An electron beam lithography proximity effect correction layout energy deposition calculation method based on a fast multipole method
By employing the fast multipole method to perform quadtree partitioning and energy deposition calculations on electron beam lithography patterns, the problem of high computational complexity in proximity effect correction is solved, achieving high computational efficiency and accuracy.
Patent Information
- Application Number
- CN202210044985.2
- Authority / Receiving Office
- CN · China
- Patent Type
- Patents(China)
- Current Assignee / Owner
- Filing Date
- 2022-01-14
- Publication Date
- 2025-12-12
- Estimated Expiration
- 2042-01-14
AI Technical Summary
Existing technologies for proximity effect correction in electron beam lithography have high computational complexity, resulting in long computation time and limiting the efficiency of electron beam lithography.
The fast multipole method is used to partition the electron beam lithography pattern into a quadtree, and the energy deposition calculation is divided into far-field approximate calculation and near-field exact calculation. By constructing interactive lists and neighbor lists, the amount of computation is reduced and the computational efficiency is improved.
While ensuring accuracy, the computation time for proximity effect correction in electron beam lithography was significantly reduced, thus improving computational efficiency.
Smart Images
Figure CN116482937B_ABST
Abstract
Description
TECHNICAL FIELD
[0001] The application is a kind of calculation method of electron beam lithography proximity effect correction pattern energy deposition based on fast multipole method, which can quickly calculate electron beam lithography energy deposition under the premise of ensuring accuracy, improve calculation efficiency, and belongs to the field of computational lithography. BACKGROUND
[0002] Electron beam lithography (EBL) is a high-resolution lithography technology below 10 nanometers, which is widely used in manufacturing sensors, electronic and optical devices, micro-nano structures, etc. During electron beam exposure, the scattering of incident electrons in photoresist and substrate will change the trajectory of the electrons, which is an unavoidable physical phenomenon. These electron scattering will cause non-uniform exposure energy deposition, which will result in an unexpected exposure pattern. Because the range of electron scattering is very large, the exposure of a single shape is affected by other shapes in the vicinity, which is usually described as "proximity effect". Electron beam exposure proximity effect limits the effective pattern resolution of electron beam direct writing, which can be corrected by dose modification.
[0003] Proximity effects correction (PEC) is a convolution-based pattern optimization method. The exposure dose is convolved with the point spread function (PSF) representing the energy scattering intensity of the exposure point to the surrounding area. Through the correction function, different local exposure doses are assigned to the pattern, so that a more ideal uniform energy deposition can be achieved after exposure, and proximity effect correction is completed.
[0004] In electron beam lithography proximity effect correction, the calculation of exposure pattern energy deposition based on convolution simulation is the core step of consuming computing power. The calculation complexity of traditional convolution is O(N 2 ), and the calculation complexity of the widely used 2D-FFT is O(NlogN), and the entire pattern area needs to be calculated. In order to further speed up the calculation, the present application adopts the kernel-independent fast multipole method, and the calculation complexity of this method is O(N). The characteristics of this method are that through the operations of aggregation, transfer, divergence, etc., only the exposure area of the pattern needs to be calculated under the premise of ensuring accuracy, and the accurate calculation of all exposure points is changed into a combination of far-field approximation calculation and near-field accurate calculation, which greatly reduces the calculation amount and improves the efficiency of calculating proximity effect correction. SUMMARY
[0005] The application is a kind of electron beam lithography proximity effect correction pattern energy deposition calculation method based on fast multipole method, which aims to calculate the exposure area of the pattern under the premise of ensuring accuracy with linear complexity algorithm, greatly reduce the calculation time of pattern energy deposition in electron beam lithography proximity effect correction, so as to improve the efficiency of electron beam lithography proximity effect correction.
[0006] The technical solution of the application is: by dividing the electron beam lithography pattern with quadtree, the calculation of exposure point energy deposition is divided into approximate calculation of far field and accurate calculation of near field under the premise of ensuring accuracy, and the steps of the application are as follows:
[0007] Step S1: pattern grid division and normalization;
[0008] The original electron beam lithography pattern is read in, and the part of the pattern that needs to be exposed is evenly divided into square grids at equal intervals, each grid is an exposure point, and the initial exposure dose of each exposure point is the exposure dose of the exposure point center in the original pattern. The position of the exposure point is the position of the grid center, and the position of the exposure point is normalized to the area of [-1, 1] x [-1, 1] to facilitate the subsequent calculation.
