CN111539167B - Layout method of ultra-large-scale integrated circuit considering atomization and proximity effect - Google Patents

Layout method of ultra-large-scale integrated circuit considering atomization and proximity effect Download PDF

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CN111539167B
CN111539167B CN202010329234.6A CN202010329234A CN111539167B CN 111539167 B CN111539167 B CN 111539167B CN 202010329234 A CN202010329234 A CN 202010329234A CN 111539167 B CN111539167 B CN 111539167B
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atomization
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CN111539167A (en
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陈建利
林智峰
黄志鹏
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Fuzhou Lixin Technology Co ltd
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    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F30/00—Computer-aided design [CAD]
    • G06F30/30—Circuit design
    • G06F30/32—Circuit design at the digital level
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    • G06F30/3308—Design verification, e.g. functional simulation or model checking using simulation
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F30/00—Computer-aided design [CAD]
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    • G06F30/39—Circuit design at the physical level
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Abstract

The invention relates to a layout method of a very large scale integrated circuit (VLSI) considering atomization and proximity effect, which comprises the following steps: (1) Establishing an energy distribution model for atomization and proximity effect in the electron beam lithography EBL and describing the change of the energy distribution model; (2) Optimizing a model and a function in the global layout and determining a smooth target function; (3) Introducing new variables solves the unconstrained minimization problem into a separable minimization problem with linear constraints, CSMP; (4) Determining a neighbor set ADMM algorithm and an iterative formula for the separable minimization problem; (5) solving two sub-problems in the neighbor set iteration; and (6) carrying out convergence analysis on the adjacent group ADMM algorithm. The method is beneficial to reducing adverse effects caused by atomization and proximity effects and improving layout efficiency.

Description

Layout method of ultra-large-scale integrated circuit considering atomization and proximity effect
Technical Field
The invention belongs to the technical field of design of a very large scale integrated circuit, and particularly relates to a layout method of the very large scale integrated circuit considering atomization and proximity effect.
Background
Since EBL can print fine patterns, it is used for sub-22 nm process nodes and above. The electron gun directly emits electrons through a set of lenses and apertures to pattern the wafer. When the primary electron beam emitted by the electron gun hits the resistor and the substrate, the electrons may scatter. The scattered electrons produce backscattered electrons that may hit the bottom of the objective lens. Thus, the impact may generate next generation electrons, so-called re-scattered electrons. These scattered, backscattered and re-scattered electrons may cause unwanted exposure, leading to proximity effects and fogging effects.
The proximity effect is produced by scattered and backscattered electrons. More specifically, the scattered electrons (forward scattered electrons) cause a forward proximity effect, i.e., small angular deflection of electrons as they enter the photoresist and substrate. In contrast, backscattered electrons (called backscattered electrons) produce a backward proximity effect, which is usually deflected at large angles. In addition, the backscattered electrons may leave the resistor, strike the bottom of the objective lens, and bounce back into the resistor again, thereby generating backscattered electrons. The re-scattered electrons are dispersed a few millimeters away from the main exposure point. These re-scattered electrons can cause overexposure, resulting in a change in the layout pattern, known as a fogging effect.
Some published papers address the critical fogging and proximity effects, but most of them deal with these two effects at the manufacturing process or post-layout stage. Shimomura et al address the fogging effect by adding a scattered electron absorbing plate. Hudek and Bayer use a series of experimental data to determine the optimal parameters for the proximity effect, and a software tool (called "PROX-In") was developed to determine an optimal control point spread function to correct for the proximity effect. However, post-layout corrections are very time consuming. In addition to long run times, to ensure accuracy in critical dimensions, a large amount of data in bitmaps is required.
To address the fogging effect earlier, huang and Chang proposed the first chip-level layout algorithm to minimize the fogging variation on the chip. The basic idea is to arrange the intervals under the guidance of the atomization model to minimize the atomization variation during the arrangement. The atomization model is an accurate evaluation scheme for estimating the atomization effect by using a fast gaussian transformation. The experimental results show that the methods achieve high resolution. However, this work does not take into account the critical proximity effect. Another problem is that the run time is 2.44 times longer than for the layout method without taking into account the fogging effect. Therefore, an efficient algorithm needs to be designed to handle both critical fogging effects and proximity effects.
