阅读说明：本技术 一种考虑电子束雾化效应的解析布局方法 (Analytic layout method considering electron beam atomization effect ) 是由 陈建利 黄志鹏 于 2020-04-23 设计创作，主要内容包括：本发明涉及一种考虑电子束雾化效应的解析布局方法,包括如下步骤：1、布局框架：使用Λ型多层框架来处理布局中的大规模设计,以进行全局布局；2、雾化变化建模：对评估点进行采样,这些评估点均匀分布在整个布局中；利用雾源模型的评估点,通过带有Hermite展开的快速高斯变换估计每个评估点的雾化效果；3、设定一个同时考虑线长优化和雾化变化优化的目标函数；4、通过共轭梯度算法来优化布局线长长度；5、删除重叠的单元格部分,并使标准单元格对齐,以尽可能保留全局布局获得的布局结果；6、详细布局,通过计算交换面积变化系数,以决定是否在选定的标准单元进行交换。该方法有利于优化布局质量,在优化线长的同时减小雾化变化。(The invention relates to an analytic layout method considering electron beam atomization effect, which comprises the following steps of 1, a layout frame, a model Λ multilayer frame, a sampling method, an atomization change modeling method, a calculation method and a calculation method, wherein the large-scale design in the layout is processed through the Λ multilayer frame to carry out overall layout, the atomization change modeling method comprises the steps of sampling evaluation points which are uniformly distributed in the whole layout, estimating the atomization effect of each evaluation point through quick Gaussian transform with Hermite expansion by utilizing the evaluation points of a fog source model, 3, setting an objective function which simultaneously considers line length optimization and atomization change optimization, 4, optimizing the length of a layout line through a conjugate gradient algorithm, 5, deleting overlapped cell parts, aligning standard cells to keep the layout result obtained by the overall layout as much as possible, and 6, carrying out detailed layout, and determining whether to exchange is carried out on the selected standard cells or not through calculating exchange area change coefficients.)

1. An analytic layout method considering electron beam atomization effect, comprising the steps of:

(1) a layout framework, wherein an Λ type multi-layer framework is used for processing large-scale design in the layout so as to carry out global layout;

(2) modeling atomization change: sampling evaluation points which are uniformly distributed in the whole layout, wherein the distance between two adjacent evaluation points is constant; estimating the atomization effect of each evaluation point by using the evaluation points of the fog source model through a fast Gaussian transform with Hermite expansion;

(3) setting an objective function considering both line length optimization and atomization change optimization;

(4) optimizing the length of the layout line by a conjugate gradient algorithm;

(5) deleting the overlapped cell part and aligning the standard cells so as to keep the layout result obtained by the global layout as much as possible;

(6) and (4) detailed layout, namely calculating a change coefficient of the exchange area to determine whether to exchange in the selected standard cell.

2. An analytical layout method considering electron beam fogging effect according to claim 1, characterised in that in step (1) the Λ -type multi-layer framework comprises three main stages, clustering, initial layout and clustering, during clustering, standard cells are iteratively clustered according to cell area and connectivity first until the number of clusters is within a threshold value to effectively layout the clusters, then initial layout is performed on the clusters, and finally the clusters of standard cells are progressively de-clustered and their positions are determined iteratively.

3. The analytical layout method considering electron beam fogging effect as claimed in claim 1, wherein in step (2), the method for estimating fogging effect of each evaluation point by fast gaussian transformation with Hermite expansion is: given a set of atomization sourcesAnd a set of evaluation pointsCalculating an evaluation point by the formula (1)t _i The atomization effect of (A):

（1）

wherein x and y are the x and y coordinates of the mist source, respectively; therefore, the change in atomization effect is calculated by equation (2):

（2）。

4. the analytical layout method considering electron beam fogging effect as claimed in claim 1, wherein in step (3), the objective function is set as:

（3）

wherein λ is₁、λ₂And λ₃Is a weight, W_{(x, y)}Is a weighted average model, D_{b(x, y)}Is a bell-shaped density model; in iteratively finding the optimal positions of all circuit blocks, the weight λ₁、λ₂And λ₃Continuously updating;

in the objective function, the values of the three weights vary iteratively, first using a weight λ greater than the set value₁To optimize the line length, the weight is reduced with iteration, and then the weight lambda of the density model is gradually increased₂To expand the standard cell and then increase the weight λ of the change in atomization in subsequent iterations₃To obtain better variation in atomization by modifying the cell distribution;

weight λ₁、λ₂And λ₃Is based on Gompertz curves:

（4）

where β is the displacement along the x-axis, i.e., the graph is translated to the left or right, γ is the growth rate, and k represents the number of iterations.

