synthesize k pixels in the neighborhood of a pixel into a new pixel unit. After the search range is downsampled, the space complexity is reduced from O(N 2 ) is reduced to O((N / k) 2 ). Each solution in the population contains the selection state of each pixel unit, expressed as where θ i The value is 0 or 1.

[0066] Step 6. Calculate the population fitness. The solution in the population is upsampled and restored, and superimposed with the optimized layout in step 4. The energy deposition distribution and the developed pattern matrix are calculated using equations (2) and (5). The main criteria for measuring pattern quality are the intersection of union (IoU) e1 and the mean square error (MSE) e2 between the developed pattern H and the target pattern P, which are expressed as follows:

[0067]

[0068] The fitness function used in this invention is a weighted superposition of IoU and MSE, and the specific expression is as follows:

[0069] Fitness=s1(t)(1-e1)+s2(t)e2 (29)

[0070] Where t is the current number of iterations, s1,s2∈[0,1] are the weights of the two errors, and the weights are dynamically adjusted in subsequent iterations to accelerate convergence. The dynamic adjustment strategy used in the present invention is as follows:

[0071]

[0072] s2(t)=1-s1(t) (31)

[0073] Where t0 is the number of iterations of the transition point, which is set to 60, and k is the variation coefficient, which is set to 0.2.

[0074] Step 7. Generate a new population through the selection, crossover, and mutation operations of the genetic algorithm. The selection operation refers to retaining the individuals closest to the ideal solution from the current population to the new population. The method used in the present invention is the optimal retention method. First, all individuals are substituted into formula (8) to calculate the fitness, and then sorted in ascending order. The individuals with the highest ranking are selected and copied to the new population to ensure that the best individuals are retained in each iteration. The crossover operation refers to randomly selecting two individuals from the population, and through the exchange combination of two chromosomes, the excellent characteristics of the parent individuals p1 and p2 are inherited to the offspring individuals c1 and c2, thereby generating excellent individuals. The crossover method used in the present invention is the uniform crossover method, which exchanges all chromosomes in the two parent generations according to the probability q0, which is set to 0.5 here, as shown in the following formula:

[0075]

[0076] Where, Represents the i-th chromosome in the parent and offspring, r i is a random number between 0 and 1. After the crossover, the offspring individuals are added to the next generation population. The mutation operation refers to modifying the chromosomes of some individuals to prevent falling into the local optimal solution during the iteration process. The mutation method used in this paper is the global uniform position flipping method, which flips each chromosome in the individual with probability q1, which is set to 0.5 here. The specific expression is as follows:

[0077]

[0078] Where θ i Represents the i-th chromosome in the individual. After the genetic algorithm selection, crossover and mutation operations are performed on the original population, new individuals will be generated to form a new population for the next round of iterative calculation.

[0079] Step 8. Fine-tune the layout until the genetic algorithm terminates and the optimal solution is obtained. When the termination condition is met, the algorithm stops iterating and outputs the current optimal solution. Termination conditions include: the current number of iterations has reached the set maximum number of iterations, the iteration error has converged, and the iteration error has met the current accuracy requirement, that is, the fitness is less than the set error value δ.

[0080] Figure 4 This shows the graphics after direct development using an uncorrected layout. Figure 5 The corrected layout and developed pattern using this method are shown separately. It can be seen that the corrected pattern has no adhesion, effectively reducing the proximity effect. Table 1 compares the correction errors using existing technologies and this method. It can be seen that the present invention has a better effect on proximity effect correction and has high-precision calculation characteristics.

[0081] Table 1 Comparison of the accuracy of the present invention and the prior art

[0082] Calibration method Number of error pixels MSE IoU Existing technology 7675 0.77％ 87％ The present invention 2020 0.202％ 96.2％

[0083] It should be noted that the above result is one of the better examples of the effect of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions and improvements made within the spirit and principles of the present invention should be included in the scope of protection of the present invention.

Claims

1. A method for grading correction of electron beam lithography pattern based on genetic algorithm, characterized in that: The following steps are involved: Step 1. Layout data preprocessing: Discretize the layout and obtain the coordinate list of each polygon vertex of the input layout V = {V1, V2, ..., V n }, where V i ={(x1,y1),(x2,y2),...,(x m ,y m )}; and map the vertex coordinates according to the preset pixel size, update the vertex list, and create an edge list; Step 2. Polygon segmentation: Use the scan line algorithm to scan and segment the horizontal edges of the edge list S, update the vertex list and V edge list S, and extract the midpoint list O of each edge of the segmented polygon. Step 3. Calculate the exposure energy deposition distribution: Set the initial dose d0 for the exposure area and the dose for the non-exposure area to 0, generate a two-dimensional dose matrix D(x,y); and calculate the layout energy deposition matrix through convolution operation; Step 4. Perform a simple global correction on the layout: dynamically adjust the horizontal and vertical lengths of the rectangles based on the relationship between the midpoint energy deposition and the development threshold, and repeatedly calculate the energy deposition distribution until all rectangle side lengths no longer need to be adjusted; Step 5. Confirm the optimal solution space: Extract the pixel area between the simply corrected graph and the target graph as the solution space, and downsample the coordinates of the solution space to narrow the search range and generate the initial population; Step 6. Calculate the population fitness: Take the weighted superposition of the intersection-over-union ratio e1 and the mean square error e2 of the development graph H and the target graph P as the fitness function, and dynamically adjust the weight to accelerate convergence; Step 7. Generate a new population through selection, crossover, and mutation operations of the genetic algorithm; Step 8. Fine-tune the layout until the termination condition is met and output the optimal solution.

2. The method for correcting electron beam lithography patterns based on genetic algorithm according to claim 1, characterized in that: In step 1, the mapping formula of vertex coordinates is: Where dx and dy are the pixel sizes in the x and y directions respectively.

3. The method for correcting electron beam lithography patterns based on genetic algorithm according to claim 1, characterized in that: The calculation of the energy deposition matrix E(x,y) in step 3 uses a double Gaussian function as the energy distribution point spread function f(r) of a single electron beam, and the expression is: Where r is the distance from the current pixel to the exposure point; α is the forward scattering coefficient, β is the backscattering coefficient, and η is the ratio of the electrons consumed by forward scattering to backscattering.

4. The method for electron beam lithography pattern grading correction based on genetic algorithm according to claim 1, characterized in that: The formula for adjusting the side length of the rectangle in step 4 is: Horizontal edges: Where, L h is the current horizontal side length, E h is the energy deposition value at the midpoint of the edge, T is the development threshold, and T0 is the allowable error; Vertical edges: Where, L w is the current vertical side length, E w is the energy deposition value at the midpoint of the edge.

5. The method for correcting electron beam lithography pattern gradation based on genetic algorithm according to claim 1, characterized in that: The fitness function in step 6 is a weighted superposition of the intersection over union (IoU) and the mean square error (MSE), and its expression is: Fitness=s1(t)(1-e1)+s2(t)e2 (6) Where s1,s2∈[0,1] are the weights of the two errors, and the weights are dynamically adjusted in the subsequent iteration process.

6. The method for correcting electron beam lithography patterns based on genetic algorithm according to claim 1, characterized in that: The termination conditions in step 8 include: reaching the maximum number of iterations, the iteration error converges, or the fitness is less than a preset error value.

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