Disclosure of Invention
The invention aims to provide a layout method of a very large scale integrated circuit considering the atomization and proximity effects, which is beneficial to reducing the adverse effects caused by the atomization and proximity effects and improving the layout efficiency.
In order to achieve the purpose, the invention adopts the technical scheme that: a method for placement of a very large scale integrated circuit (vlsi) that accounts for blooming and proximity effects, comprising the steps of:
(1) Establishing an energy distribution model for atomization and proximity effect in the electron beam lithography EBL and describing the change of the energy distribution model;
(2) Optimizing models and functions in the global layout and determining a smooth target function F (x, y);
(3) Introduction of new variables minimizes the problem without constraints
Figure BDA0002464351910000021
To the separable minimization problem CSMP with linear constraints;
(4) Determining a neighbor set ADMM algorithm and an iterative formula for the separable minimization problem;
(5) Solving two sub-problems in the neighbor set iteration;
(6) Convergence analysis was performed on the neighborhood set ADMM algorithm.
Further, in the step (1), in the EBL, the fogging effect and the proximity effect are modeled as a gaussian distribution and a double gaussian function, respectively, f fog (d) Fogging effect and proximity effect f pro (d) The mathematical models of (a) are respectively:
Figure BDA0002464351910000022
Figure BDA0002464351910000023
wherein the content of the first and second substances,
Figure BDA0002464351910000024
is the distance from the point of incidence, beta F 、β f And beta b The atomization effect action range, the forward proximity effect and the backward proximity effect; v. of F Is a parameter related to the weight of the fogging effect; η is the ratio of the backscattered energy to the forward energy;
for the fogging effect, an evaluation point t is given i And a set of sources affected by the point
Figure BDA0002464351910000025
The evaluation scheme was calculated by fast gaussian transformation as:
Figure BDA0002464351910000026
wherein q is j Is a j Weight of (d), δ a Is a normal number; also, the forward proximity effect and the backward proximity effect are calculated by:
Figure BDA0002464351910000031
Figure BDA0002464351910000032
in the formula (I), the compound is shown in the specification,
Figure BDA00024643519100000310
and
Figure BDA00024643519100000311
are the set of sources of occurrence caused by forward and backward directions, respectively, at the evaluation point t i Has an approximate effect;
the above evaluation scheme was used to estimate the FPE: let the evaluation point be
Figure BDA0002464351910000033
They are evenly distributed over the entire layout and the variation in the fogging effect is calculated by:
Figure BDA0002464351910000034
the change in proximity effect is calculated by:
Figure BDA0002464351910000035
wherein, x and y is the sum of x and of the generation source y And (4) coordinates.
Further, in the step (2), in the global layout, the line length and the density function are not smooth, and a logarithm model and a line length model are used
Figure BDA0002464351910000036
And bell function
Figure BDA0002464351910000037
Approximate total semi-perimeter line length and smooth density function respectively; the global layout problem is formulated as a smooth constraint minimization problem, as follows:
Figure BDA0002464351910000038
Figure BDA0002464351910000039
wherein, M b Is the maximum allowable area of the mobile unit in the binb;
to optimize variation in atomization and proximity effects while maintaining good layout line length and density, the objective function of the layout is defined as:
Figure BDA0002464351910000041
wherein λ is 1 、λ 2 、λ 3 And λ 4 As weights, the weights are continuously updated to search for the optimal positions of all circuit units in the iterative process;
by introducing two new variables (g, h), the problem of unconstrained minimization is solved
Figure BDA0002464351910000042
Further formulated as a separable minimization problem with linear constraints as follows:
Figure BDA0002464351910000043
wherein the content of the first and second substances,
Figure BDA0002464351910000044
representing line length and density constraints, theta 2 (g，h)＝λ 3 S f (g，h)+λ 4 S p (q, h) represents atomization and proximity change; this formula separates the atomization and proximity variations from line length and density constraints.