5. The analytic layout method considering electron beam atomization effect of claim 1 wherein in step (4), the unconstrained minimization problem is not solved by a precise line search method, and the conjugate optimization algorithm first initializes iteration number i to 0 and then calculates gradient g in line 1_kAnd the direction of conjugate gradient d_k(ii) a Then determining the step size of the kth iteration in the 3 rd row; the accurate step size is obtained by solving the following optimization problem:

（5）。

6. the analytical layout method considering electron beam fogging effect as claimed in claim 1, wherein in step (5), given the global layout result, the cell overlap with the minimum standard cell displacement is eliminated by extending the Abacus algorithm and the minimum fogging variation is retained; firstly, sorting all standard cells according to an x coordinate, and then legalizing the standard cells according to the x coordinate of the standard cells; after making the standard cell S_iWhile legalizing, inserting the standard cell into each row, and finding the best row of the standard cell; when the standard cell S is used_iInserted into a certain row r_jThe inserted standard cell S is calculated with a minimum displacement by dynamic programming_iAnd has been laid out to that row r_jThe location of other standard cells; the standard cell S is then put at minimum cost_iInserting into a row; validating the global layout result and minimizing the total displacement of the standard cells by validating the standard cells; the method of determining the position of the standard cell in a row with the smallest displacement is as follows:

（6）

（7）

wherein N is_rIs the number of standard cells in the row, k_iIs a standard pixel element S_iWeight of (1), x_iIs a picture element S_iIs determined by the x-coordinate of the user,is a picture element S_iOriginal x-coordinate of (1), w_iIs a picture element S_iThe width of the pixel; in order to minimize both displacement and variation in the fogging effect obtained from the global layout, the standard cell S is used_iWeight k of_iSet as equation (6), proportional to intensity; by modifying the formula (6)Moving critical mist sources has higher losses than moving conventional mist sources to better preserve minimal variation in atomization during legitimization.

7. The analytic layout method of claim 1 wherein in step (6), the exchange area variation coefficient is calculated to determine whether pixel exchange is performed between selected standard pixels, wherein the exchange area variation coefficient ξ between the set C of selected standard pixels is defined by the following formula:

（8）

wherein N is_cIs the number of standard cells in C, a_iIs a standard cell c_i∈ C, calculating coefficients, if ξ is less than the user-defined threshold ξ_tThen the cell swap is allowed, otherwise the swap is prohibited.

Technical Field

The invention belongs to the technical field of super-large-scale integrated circuit design, and particularly relates to an analytic layout method considering an electron beam atomization effect.

Background

In NG L, electron beam (e-beam) lithography (EB L) is a promising technology, electron beam can be focused to nanometer diameters by using electromagnetic or electrostatic lenses, and furthermore, direct writing allows EB L to print features directly on the wafer without a mask, thus not being limited by light diffraction.

The electron beam lithography EB L manufacturing process is illustrated in FIG. 2, as illustrated in FIG. 2 (a), electrons from an electron source pass through a set of lenses and apertures and then write a pattern directly on the wafer, the pattern is created by exposing the resist, evaporating the metal and dissolving unwanted parts, however, at this time, electrons may scatter in the resist and silicon substrate, in FIG. 2 (b), the electrons scatter as they reach the resist and silicon substrate, these scattered electrons create backscattered electrons, and these backscattered electrons may strike the bottom of the objective lens, the next generation of electrons may be created, called re-scattered electrons, and the re-scattered electrons are then spread to a location that may be far from the point of initial exposure, all these scattered electrons cause unwanted additional exposure, causing layout pattern distortion around the primary electron beam, called proximity effects and fogging effects.