[0009] Step S2: construct quadtree;
[0010] The root node of the quadtree is the area containing all exposure points, that is, the [-1, 1] x [-1, 1] area normalized in step S1. Starting from the root node of the quadtree, for each quadtree node I, if the number of exposure points in the area represented by node I exceeds the preset maximum number of exposure points q in the leaf node, node I is divided into four child nodes J1, J2, J3, J4 of the same size (such as Figure 2 ), and the process is recursively performed until the number of exposure points in the area represented by the quadtree node is less than q, and the construction of the quadtree is completed.
[0011] Step S3: construct interaction list and proximity list for each quadtree node;
[0012] The quadtree is pre-ordered, and the interaction list and proximity list are constructed for each quadtree node. If two nodes are in the same level in the quadtree and have common edges or common points, they are called proximate (a node is proximate to itself). The interaction list associated with each node I is composed of the child nodes of the nodes in the parent node proximity list of I, which are in the same level as node I, but have no common edges or common points (i.e. not proximate). In Figure 3 , the nodes contained in the proximity list U and the interaction list V of the dark node U are marked with U or V.
[0013] Step S4: Source points in node I are aggregated to the Chebyshev interpolation points of node I.
[0014] The aggregation is an up process, in which the source points in node I are aggregated to the Chebyshev interpolation points of node I, and the up equivalent density of node I is obtained (as shown in formula (1) Figure 4 The aggregation). For a node J in the non-adjacent list of node I, the up equivalent density of node I at its Chebyshev interpolation points can be used to replace the effect of the exposure points contained in node I on the exposure points in node J. This process is divided into 1) aggregation of exposure points to the Chebyshev interpolation points of leaf node I, as shown in formula (1); and 2) aggregation of the Chebyshev interpolation points of the child nodes of node I to the Chebyshev interpolation points of the parent node, since the Chebyshev interpolation points of the child nodes can be regarded as the source points of node I. The four-ary tree is traversed in postorder, and the source points of each node are aggregated to the Chebyshev interpolation points of the parent node (except for the root node), as shown in formula (2).
[0015]
[0016]
[0017] wherein represents the up equivalent density of node I at the interpolation point h, h = 1, 2, …, n 2 , S n (x, y) is an n-order Chebyshev interpolation polynomial, and since the layout is two-dimensional, the n-order Chebyshev interpolation will have n 2 interpolation points in total, is the position of the hth Chebyshev interpolation point in node I. r j is the position of the exposure point contained in leaf node I, and σ(r j ) is the exposure dose of the exposure point.
[0018] Step S5: Interpolation point transfer between nodes in the same layer;
[0019] The transfer is the effect of the nodes in the interaction list of node I on each other (as shown in formula (3) Figure 4 Transfer), that is, the approximate effect of the far-field exposure points on the exposure points in node I. For the root node, the down equivalent density and the up equivalent density are equal in value. The four-ary tree is traversed in preorder, and for each non-root node, the down equivalent density of node I is obtained by formula (3):
[0020]
[0021] wherein, is the down equivalent density of node I at the Chebyshev interpolation point h, h = 1, 2, …, n 2 . P(x, y) is the value of the point spread function at r = |x - y|.
[0022] Step S6: parent node interpolation point divergence to child node interpolation point;
[0023] Divergence is from the Chebyshev interpolation point of node K to its source point (e.g. Figure 4 Divergence). If node K is a quaternary tree internal node, its source point is the Chebyshev interpolation point of its child node I, calculated by formula (4); if node K is a quaternary tree leaf node, its source point is the exposure point contained in it, at this time the exposure point r i The energy deposition of the far-field exposure point at its position is obtained, calculated by formula (5).
[0024]
[0025] Wherein is the downward equivalent density of node I at the Chebyshev interpolation point h, h = 1, 2, …, n 2 .
[0026]
[0027] Wherein E(r i ) is the energy deposition at the position of r i .
[0028] Step S7: neighbor list calculation;
[0029] For the exposure point r i in the leaf node I, the energy deposition of all exposure points in the neighbor list of the leaf node I at r i is accurately calculated, as formula (6).
[0030]
[0031] Step S8: update the exposure dose of the layout. BRIEF DESCRIPTION OF DRAWINGS
[0032] Figure 1 is the operation flowchart of the present application;
[0033] Figure 2 is the schematic diagram of node recursive division;
[0034] Figure 3 is the schematic diagram of interaction list and neighbor list;
[0035] Figure 4 is the schematic diagram of fast multipole method operation. DETAILED DESCRIPTION
[0036] The present application will be further described in detail below in combination with the drawings and specific examples of the specification.