Further, in the step (3), the lagrangian function related to the minimization problem of equation (5) is:
Figure BDA0002464351910000045
wherein, the augmented Lagrange multiplier method is as follows:
Figure BDA0002464351910000046
the minimization problem of equation (5) is solved with the near end set ADMM: given a
Figure BDA0002464351910000047
The near-end set of minimization problems ADMM of equation (5) generates the next iteration based on the enhanced Lagrangian function (6) by the following equation
Figure BDA0002464351910000048
Figure BDA0002464351910000049
Figure BDA00024643519100000410
Figure BDA0002464351910000051
In the formula (I), the compound is shown in the specification,
Figure BDA0002464351910000052
and
Figure BDA0002464351910000053
is an approximation term; the method divides original variables into two types of (x, y) and (g, h), and then adopts near-end ADMM to solve two solving problems; in each iteration of the near-end set ADMM, there are two sub-problems to be solved, sub-problem of equation (7) and sub-problem of equation (8), respectively;
for the sub-problem of equation (7), the steepest descent method is used to minimize line length and density constraint violations; according to the first order requirement for optimality, it follows from equation (7):
Figure BDA0002464351910000054
and
Figure BDA0002464351910000055
this results in the steepest descent step of the form:
Figure BDA0002464351910000056
Figure BDA0002464351910000057
order:
Figure BDA0002464351910000058
and
Figure BDA0002464351910000059
the subproblems of formula (8) correspond to min φ (g) and
Figure BDA00024643519100000510
a first order optimality condition of
Figure BDA00024643519100000511
And
Figure BDA00024643519100000512
the gradient is calculated in the same manner
Figure BDA00024643519100000513
And
Figure BDA00024643519100000514
firstly, there are:
Figure BDA0002464351910000061
the calculation of equation (12), namely:
Figure BDA0002464351910000062
since both the fogging and the proximity effect are the sum of the associated gaussian distribution functions, S is obtained by the same method f (g，h t ) And S p (g，h t ) Partial derivatives of (d); s f (g，h t ) The partial derivative of (a) is described as:
Figure BDA0002464351910000063
the average is divided into two parts as above: one is the average of the fraction that is mainly affected by the fogging effect and the other is the average of the fraction that is mainly unaffected by the fogging effect; since the average value is not mainly affected by the atomization effect, the average value is regarded as a constant and is expressed as E, namely, a surrounding grid is selected to calculate an average effect change value; then, there are:
Figure BDA0002464351910000064
Figure BDA0002464351910000065
wherein, the first and the second end of the pipe are connected with each other,
Figure BDA0002464351910000066
is the evaluation point t i The coordinates of (a);
will be evaluated at point t i The window of influence is denoted Γ; the exponential function is approximated using an integration method to yield:
Figure BDA0002464351910000071
wherein w Γ 、h Γ And R Γ The width, height and boundaries of the window Γ, respectively; t is t i Is an evaluation point, a i Located in its area of influence;
to estimate in equation (15)
Figure BDA0002464351910000072
The value of (A) is determined by the boundary determining principle, and comprises the following steps:
Figure BDA0002464351910000073
substituting equations (16), (17) and (18) into (15) yields:
Figure BDA0002464351910000074
Figure BDA0002464351910000075
and also,
Figure BDA0002464351910000076
following the procedures of equations (14) to (20), the following are calculated:
Figure BDA0002464351910000077
combining formulae (12), (13), (19) and (21), and
Figure BDA0002464351910000078
obtaining:
Figure BDA0002464351910000079
namely:
Figure BDA00024643519100000710
also, the following were obtained:
Figure BDA0002464351910000081
compared with the prior art, the invention has the following beneficial effects: 1) The present invention represents the global layout problem that reduces the effects of fogging and proximity effects as a separable linear constraint minimization problem, which greatly reduces the complexity of the optimization problem. 2. The invention provides a new adjacent group ADMM algorithm to solve the separable minimization problem. To take advantage of the separable-recurrence formula, two sub-problems are solved at a lower computational cost in each iteration of the method. 3) The invention guarantees the global convergence to the first-order critical point under two reliable assumptions, and provides theoretical guarantee for the quality of the generated solution. 4. The invention can effectively solve the VLSI layout problem considering atomization and proximity effect. Experimental results show that the method is effective and efficient for solving the problems. Compared with the most advanced algorithm at present, the method not only reduces the influence caused by 13.4% of atomization effect and 21.4% of proximity effect, but also accelerates by 1.65 times.