Disclosure of Invention

The invention aims to provide an analytic layout method considering the electron beam atomization effect, which is beneficial to optimizing the layout quality and reducing the atomization change while optimizing the line length.

In order to achieve the purpose, the invention adopts the technical scheme that: an analytic layout method considering electron beam atomization effect comprises the following steps:

(1) a layout framework, wherein an Λ type multi-layer framework is used for processing large-scale design in the layout so as to carry out global layout;

(2) modeling atomization change: sampling evaluation points which are uniformly distributed in the whole layout, wherein the distance between two adjacent evaluation points is constant; estimating the atomization effect of each evaluation point by using the evaluation points of the fog source model through a fast Gaussian transform with Hermite expansion;

(3) setting an objective function considering both line length optimization and atomization change optimization;

(4) optimizing the length of the layout line by a conjugate gradient algorithm;

(5) deleting the overlapped cell part and aligning the standard cells so as to keep the layout result obtained by the global layout as much as possible;

(6) and (4) detailed layout, namely calculating a change coefficient of the exchange area to determine whether to exchange in the selected standard cell.

Further, in the step (1), the Λ -type multi-layer framework comprises three main stages of clustering, initial layout and clustering, during clustering, standard cells are iteratively clustered according to cell areas and connectivity until the number of clusters is within a threshold value so as to effectively layout the clusters, then initial layout is performed on the clusters, and finally the clusters of the standard cells are progressively de-clustered and the positions of the standard cells are determined iteratively.

Further, in the step (2), the method for estimating the atomization effect of each evaluation point through a fast gaussian transform with a Hermite expansion comprises the following steps: given a set of atomization sourcesAnd a set of evaluation pointsThe atomization effect at the evaluation point is calculated by equation (1):

（1）

wherein x and y are the x and y coordinates of the mist source, respectively; therefore, the change in atomization effect is calculated by equation (2):

（2）。

further, in the step (3), the objective function is set as:

（3）

wherein λ is₁、λ₂And λ₃Is a weight, W_{(x, y)}Is a weighted average model, D_{b(x, y)}Is a bell-shaped density model; in iteratively finding the optimal positions of all circuit blocks, the weight λ₁、λ₂And λ₃Continuously updating;

in the objective function, the values of the three weights vary iteratively, first using a weight λ greater than the set value₁To optimize the line length, the weight is reduced with iteration, and then the weight lambda of the density model is gradually increased₂To expand the standard cell and then increase the weight λ of the change in atomization in subsequent iterations₃To obtain better variation in atomization by modifying the cell distribution;

weight λ₁、λ₂And λ₃Is based on Gompertz curves:

（4）

where β is the displacement along the x-axis, i.e., the graph is translated to the left or right, γ is the growth rate, and k represents the number of iterations.

Further, in the step (4), the unconstrained minimization problem is not solved by a precise line search method, and the conjugate optimization algorithm firstly initializes the iteration number i to 0 and then calculates the gradient g in the 1 st row_kAnd the direction of conjugate gradient d_k(ii) a Then determining the step size of the kth iteration in the 3 rd row; the accurate step size is obtained by solving the following optimization problem:

（5）。

further, in the step (5), given a global layout result, the cell overlap with the minimum standard cell displacement is eliminated and the minimum atomization change is reserved by expanding the Abacus algorithm; firstly, sorting all standard cells according to an x coordinate, and then legalizing the standard cells according to the x coordinate of the standard cells; after making the standard cell S_iWhile legalizing, inserting the standard cell into each row, and finding the best row of the standard cell; when the standard cell S is used_iInserted into a certain row r_jThe inserted standard cell S is calculated with a minimum displacement by dynamic programming_iAnd has been laid out to that row r_jThe location of other standard cells; the standard cell S is then put at minimum cost_iInserting into a row; validating the global layout result and minimizing the total displacement of the standard cells by validating the standard cells; the method of determining the position of the standard cell in a row with the smallest displacement is as follows:

（6）

（7）

wherein N is_rIs the number of standard cells in the row, k_iIs a standard pixel element S_iWeight of (1), x_iIs a picture element S_iIs determined by the x-coordinate of the user,is a picture element S_iOriginal x-coordinate of (1), w_iIs a picture element S_iThe width of the pixel; in order to minimize both displacement and variation in the fogging effect obtained from the global layout, the standard cell S is used_iWeight k of_iSet as equation (6), proportional to intensity; by modifying the weights in equation (6), moving critical mist sources has higher losses than moving conventional mist sources to better preserve minimal variation in misting during legitimization.