[0037] The application is a kind of calculation method of electron beam lithography proximity effect correction layout energy deposition based on fast multipole method, which aims to calculate the exposure area of the layout with linear complexity algorithm under the premise of ensuring accuracy, greatly reduce the calculation time of layout energy deposition in electron beam lithography proximity effect correction, and improve the efficiency of electron beam lithography proximity effect correction. Figure 1 The steps of the application are shown in the following eight steps: layout grid division and normalization; constructing quadtree; constructing interaction list and proximity list for each quadtree node; gathering node interpolation points for intra-node source points; transferring interpolation points between nodes in the same layer; diverging interpolation points of parent nodes to interpolation points of child nodes; proximity list calculation; updating layout exposure dose. The technical solution of the application is as follows: by dividing the electron beam lithography layout with quadtree, the calculation of exposure point energy deposition is divided into approximate calculation of far field and accurate calculation of near field under the premise of ensuring accuracy, and the specific implementation steps are as follows:
[0038] Step S1: layout grid division and normalization;
[0039] The original electron beam lithography layout is read in, and the part of the layout that needs to be exposed is evenly divided into square grids at equal intervals, each grid being an exposure point, and the initial exposure dose of each exposure point being the exposure dose of the center of the exposure point in the original layout. The position of the exposure point is the position of the grid center, and the position of the exposure point is normalized to the area of [-1, 1] x [-1, 1] to facilitate subsequent calculation.
[0040] Step S2: constructing quadtree;
[0041] The root node of the quadtree is the area containing all exposure points, i.e. the [-1, 1] x [-1, 1] area after normalization in step S1. Starting from the root node of the quadtree, for each quadtree node I, if the number of exposure points in the area represented by node I exceeds the preset maximum number of exposure points in the leaf node q, node I is divided into four child nodes J1, J2, J3, J4 of the same size (as shown in Figure 2 ), and the process is recursively performed until the number of exposure points in the area represented by the quadtree node is less than q, and the construction of the quadtree is completed.
[0042] Step S3: constructing interaction list and proximity list for each quadtree node;
[0043] The pre-order traversal of the quadtree is performed, and the interaction list and the neighbor list are constructed for each quadtree node. Two nodes are called neighbors if they are at the same level in the quadtree and have a common edge or a common point (a node is a neighbor of itself). The interaction list associated with each node I is composed of the children of the nodes in the parent node neighbor list of I that are at the same level as I but do not have a common edge or a common point (i.e., are not neighbors). In Figure 3 , the nodes contained in the neighbor list U and the interaction list V of the dark node U are marked with U or V.
[0044] Step S4: The source points in a node are aggregated to the interpolation points of the node.
[0045] The aggregation is an up-ward process, and the source points in a node I are aggregated to the Chebyshev interpolation points of the node I to obtain the up-ward equivalent density of the node I (as shown in Figure 4 ). For a node J in the non-neighbor list of the node I, the up-ward equivalent density of the node I at its Chebyshev interpolation points can be used to replace the effect of the exposure points contained in the node I on the exposure points in the node J. This process is divided into 1) the aggregation of the exposure points to the Chebyshev interpolation points of the leaf node I, as shown in equation (1); and 2) since the Chebyshev interpolation points of the child nodes of the node I can be regarded as the source points of the node I, the aggregation of the child node Chebyshev interpolation points to the parent node Chebyshev interpolation points. The post-order traversal of the quadtree is performed, and the source points of each node are aggregated to the Chebyshev interpolation points of the parent node (except for the root node), as shown in equation (2).
[0046]
[0047]
[0048] wherein represents the up-ward equivalent density of the node I at the interpolation point h, h = 1, 2, …, n 2 , S n (x, y) is an n-order Chebyshev interpolation polynomial, and since the layout is two-dimensional, the n-order Chebyshev interpolation will have n 2 interpolation points in total, is the position of the hth Chebyshev interpolation point in the node I. r j is the position of the exposure point contained in the leaf node I, and σ(r j ) is the exposure dose of the exposure point.
[0049] Step S5: Interpolation point transfer between nodes at the same level.
[0050] The transfer is the effect of the nodes in the interaction list of the node I on the node I (as shown in Figure 4The down equivalent density of node I is obtained by equation (3) for each non-root node in the pre-order traversal of the quadtree:
[0051]
[0052] where, is the down equivalent density of node I at Chebyshev interpolation point h, h = 1, 2,..., n 2 P(x, y) is the value of the point spread function at r = |x - y|.