Drawings
FIG. 1 is a flow chart of a method implementation of an embodiment of the present invention.
Detailed Description
The invention is further described with reference to the following figures and specific embodiments.
The invention provides a layout method of a very large scale integrated circuit considering atomization and proximity effects, which solves the VLSI layout problem considering the atomization and proximity effects, further reduces adverse effects brought by the atomization and proximity effects and improves layout efficiency. The basic idea of this approach is to first formulate the global layout problem as a separable linear constraint minimization problem. According to the energy model of the atomization and the proximity effect, the targets in the energy model are solved one by one in an alternating mode, the proximity group is written out and solved, the complexity of the optimization problem is greatly reduced, and meanwhile, the calculation cost is reduced, so that the influence caused by the atomization and the proximity effect is effectively reduced. In each iteration of the method, two sub-problems are solved with low computational cost. The first sub-problem (mainly related to line length and density) is solved by the steepest descent algorithm of wireless search, while the second sub-problem (mainly related to atomization and proximity effects) is solved approximately by some approximation technique with analytical solutions. The method specifically comprises the following steps:
(1) The energy distribution is modeled and its variation is described for the fogging and proximity effects in Electron Beam Lithography (EBL).
(2) Models and functions in the global layout are optimized, and a smooth objective function F (x, y) is determined.
(3) Introduction of newProblem of variable unconstrained minimization
Figure BDA0002464351910000082
A separable minimization problem (CSMP) with linear constraints.
(4) A neighbor set ADMM algorithm and an iterative formula that can separate the minimization problem are determined.
(5) Two sub-problems in the neighbor set iteration are solved.
(6) Convergence analysis was performed on the neighborhood group ADMM algorithm.
Referring to fig. 1, fig. 1 is a flow chart of the method of the present invention. The mathematical model of the method is described as follows:
in EBL, the fogging effect and the proximity effect are modeled as a Gaussian distribution and a double Gaussian function, respectively, f fog (d) Fogging effect and proximity effect f pro (d) The mathematical models of (a) are respectively:
Figure BDA0002464351910000091
Figure BDA0002464351910000092
wherein the content of the first and second substances,
Figure BDA0002464351910000093
is the distance from the point of incidence, beta F 、β f And beta b Are the fogging effect coverage, forward proximity effect and backward proximity effect. v. of F Is a parameter related to the weight of the fogging effect. η is the ratio of the backscattered energy to the forward energy. For the smallest size correctable, β F Is 20000 mu m, beta F Is 0.06 μm, beta b And 30 μm. The present invention proposes an effective and accurate evaluation scheme to estimate the fogging effect by fast gaussian transformation. In this scheme, each standard cell is considered as a source of occurrence, and evaluation points uniformly distributed over the entire layout are selected to estimate the variation of both effects (two adjacent evaluation points)The spacing therebetween is a constant, e.g., 5 μm).
For the fogging effect, an evaluation point t is given i And a set of sources affected by the point
Figure BDA0002464351910000094
The evaluation scheme is calculated by fast gaussian transformation as:
Figure BDA0002464351910000095
wherein q is j Is a j Weight of δ a Is a normal number. Also, the forward proximity effect and the backward proximity effect are calculated by the following equation:
Figure BDA0002464351910000096
Figure BDA0002464351910000097
in the formula (I), the compound is shown in the specification,
Figure BDA0002464351910000098
and
Figure BDA0002464351910000099
are the set of sources of occurrence caused by forward and backward directions, respectively, at the evaluation point t i Has an approximate effect.