Further, in step (6), for the problem that exchanging standard pixels may change the atomization change, an exchange area change coefficient is calculated to determine whether pixel exchange is performed between selected standard pixels, wherein the exchange area change coefficient ξ between the set C of selected standard pixels is defined by the following formula:

（8）

wherein N is_cIs the number of standard cells in C, a_iIs a standard cell c_i∈ C, calculating coefficients, if ξ is less than the user-defined threshold ξ_tThen the cell swap is allowed, otherwise the swap is prohibited.

Compared with the prior art, the method has the advantages that 1, the atomization effect is reduced in the layout stage in advance according to the influence of unit layout on the atomization effect in the global layout period, 2, a specific fog source analysis model is established, meanwhile, the time of Gaussian operation is greatly shortened by using Hermite expansion, 3, the layout model with the exchange area coefficient as the model in the detailed layout stage can further reduce the length of the layout line, 4, the method can effectively solve the legalization problem of the V L SI mixed height standard unit and provide a practical legalization result, compared with NTUpalace 4dr, the method can effectively reduce the atomization effect by 35.4%, and meanwhile, higher line length quality is kept, compared with ICCAD' 18, the atomization effect is reduced by 8.5%, the line length is shortened by 3.3%, and meanwhile, the operation time is shortened by 35%.

Drawings

FIG. 1 is a flow chart of a method implementation of an embodiment of the present invention.

Fig. 2 is a schematic diagram of a prior art EB L fabrication process.

Fig. 3 is a weight variation curve in the embodiment of the present invention.

FIG. 4 is a flow chart of a conjugate gradient algorithm in an embodiment of the present invention.

FIG. 5 is an exemplary diagram of a unit swap in an embodiment of the invention.

Fig. 6 is a schematic diagram of cell switching in an embodiment of the present invention.

Fig. 7 is a representation of a parameter symbol used in an embodiment of the present invention.

Fig. 8 is a layout diagram in an embodiment of the present invention.

FIG. 9 is a model of the deposition energy produced by atomization in an embodiment of the present invention.

FIG. 10 shows two Gaussian distributions of atomization in an embodiment of the present invention

FIG. 11 is a schematic diagram of converting each standard cell to a source of the fogging effect in an embodiment of the invention.

FIG. 12 is a diagram illustrating an embodiment of the present invention in which the standard cell is considered as a movable source of atomization.

Fig. 13 is a diagram of a fast gaussian transform in an embodiment of the present invention.

FIG. 14 is a grid diagram in a fast Gaussian transform with Hermite expansion in an embodiment of the present invention.

FIG. 15 is a diagram illustrating how the sum of all Gaussian distributions on a target is calculated with low temporal complexity by the fast Gaussian transform with Hermite expansion in an embodiment of the present invention.

Detailed Description

The invention is further described with reference to the following figures and specific embodiments.

FIG. 1 is a flow chart of an implementation of the analytic layout method of the present invention considering the effect of electron beam atomization. The invention provides an analytic layout method considering electron beam atomization effect, which comprises the following steps:

(1) layout framework, i.e. "Global layout with fog sense" section of FIG. 1

A large-scale design in a layout is processed using an Λ -type multi-layer framework for global layout.

The Λ -type multi-layer framework comprises three main stages of clustering, initial placement and clustering, during clustering, standard cells are iteratively clustered according to cell area and connectivity until the number of clusters is within a threshold value so that the clusters can be effectively placed, initial placement is performed on the clusters, and finally the clusters of the standard cells are progressively de-clustered and the positions of the standard cells are determined iteratively.

In order to address the fogging effect, the invention also considers the pattern regions of the standard cells during clustering. The present invention sets an upper limit for the total pattern area in one cluster that is slightly larger than the average pattern area in all clusters. As a result, the total pattern area of the clusters is similar to the other pattern areas.