[0053] Step S6: Parent interpolation points diverge to child interpolation points;
[0054] Divergence is the divergence from the Chebyshev interpolation points of node K to the source points therein (e.g. Figure 4 Divergence). If node K is an internal node of the quadtree, its source points are the Chebyshev interpolation points of its child nodes I, which are calculated by equation (4); if node K is a leaf node of the quadtree, its source points are the exposure points contained therein, in which case the exposure points r i get the approximation of the energy deposition of the far-field exposure point at its position, which is calculated by equation (5).
[0055]
[0056] where, is the down equivalent density of node I at Chebyshev interpolation point h, h = 1, 2,..., n 2 .
[0057]
[0058] where E(r i ) is the energy deposition at r i position.
[0059] Step S7: Neighboring list calculation;
[0060] For the exposure point r i in leaf node I, the energy deposition of all exposure points in the neighboring list of leaf node I at r i is calculated accurately, as equation (6).
[0061]
[0062] Step S8: Update the layout exposure dose.
[0063] As shown in Table 1, the present application is a kind of electron beam lithography proximity effect correction layout energy deposition calculation method based on fast multipole method, when the layout exposure area proportion ρ layout less than a certain proportion (such as Figure 1 less than 80%, the fast multipole method used in the present application is faster than two-dimensional fast Fourier transform. Since, for the lithography layout, the layout exposure area proportion is between 50% and 60% on average, the method of the present application can be well applied to the actual situation.
[0064] Table 1
[0065] Layout exposure area ratio (%) 20 40 60 80 100 Fast multipole method (s) 6..9 13.8 19.9 27.2 32.5 Two-dimensional Fourier transform (s) 26.7 26.7 26.7 26.7 26.7
[0066] Finally, it should be noted that the above examples are only used to illustrate the technical solutions of the present application, and are not a limitation on the scope of protection of the present application. With reference to the description of the examples, those of ordinary skill in the art should be able to understand and make relevant modifications or replacements to the technical solutions of the present application without departing from the essence and scope of the present application.
Claims
1. A method for calculating the energy deposition of electron beam lithography proximity effect-corrected layouts based on the fast multipole method, characterized in that, Includes the following steps: Step S1: Perform mesh division and normalization on the original electron beam lithography pattern, dividing the exposure area into equally spaced square meshes and normalizing them to [-1,1]×[-1,1]; Step S2: Construct a quadtree based on the exposure point number threshold, with the normalized region containing all exposure points as the root node. When the number of exposure points contained in a node exceeds the preset threshold q, it is recursively divided into four child nodes until the number of exposure points in each leaf node does not exceed q. Step S3: Construct an interaction list and a neighbor list for each quadtree node. A neighbor node is defined as a node at the same level that has a common edge or a common point, including the node itself. The interaction list consists of the child nodes at the same level as the current node but not neighboring the current node in the neighbor list of the parent node. Step S4: Aggregate the source points within the node to the Chebyshev interpolation points of the node to obtain the upward equivalent density. The aggregation process includes the aggregation of exposure points to leaf node interpolation points and the aggregation of child node interpolation points to parent node interpolation points. Step S5: Based on the interaction list, perform interpolation point transfer between nodes on the same layer, and use the point spread function P(x,y) to calculate the approximate effect of the far-field exposure point to obtain the downward equivalent density; Step S6: Diverge the Chebyshev interpolation points of the parent node to the interpolation points of the child nodes. When the node is an internal node, diverge to the interpolation points of its child nodes. When the node is a leaf node, diverge to the exposure points it contains, thereby obtaining an approximate value of energy deposition. Step S7: Perform precise energy deposition calculations between the exposure points within the leaf node and their neighboring exposure points in the list. Step S8: Update the exposure dose of the layout based on the calculation results.
2. The method for calculating the energy deposition of electron beam lithography proximity effect correction layout based on the fast multipole method as described in claim 1, characterized in that: In step S1, the original electron beam lithography pattern is read in, and the part of the pattern that needs to be exposed is evenly divided into square grids at equal intervals. Each grid corresponds to an exposure point. The initial exposure dose of the exposure point is the exposure dose of the center of the exposure point on the original pattern. The position of the exposure point is the position of the grid center. The position of the exposure point is normalized to the region of [-1,1]×[-1,1] to facilitate subsequent calculations.
3. The method for calculating the energy deposition of electron beam lithography proximity effect correction layout based on the fast multipole method as described in claim 1, characterized in that: In step S2, the root node of the quadtree is the region containing all exposure points, which is the normalized [-1,1]×[-1,1] region in step S1. Starting from the root node of the quadtree, for each quadtree node I, if the number of exposure points in the region represented by node I exceeds the preset maximum number of exposure points q in the leaf node, then node I is divided into four child nodes of the same size. This process is repeated recursively until the number of exposure points in the region represented by the quadtree node is less than q, thus completing the construction of the quadtree.