In the present invention, the above efficient and accurate evaluation scheme is used to estimate the FPE: let the evaluation point be
Figure BDA0002464351910000101
Figure BDA0002464351910000102
They are evenly distributed over the entire layout and the variation in the fogging effect is calculated by:
Figure BDA0002464351910000103
the change in proximity effect is calculated by:
Figure BDA0002464351910000104
where x and y are the x and y coordinates of the generating source, such as the location of a standard cell.
In a global layout, both line length and density functions are not smooth, using log and line length models
Figure BDA0002464351910000105
And bell function
Figure BDA0002464351910000106
Approximate total half perimeter line length (HPWL) and smooth density functions, respectively. The global layout problem is formulated as a smooth constraint minimization problem, as follows:
Figure BDA0002464351910000107
Figure BDA0002464351910000108
wherein M is b The maximum allowable area of the movable unit in the bin b.
To optimize variation in atomization and proximity effects while maintaining good layout line length and density, the objective function of the layout is defined as:
Figure BDA0002464351910000109
wherein λ is 1 、λ 2 、λ 3 And λ 4 For the weights, the weights are continuously updated to search all of them in an iterative processThe optimal position of the circuit unit.
By introducing two new variables (g, h), the unconstrained minimization problem is solved
Figure BDA00024643519100001010
Further formulated as a separable minimization problem with linear constraints, as follows:
Figure BDA0002464351910000111
wherein the content of the first and second substances,
Figure BDA0002464351910000112
denotes line length and density constraints, θ 2 (g，h)＝λ 3 S f (g，h)+λ 4 S p (q, h) represents atomization and proximity change. This formula separates the atomization and proximity variation from line length and density constraints. In particular, the two objective functions can be optimized one by one in an alternating manner, which can significantly speed up the optimization process.
The ADMM algorithm is an efficient algorithm for the optimization problem of separable structures, especially for separable optimization problems with linear constraints. The present invention further proposes a novel near-end set of ADMMs to solve the minimization problem of equation (5) and to prove that the iterative sequence generated by the method converges to the first-order critical point of the minimization problem of equation (5).
The "Proximal group ADMM" section in FIG. 1 is specifically as follows:
in the step (3), the lagrangian function related to the minimization problem of equation (5) is:
Figure BDA0002464351910000113
wherein, the augmented Lagrange multiplier method is as follows:
Figure BDA0002464351910000114
the near end set ADMM is used to solve the minimization problem of equation (5): given the
Figure BDA0002464351910000115
The near-end set of minimization problems ADMM of equation (5) generates the next iteration based on the enhanced Lagrangian function (6) by the following equation
Figure BDA0002464351910000116
Figure BDA0002464351910000117
Figure BDA0002464351910000118
Figure BDA0002464351910000119
In the formula (I), the compound is shown in the specification,
Figure BDA0002464351910000121
and
Figure BDA0002464351910000122
is an approximation term. The method divides original variables into two types of (x, y) and (g, h), and then adopts near-end ADMM to solve two solving problems. In each iteration of the near-end set ADMM, there are two sub-problems to be solved, sub-problem of equation (7) and sub-problem of equation (8).