(2) Modeling atomization change: sampling evaluation points, wherein the evaluation points are uniformly distributed in the whole layout, and the distance between two adjacent evaluation points is a constant (for example, 5 mu m); the fogging effect of each evaluation point was estimated by a fast gaussian transform with Hermite expansions using the evaluation points of the fog source model.

The method for estimating the fogging effect of each evaluation point by a fast gaussian transform with Hermite expansion is: given a set of atomization sourcesAnd a set of evaluation pointsCalculating an evaluation point by the formula (1)t _i The atomization effect of (A):

（1）

wherein x and y are the x and y coordinates of the mist source, respectively; therefore, the change in atomization effect is calculated by equation (2):

（2）

(3) an objective function is set which simultaneously considers optimization of the line length and optimization of the atomization change, so that the atomization change is reduced to the maximum extent, and meanwhile, a good layout line length is kept.

To minimize haze variation while maintaining good layout line length, the objective function of the analysis layout of the present invention is:

（3）

wherein λ is₁、λ₂And λ₃Is a weight, W_{(x, y)}Is a weighted average model, D_{b(x, y)}Is a bell-shaped density model; in iteratively finding the optimal positions of all circuit blocks, the weight λ₁、λ₂And λ₃And continuously updating.

In the objective function, first the line length optimization in the early iteration of the global layout is focused on, and then the density weight λ is gradually increased₂Finally, the atomization variation weight λ is increased in subsequent iterations₃Thus providing a smoother optimization over the three design criteria. In early iterations, the total pattern area in each block was used to roughly guide the motion of the standard cell. The present invention attempts to distribute the pattern area evenly throughout the block, thereby evenly distributing the mist source.

The values of the three weights vary from iteration to iteration. The invention is first of all obtained by using a larger lambda₁The wire length is optimized by a weight that decreases with iteration. Then gradually increasing the weight lambda of the density model₂With expanded standard cells, and varying weight λ of fog₃Only in subsequent iterations is significantly increased to obtain better variation in atomization by modifying the cell distribution.

Weight λ₁、λ₂And λ₃Is based on Gompertz curves:

（4）

where β is the displacement along the x-axis, i.e., the graph is shifted left or right, γ is the rate of increase, k represents the number of iterations₁=1-f (λ), where β =20, γ = 0.04, λ for density control weight₂= f (λ) where β = 2000, γ = 0.07, λ, for the atomization effect weight₃= f (λ), where β =20, γ = 0.045 with this optimization we can minimize the fogging variation without sacrificing line length_fThe term (x, y), the standard cell may be excessively scattered to a bad position having a large line length. Thus, the pattern region is used as a guide in the early iteration, S_f(x, y) weight of fog change λ₃Only in later iterations is substantially increased to achieve better simultaneous atomization changes and line length minimization.

(4) The layout line length is optimized by a conjugate gradient algorithm.

In conjugate gradient algorithms, the exact line search method is sometimes a useful and efficient technique to calculate the step size. However, each iteration requires a line search process, resulting in a large number of computations. To reduce the evaluation of the objective function and the gradient, the use of line search procedures should be avoided in the algorithm design. In each clustering level, let ν = (x, y). Algorithm 1 in fig. 4 shows the conjugate gradient algorithm of the present invention. The present invention does not solve the unconstrained minimization problem by an accurate line search method. The number of iterations i is first initialized to 0 and then the gradient g is calculated in row 1_kAnd the direction of conjugate gradient d_k(ii) a Then determining the step size of the kth iteration in the 3 rd row; the accurate step size is obtained by solving the following optimization problem:

（5）

for some constant c₁∈ (0,1), since the layout problem is very large in scale, accurate line search is usually time consuming, and it is difficult to incorporate the accurate line search method directly into the conjugate gradient to solve the unconstrained minimization problem of the present invention.

(5) Section entitled "layout validation with fog feeling" in fig. 1

The overlapping cell portions are removed and the standard cells are aligned to preserve as good a layout result as possible from the global layout.