4. The method for calculating the energy deposition of electron beam lithography proximity effect correction layout based on the fast multipole method as described in claim 1, characterized in that: In step S3, a preorder traversal is performed on the quadtree to construct an interaction list and a neighbor list for each quadtree node. A neighbor node is defined as a node at the same level that has a common edge or a common point. Each node also includes itself as a neighbor node. The interaction list associated with each node I consists of the child nodes of the nodes in the neighbor list of I's parent node. These nodes are at the same level as node I but have no common edge or common point.
5. The method as described in claim 1, characterized in that: In step S4, aggregation is an upward process, which refers to aggregating the source points within node I to the Chebyshev interpolation points of node I to obtain the upward equivalent density of node I. For node J in the non-nearest neighbor list of node I, the upward equivalent density of node I at its Chebyshev interpolation point can be used to approximate the effect of the exposure points contained in node I on the exposure points in node J. This includes two processes: aggregation of exposure points to leaf node interpolation points and aggregation of child node interpolation points to parent node interpolation points, as shown in equation (1). Finally, the source points of each node are aggregated to the interpolation points of its parent node through a post-order traversal of the quadtree until the root node is reached, as shown in equation (2). This represents the upward equivalent density of node I at the interpolation point h, where h = 1, 2, ..., n 2 S n (x, y) is an nth-order Chebyshev interpolation polynomial. Since the layout is two-dimensional, there will be a total of n interpolation points for the nth-order Chebyshev interpolation. It is the position of the h-th Chebyshev interpolation point within node I. It is the h-th node of node J ′ The location of each Chebyshev interpolation point, r j It is the location of the exposure point contained in leaf node I, σ(r) j ) is the exposure dose at the exposure point. It is the h-th child node J ′ The upward equivalent density at each interpolation point.
6. The method for calculating the energy deposition of electron beam lithography proximity effect correction layout based on the fast multipole method as described in claim 1, characterized in that: In step S5, the interaction between nodes in the node I interaction list is transformed into the effect of the nodes on the node, that is, the approximate effect of the far-field exposure point on the exposure point within node I. For the root node, the downward equivalent density and the upward equivalent density are numerically equal. The quadtree is traversed in preorder. For each non-root node, the downward equivalent density of node I is obtained by equation (3), where, Let h be the downward equivalent density of node I at the Chebyshev interpolation point h, where h = 1, 2, ..., n 2 P(x,y) is the value of the point spread function at r = |xy|. It is the position of the h-th Chebyshev interpolation point within node I. It is the h-th node of node J ′ The location of the Chebyshev interpolation points It is the h-th child node J ′ The upward equivalent density at each interpolation point.
7. The method for calculating the energy deposition of electron beam lithography proximity effect correction layout based on the fast multipole method as described in claim 1, characterized in that: In step S6, the divergence is from the Chebyshev interpolation point of node K to its source point. If node K is an internal node of the quadtree, its source point is the Chebyshev interpolation point of its child node I, calculated by equation (4); if node K is a leaf node of the quadtree, its source point is the exposure point it contains, where Let h be the downward equivalent density of node I at the Chebyshev interpolation point h, where h = 1, 2, ..., n 2 , It is the position of the h-th Chebyshev interpolation point within node I. It is the h-th node of node K ′ The location of the Chebyshev interpolation points For node K at Chebyshev interpolation point h ′ The downward equivalent density at point r is used to approximate the energy deposition at the far-field exposure point at its location. E(r) is calculated using equation (5). i ) is r i Energy deposition at location, S n (x,y) is an nth-order Chebyshev interpolation polynomial. It is the position of the h-th Chebyshev interpolation point of node K. Let be the downward equivalent density of node K at the Chebyshev interpolation point h.
8. The method for calculating the energy deposition of electron beam lithography proximity effect correction layout based on the fast multipole method as described in claim 1, characterized in that: In step S7, for an exposure point r within leaf node I, the energy deposition at r for all exposure points in the neighboring list of leaf node I is precisely calculated, as shown in equation (6), where E(r) is the energy deposition at position r, P(x,y) is the point spread function at r = |xy|, and σ(r) is the energy deposition at position r. j ) is the exposure dose at the exposure point. After energy deposition is obtained, the exposure dose of the pattern is updated.