"wire & diversity Subproblem (7) in FIG. 1; solve (10) (11) by steeest facility method ", the concrete way is as follows:
for the sub-problem of equation (7), the steepest descent method is used to minimize line length and density constraint violations. According to the first order requirement for optimality, it follows from equation (7):
Figure BDA0002464351910000123
and
Figure BDA0002464351910000124
this results in the steepest descent step of the form:
Figure BDA0002464351910000125
Figure BDA0002464351910000126
"Fogging & promotion effects Subproblem (8) in FIG. 1; the Analytical solution of via (22) - (23) "section, in a specific manner is as follows:
order:
Figure BDA0002464351910000127
and
Figure BDA0002464351910000128
the subproblems of formula (8) correspond to min φ (g) and
Figure BDA0002464351910000139
is a first order optimality condition of
Figure BDA0002464351910000131
And
Figure BDA0002464351910000132
the gradient is calculated in the same manner
Figure BDA0002464351910000133
And
Figure BDA0002464351910000134
firstly, there are:
Figure BDA0002464351910000135
the calculation of equation (12), namely:
Figure BDA0002464351910000136
since both the fogging and the proximity effect are the sum of the associated gaussian distribution functions, S is obtained by the same method f (g，h t ) And S p (g，h t ) The partial derivative of (c). S f (g，h t ) The partial derivative of (a) is described as:
Figure BDA0002464351910000137
the average is divided into two parts as above: one is the average of the fraction that is mainly affected by the fogging effect and the other is the average of the fraction that is mainly unaffected by the fogging effect. Since the mean value is largely unaffected by the fogging effect, it is considered constant and denoted as E, i.e. the surrounding grid is selected to calculate the mean effect variation value. Then, there are:
Figure BDA0002464351910000138
Figure BDA0002464351910000141
which is composed of
Figure BDA0002464351910000142
Is the evaluation point t i The coordinates of (a). The last equation holds because
Figure BDA0002464351910000143
Independent of g i 。
Will be evaluated at the point t i The window of influence is denoted Γ. The exponential function is approximated using an integration method to yield:
Figure BDA0002464351910000144
wherein, w Γ 、h Γ And R Γ Respectively the width, height and boundaries of the window Γ. t is t i Is an evaluation point, a i Located in its area of influence.
To estimate in equation (15)
Figure BDA0002464351910000145
The value of (A) is determined by the boundary determining principle, and comprises the following steps:
Figure BDA0002464351910000146
substituting formulae (16), (17), and (18) into (15) yields:
Figure BDA0002464351910000147
Figure BDA0002464351910000148
and also,
Figure BDA0002464351910000149
following the procedures of equations (14) to (20), the following are calculated:
Figure BDA0002464351910000151
here Q can be derived in a similar way. Combining formulae (12), (13), (19) and (21), and
Figure BDA0002464351910000152
obtaining:
Figure BDA0002464351910000153
namely:
Figure BDA0002464351910000154
also, the following were obtained:
Figure BDA0002464351910000155
"legacy & modified placement" in FIG. 1; the Placement result section, in detail, follows:
for each illegal unit, the nearest and legal location is found to place the unit.
In the step (5), the critical point of the problem (5) satisfies the following equation by a first-order optimal condition:
Figure BDA0002464351910000156
by the neighbor set ADMM algorithm (alternating direction multiplier method) proposed earlier, the equation is obtained:
Figure BDA0002464351910000157
Figure BDA0002464351910000158
Figure BDA0002464351910000159
Figure BDA00024643519100001510
Figure BDA00024643519100001511
Figure BDA0002464351910000161
the following assumptions are made: 1) L is A Underlying within the chip area under consideration. 2)
Figure BDA0002464351910000162
And
Figure BDA0002464351910000163
is continuous over (x, y) by Lipschitz, and
Figure BDA0002464351910000164
and
Figure BDA0002464351910000165
is continuous over (g, h) by Lipschitz. Represent
Figure BDA0002464351910000166
Then we have the following reasoning:
lesion 1) hypothetical sequence z t Is generated by the proposed method. Then, if the penalty parameters β > 0 and γ > 0 are sufficiently large, the function value
Figure BDA0002464351910000167
In a sequence of (2), whereinAs long as L A With the lower bound, we can get the equation:
Figure BDA0002464351910000168
and
Figure BDA0002464351910000169
theorem 1) Generation of z from neighboring sets of ADMMs t It converges to the critical point of the problem.
And (3) proving that: by the introduction of 1, the method has the advantages that,
Figure BDA00024643519100001610
H 0 is a place of a bounded set. Thus, { z t The are converged, with at least one converged subsequence. Suppose that
Figure BDA00024643519100001611
Is a convergent subsequence, and
Figure BDA00024643519100001612
t i →+∞。
the following equations (25) to (30) and (31) to (32) give:
Figure BDA00024643519100001613
and
Figure BDA00024643519100001614
in the same way, the following can be obtained:
Figure BDA0002464351910000171
Figure BDA0002464351910000172
and
Figure BDA0002464351910000173
note that for any t, 1,t or more i Belonging to natural numbers, all the equations are true, and the two sides of the equation (25) - (30) are simultaneously paired with t i Take a limit, and, note when t i When it approaches 0
Figure BDA0002464351910000174
Toward z, we have a limit point z that satisfies condition 24, so it is the dwell point for problem (5).