The object of the invention is to minimize the displacement of the standard cell. Given the global layout results, the present invention eliminates the cell overlap with the minimum standard cell displacement and preserves the minimum variation in atomization by extending the Abacus algorithm. Firstly, sorting all standard cells according to an x coordinate, and then legalizing the standard cells according to the x coordinate of the standard cells; after making the standard cell S_iWhile legalizing, inserting the standard cell into each row, and finding the best row of the standard cell; when the standard cell S is used_iInserted into a certain row r_jThe inserted standard cell S is calculated with a minimum displacement by dynamic programming_iAnd has been laid out to that row r_jThe location of other standard cells; the standard cell S is then put at minimum cost_iInserting into a row; validating the global layout result and minimizing the total displacement of the standard cells by validating the standard cells; the method of determining the position of the standard cell in a row with the smallest displacement is as follows:

（6）

（7）

wherein N is_rIs the number of standard cells in the row, k_iIs a standard pixel element S_iWeight of (1), x_iIs a picture element S_iIs determined by the x-coordinate of the user,is a picture element S_iOriginal x-coordinate of (1), w_iIs a picture element S_iThe width of the pixel. Abacus legalizes standard cell blocks with a minimum of displacement. Minimizing the displacement implicitly preserves the variation in fogging achieved during the overall layout. Because fogging is a global effect, local movement of standard cells does not significantly affect the fogging effect of the design. However, if all standard cells move slightly, cumulative impact may reduce minimized fogging variation. Thus, the present invention meets our goal by modifying Abacus to maintain minimal fogging variation from the overall layout. Since standard cells are inevitably moved during legitimization, we avoid moving critical atomisation sources. An atomisation source is said to be of critical importance if its movement has a greater effect on the atomisation effect than other standard cells, i.e. standard cells with greater sensitivity to atomisation with respect to displacement. It is clear that a higher strength mist source has a greater effect on the mist effect than a lower strength mist source for the same unit displacement. Therefore, in order to minimize both displacement and variation in the fogging effect obtained from the global layout, the standard cell S is used_iWeight k of_iSet as equation (6), proportional to the intensity.

By modifying the weights in equation (6), moving critical mist sources has higher losses than moving conventional mist sources. Thus, the critical atomization source tends not to move as vigorously, thereby better preserving minimal atomization variation during legitimization.

(6) Detailed layout with fog feeling in FIG. 1

And calculating the exchange area change coefficient to determine whether to exchange in the selected standard cell.

Cell swapping is a popular technique to minimize wire length during detailed layout, as shown in fig. 5. Fig. 5(a) is three selected standard cells, and fig. 5(a) is all six permutations of the selected standard cells. For cell swapping, a standard cell window is first selected. For a branch-and-bound scheme of all selected standard cells, or a two-way matching scheme between standard cells and the desired location, the desired solution with the smallest wire length may be selected.

The present invention derives a new scheme to keep the fogging variation during cell swapping to a minimum. The main problem is that exchanging standard picture elements may change the fogging change, especially when the exchanged picture elements have different pattern areas. Taking into account the position x_iStandard cell c with large pattern area_iAnd the cell is occupied with another cell c_jAnd (4) exchanging. If c is_jIs smaller, then from position x_iThe fogging intensity of the distribution will be lower than that of the non-exchanged configuration. Once from position x_iDecrease in the fogging Strength, x_iThe surrounding atomization effect will decrease, which may lead to a change in atomization. The present invention therefore modifies the cell swap to prevent standard cell swaps with large pattern area variations.

Aiming at the problem that the exchange standard image elements can change the atomization change, the invention calculates the exchange area change coefficient to decide whether to exchange the image elements among the selected standard image elements, wherein the exchange area change coefficient ξ among the set C of the selected standard image elements is defined by the following formula:

（8）

wherein N is_cIs the number of standard cells in C, a_iIs a standard cell c_i∈ C, calculating coefficients, if ξ is less than the user-defined threshold ξ_tThen cell swapping is allowed, otherwise swapping is prohibited ξ_tThe larger the improvement in line length (because more cells can be considered for exchange) and the greater the impact on haze variation (because greater dimensional variation is allowed).

The cells are swapped according to fig. 6 to further reduce the line length during detailed layout. However, swapping between a group of standard cells having a large change coefficient of the swapping area is prohibited to keep the fogging variation to a minimum.

The following further explains the related contents related to the present invention.