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.

Claims (1)

1. A method for placement of a very large scale integrated circuit (vlsi) that accounts for fogging and proximity effects, comprising the steps of:
(1) Establishing an energy distribution model for atomization and proximity effects in the electron beam lithography EBL and describing the change of the energy distribution model;
(2) Optimizing models and functions in the global layout and determining a smooth target function F (x, y);
(3) Introduction of new variables minimizes the problem without constraints
Figure FDA0003879766370000011
To the separable minimization problem with linear constraints, CSMP;
(4) Determining a neighbor set ADMM algorithm and an iterative formula for the separable minimization problem;
(5) Solving two sub-problems in the neighbor set iteration;
(6) Carrying out convergence analysis on the adjacent group ADMM algorithm;
in the step (1), in EBL, the atomization effect and the proximity effect are respectively modeled as Gaussian distribution and double Gaussian function, F fog (d) Fogging effect and proximity effect f pro (d) The mathematical models of (a) are respectively:
Figure FDA0003879766370000012
Figure FDA0003879766370000013
wherein the content of the first and second substances,
Figure FDA0003879766370000014
is the distance from the point of incidence, beta F 、β f And beta b The atomization effect action range, the forward proximity effect and the backward proximity effect; upsilon is F Is a parameter related to the weight of the fogging effect; η is the ratio of the backscattered energy to the forward energy;
for the fogging effect, an evaluation point t is given i And a set of sources affected by the point
Figure FDA0003879766370000015
The evaluation scheme is calculated by fast gaussian transformation as:
Figure FDA0003879766370000016
wherein q is j Is a j Weight of δ a Is a normal number; also, the forward proximity effect and the backward proximity effect are calculated by:
Figure FDA0003879766370000017
Figure FDA0003879766370000021
in the formula (I), the compound is shown in the specification,
Figure FDA0003879766370000022
and
Figure FDA0003879766370000023
are a set of sources of occurrence caused by the forward and backward directions, respectively, at an evaluation point t i Has an approximate effect;
the above evaluation scheme was used to estimate the FPE: let the evaluation point be
Figure FDA0003879766370000024
They are evenly distributed over the entire layout and the variation in the fogging effect is calculated by:
Figure FDA0003879766370000025
the change in proximity effect is calculated by:
Figure FDA0003879766370000026
wherein x and y are the x and y coordinates of the generating source;
in the step (2), in the global layout, the line length and the density function are not smooth, and a logarithm model and a line length model are used
Figure FDA0003879766370000027
And bell function
Figure FDA0003879766370000028
Respectively approximating a total semi-perimeter line length and a smooth density function; the global layout problem is formulated as a smooth constraint minimization problem, as follows:
Figure FDA0003879766370000029
Figure FDA00038797663700000210
wherein, M b Is the maximum allowable area of the movable unit in bin b;
to optimize the variation in atomization and proximity effects while maintaining good layout line length and density, the objective function of the layout is defined as:
Figure FDA00038797663700000211
wherein λ is 1 、λ 2 、λ 3 And λ 4 The weights are continuously updated to search the optimal positions of all circuit units in the iterative process;
by introducing two new variables (g, h), the problem of unconstrained minimization is solved
Figure FDA0003879766370000031
Further formulated as a separable minimization problem with linear constraints as follows:
Figure FDA0003879766370000032
wherein the content of the first and second substances,
Figure FDA0003879766370000033
denotes line length and density constraints, θ 2 (g，h)＝λ 3 S f (g，h)+λ 4 S p (q, h) represents atomization and proximity change; this formula separates the atomization and proximity variations from line length and density constraints;
in the step (3), the lagrangian function related to the minimization problem of equation (5) is:
Figure FDA0003879766370000034
wherein, the augmented Lagrange multiplier method is as follows:
Figure FDA0003879766370000035
the minimization problem of equation (5) is solved with the near end set ADMM: given the
Figure FDA0003879766370000036
The near-end set of minimization problems ADMM of equation (5) generates the next iteration based on the enhanced Lagrangian function (6) by the following equation
Figure FDA0003879766370000037
Figure FDA0003879766370000038
Figure FDA0003879766370000039
Figure FDA00038797663700000310
In the formula (I), the compound is shown in the specification,
Figure FDA00038797663700000311
and
Figure FDA00038797663700000312
is an approximation term; the method divides original variables into two types of (x, y) and (g, h), and then adopts near-end ADMM to solve two solving problems; in each iteration of the near-end set ADMM, there are two sub-problems to be solved, sub-problem of equation (7) and sub-problem of equation (8), respectively;
for the sub-problem of equation (7), the steepest descent method is used to minimize line length and density constraint violations; according to the first order requirement for optimality, it follows from equation (7):
Figure FDA0003879766370000041
and
Figure FDA0003879766370000042
this results in the steepest descent step of the form:
Figure FDA0003879766370000043
Figure FDA0003879766370000044
order:
Figure FDA0003879766370000045
and
Figure FDA0003879766370000046
the subproblems of formula (8) correspond to min φ (g) and
Figure FDA00038797663700000413
is a first order optimality condition of
Figure FDA0003879766370000047
And
Figure FDA0003879766370000048
the gradient is calculated in the same manner
Figure FDA0003879766370000049
And
Figure FDA00038797663700000410
first, there are:
Figure FDA00038797663700000411
the calculation of equation (12), namely:
Figure FDA00038797663700000412
since both the fogging and the proximity effect are the sum of the associated gaussian distribution functions, S is obtained by the same method f (g，h t ) And S p (g，h t ) Partial derivatives of (d); s f (g，h t ) The partial derivatives of (c) are described as:
Figure FDA0003879766370000051
the average is divided into two parts as above: one is the average of the fraction that is mainly affected by the fogging effect and the other is the average of the fraction that is mainly unaffected by the fogging effect; since the average value is not mainly affected by the atomization effect, the average value is regarded as a constant and is expressed as E, namely, a surrounding grid is selected to calculate an average effect change value; then, there are:
Figure FDA0003879766370000052
Figure FDA0003879766370000053
wherein the content of the first and second substances,
Figure FDA0003879766370000054
is the evaluation point t i The coordinates of (a);
will be evaluated at the point t i The window of influence is denoted Γ; approximating the exponential function using an integral method yields:
Figure FDA0003879766370000055
wherein, w Γ 、h Γ And R Γ The width, height and boundaries of the window Γ, respectively; t is t i Is an evaluation point, a i Located in its area of influence;
estimating in equation (15) using the principle of boundary determination
Figure FDA0003879766370000061
The values of (c) are:
Figure FDA0003879766370000062
substituting equations (16), (17) and (18) into (15) yields:
Figure FDA0003879766370000063
Figure FDA0003879766370000064
and the number of the first and second electrodes,
Figure FDA0003879766370000065
following the procedures of equations (14) to (20), the following are calculated:
Figure FDA0003879766370000066
combining formulae (12), (13), (19) and (21), and reacting
Figure FDA0003879766370000067
Obtaining:
Figure FDA0003879766370000068
namely:
Figure FDA0003879766370000069
also, there are obtained:
Figure FDA00038797663700000610
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CN106980730A (en) * 2017-03-31 2017-07-25 福州大学 VLSI standard cell placement methods based on direct solution technology
CN107526860A (en) * 2017-03-31 2017-12-29 福州大学 VLSI standard cell placement methods based on electric field energy modeling technique
CN108763777A (en) * 2018-05-30 2018-11-06 福州大学 VLSI global wiring method for establishing model based on Poisson's equation explicit solution

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