The mathematical model of the method of the invention is described as follows:

by using a set of vertices V = { V = { (V) }₁，v₂，...，v_mAnd a set of supercedes E = { E = }₁，e₂，...，e_nModeling circuit blocks and connections between blocks, the circuit layout problem can be represented by a hypergraph H = (V, E). Let the coordinate of the center point of block vi be (x)_i，y_i) The layout area is a rectangular sheet, wherein (0, 0) and (W)_R，H_R) The lower left corner and the upper right corner, respectively. The circuit may contain some fixed x and y coordinates and immovable blocks. Ignoring possible block overlaps, the goal of the global layout is to determine the best position of each movable block without violating layout density constraints, thereby minimizing the target cost. Fig. 7 illustrates the meaning of the parameter notation used in the present invention.

One of the most common goals of the layout problem is to minimize the total half cycle length (HPW L), which can be defined as:

（9）

wherein, V_eIs a set of circuit blocks connected to network e.

Fig. 8 shows a layout. FIG. 8(a) divides the layout area into uniform bins and replaces non-overlapping constraints with bin density constraints; fig. 8(b) shows an exact overlap function.

In order to uniformly distribute the circuit blocks, the layout is divided into uniform cells as shown in fig. 8 (a). Let w_bAnd h_bRespectively, the width and height of Bin b. The density in each bin cannot exceed an upper limit to ensure that the circuit blocks are not congested.

The density function of Bin b calculates the overlapping area between Bin b and all circuit blocks, and is defined as

（10）

Wherein O is_xAnd O_yIs the overlap function of bin b and block v in the respective x and y directions.

FIG. 8(b) shows the exact overlap function O_x（b，v）It is a piece-wise linear function and therefore not trivial.

To minimize wire length and density constraints, the analytical layout problem can be expressed as:

（11）

（12）

wherein W_（x，y）Is a function of line length, D_b（x，y）Is a density function, B is a set of bin positions, A_bIs the upper density limit of bin b.

To solve the constraint minimization problem, the density constraint in inequality (12) can be put into the objective function in equation (11) using a quadratic penalty method. Thus, the global layout problem is converted to an unconstrained optimization problem, as follows:

（13）

where λ is an incremental multiplier, e.g., doubled for each iteration. Since a square violation for each mesh density constraint is added to the objective function, the quadratic penalty method can quickly expand the block by adding λ.

FIG. 9 is a model of the deposition energy produced by atomization in an embodiment of the invention, which is a Gaussian distribution centered on the primary electron beam, sufficient to describe the energy distribution caused by the fog effect.

The energy distribution caused by the atomization effect is simulated by a function:

（14）

fogging is a distant effect, so the two PSFs (n-gaussian point spread functions) of a fogging effect are concentrated at nearby points, and there is usually a small difference. As shown in FIG. 10, two PSFsf _fog (r-s ₁) Andf _fog (r-s ₂) Has a center of s₁And s₂。

As shown in fig. 10, the two gaussian distributions of the atomization effect are centered at different points. The absolute difference of the two gaussian distributions is small compared to the maximum of the original gaussian distribution.

When s1 and s2 are close enough, there is little difference, respectively. Given a library of standard cells, we can extract the metal pattern of each standard cell. For modern cell libraries, the width and height of the standard cells are on the order of microns, much smaller than the millimeter range of remote atomization effects. Thus, it can be reasonably assumed that the location on the standard cell where the exposure to create its pattern is located is at the center of the cell. In this way, we can convert each standard cell to the source of the atomization effect, as shown in FIG. 11. Then, we calculate the total pattern area (instead of the exact geometric design of each standard pixel element), modeling the standard pixel with the larger pattern area as a gaussian distribution with higher intensity of the fogging effect, as shown in fig. 11. Standard cells with larger cell areas contribute to greater fogging strength, as in fig. 11 (a); the standard cell having a smaller pattern area has lower fogging strength as shown in fig. 11 (b).

By our modeling of the nebulization source, the standard cell is considered as the nebulization source modeled by a gaussian distribution with different intensities and the same effective range, as shown in fig. 12. The position on the standard cell where exposure is performed to create its pattern is located in the center of the cell.

Fast gaussian transforms are commonly used for such approximationsAnd (6) like. Given the sourceA set of Gaussian distributions centered on, and a set of targetsThe fast gaussian transform sums these distributions as follows:

（15）

where is a user-specified normal number (e.g., 0.25 in this embodiment), and q is_jIs s_jThe weight of (c). s_jThe larger the pattern area of (a), q_jThe higher.

As shown in fig. 13, there are two targets and seven gaussian distributions centered at different points, called sources. By moving the origin and rescaling, the range containing the target and the source can be zoomed into a unit frame with a width of 1 μm. FIG. 13(a) shows the positions of two targets and seven sources in one unit box by moving the origin and rescaling. The fast gaussian transform then evaluates the sum of all gaussian distributions on different targets. Fig. 13(b) shows that on different targets, a fast gaussian transform approximates the sum of all gaussian distributions centered around different sources. The sum is directly calculated by adding all the values of the source contributions. In equation (15), the calculation associates each source with each target. Thus, the time complexity isO(N _t +N _s ) In which N is_tAnd N_sRepresenting the size of T and S, respectively.

Given the high temporal complexity in equation (15) without the need for a gaussian transform, the present invention further employs a Hermite development to reduce computation time. Based on Hermite expansion, the calculation process of the fast Gaussian transformation can be simplified. In the following description, calculations will be made from a particular source.

FIG. 14 is a trellis diagram in the fast Gaussian transform with Hermite expansion in this embodiment. In fig. 14(a), the range containing the source and the target in the unit frame is divided into smaller grids, and the width of the grid can be determined by wg = 1/Ng. In fig. 14(b), examples of the positions of the target, source, and extension points in a unit frame having four grids are shown. Each source and target belongs to a grid and each grid has an extension point.

First, the Hermite function is defined as:

（16）

for all t ∈ R, where D = D/dt and hn (t) are conventional Hermite polynomials.

（17）

By using the Hermite function and the modified generation function for the Hermite polynomial, as shown in equations (16) and (17), in the Hermite expansionBy s₀Centering can be calculated by the following equation:

（18）

the above equation gives how to use the Hermite development to calculate the value of a single Gaussian distribution source s at the target t. To summarize the Hermite expansion of the sets S and T, all elements in S are summed. In the following equation (19), at all targetst _i ∈TUpper evaluation of all sourcess _j ∈SSum of (a):

（19）

（20）

by using the Hermite expansion in equation (19), the fast Gaussian transform greatly reduces the time complexity, just toO(N _t +N _s )。

To calculate equation (8) by a fast gaussian transform with Hermite expansion, assume that the positions of the source S and target T are located in one unit box by moving the origin and rescaling. Then, as shown in fig. 14(a), the unit frame is divided into smaller grids with N on one side_gGrid, where N_gIs a minimum integer satisfying the following conditions。

Thus, can pass through w_g= 1/N_gThe width of the grid is determined. Now, each source and target belongs to a grid, and each grid has an extension point, as shown in FIG. 14 (b). For a source or target, a Hermite extension point is an extension point that is located in the same grid as the source or target.

Fig. 15 shows how the sum is calculated using the fast gaussian transform of the Hermite expansion. In FIG. 15(a), in different gridss _O1、s _O2、s _O3Ands _O4the Hermite unfolding of (a) causes the source not to be directly connected to the target. For each mesh we have one extension point. The fogging effect of each extension point can be evaluated from sources in the same grid and then can be reused. As a result, when we calculate the atomization effect of another target as shown in fig. 15(b), it is not necessary to recalculate the evaluation. Further, A in the formula (14)_nThe calculation of (c) may be used for each source. We need only evaluate each source from the extension point.

Thus, the time complexity of using equation (13) isO(N _t +N _s )。

The invention considers the analytic layout method of the electron beam atomization effect, solves the electron beam atomization effect earlier through analyzing the layout, so as to reduce the time-consuming compensation of the atomization effect at the post-layout stage, and can realize better design fusion with the EB L manufacturing process.

The above are preferred embodiments of the present invention, and all changes made according to the technical scheme of the present invention that produce functional effects do not exceed the scope of the technical scheme of the present invention belong to the protection scope of the present invention.