JP2009252140A - Cell delay library and method of creating the same, and delay analysis method - Google Patents

Abstract

<P>PROBLEM TO BE SOLVED: To enhance precision of a delay model that a cell delay library provides. <P>SOLUTION: A method of creating a cell delay library for statistical STA is provided. The creation method comprises steps of: (A) providing a delay model (linear function) which gives cell delay values depending on the variation in a model parameter of a transistor which constitutes a cell; (B) performing a circuit simulation using a plurality of simulation points of the model parameter to calculate a plurality of cell delay values that correspond to the plurality of respective simulation points; (C) determining a constant term and a sensitivity coefficient of the delay model by moment-matching the plurality of cell delay values with the delay model; and (D) creating the cell delay library which provides the constant term and the sensitivity coefficient. <P>COPYRIGHT: (C)2010,JPO&INPIT

Description

  The present invention relates to statistical STAs in the design of semiconductor integrated circuits. In particular, the present invention relates to a cell delay library used in statistical STAs, a method for creating the same, a delay analysis method using the same, and a method for designing and manufacturing a semiconductor integrated circuit using the cell delay library.

  In recent years, with the miniaturization and large scale of LSI, manufacturing variation (process variation) has increased. Variations in device manufacturing lead to variations in delay values, which greatly affects yield and the like. Therefore, it is important to estimate and analyze the delay value in consideration of manufacturing variations in the design stage. As such a delay analysis technique, static timing analysis (STA) is generally known. Among them, an STA whose analysis accuracy is improved by statistically considering manufacturing variation is “Statistical Static Timing Analysis (STA)” (Patent Document 1, Patent Document 2, Patent Document 3, Non-Patent Document). 1).

  In the statistical STA, the variation of the delay value according to the manufacturing variation of the element is estimated. For example, the delay value D of a certain logic cell varies depending on variations in the model parameters (gate length, gate width, mobility, etc.) of the transistors constituting the logic cell. It is assumed that the nominal value (design value) of a certain model parameter X is Xnom, and the variation from the nominal value Xnom is given by a random variable Xvar (X = Xnom + Xvar). For other model parameters, nominal values and random variables are defined. At this time, the delay value D of the cell is approximately given by a linear function of each variation variable as expressed by the following equation (1).

Formula (1):
D = D0 + α · Xvar + ...

  In Expression (1), D0 is a constant term, and is typically a cell delay value (nominal delay value Dnom) when all model parameters are nominal values. α · Xvar is a variation amount of the cell delay value when the model parameter X varies from the nominal value Xnom by Xvar. That is, in response to the variation Xvar of the model parameter X, the cell delay value D varies by α · Xvar. α is called a “sensitivity coefficient”. The same applies to variations in other model parameters.

  FIG. 1 conceptually shows a linear function expressed by the above formula (1). Here, for simplicity, only one model parameter X is considered. In FIG. 1, the horizontal axis indicates the variation Xvar (or model parameter X), and the vertical axis indicates the cell delay value D. A point P1 in FIG. 1 is a nominal point indicating a nominal condition, and a point P2 is a variation point indicating a certain variation condition. A straight line connecting the nominal point P1 and the variation point P2 represents a linear linear function given by the above equation (1). The slope of the straight line is the sensitivity coefficient α.

  A method for determining the sensitivity coefficient α will be described with reference to FIG. First, the nominal delay value Dnom of the cell is calculated through SPICE simulation. At this time, all model parameters are set to nominal values. Next, a cell delay value D (Xnom + ΔX) when the model parameter X changes from the nominal value Xnom by a predetermined change amount ΔX is calculated through SPICE simulation. At this time, the model parameter X is set to Xnom + ΔX, and the other model parameters are set to nominal values. Then, by using the nominal delay value Dnom, the cell delay value D (Xnom + ΔX), and the fluctuation amount ΔX, the sensitivity coefficient α is calculated by the following equation (2).

Formula (2):
α = (D (Xnom + ΔX) −Dnom) / ΔX

  Thus, the sensitivity coefficient α is simply determined as a slope of a straight line connecting the nominal point P1 and the variation point P2 by linear approximation. Sensitivity coefficients for other model parameters are determined similarly. When all the sensitivity coefficients are determined, the linear function of the above equation (1) is determined. This linear function is hereinafter referred to as a “delay model”. However, the delay model is only approximate. The actual cell delay value D responds to the variation, for example, as shown by the broken curve in FIG. The delay model is obtained by a simple linear approximation to the actual response.

  The process of determining a delay model for a certain cell is called “characterizing”. The delay model obtained as a result of cell characterization is provided as a library for use in statistical STAs. More specifically, for each cell, a nominal delay value Dnom and a sensitivity coefficient (such as α) are provided as a library. Such a library is hereinafter referred to as a “cell delay library”. In the statistical STA, by using this cell delay library, a delay analysis that statistically considers manufacturing variations is performed.

JP 2004-252831 A JP 2007-183932 A JP 2007-233550 A Statistical Static Timing Analysis Technology, FUJITSU. 58, 3, p. 267-273 (05, 2007)

  In the conventional cell delay library, the accuracy of the delay model is not always sufficient. There are mainly two reasons for this.

  The first reason is that, for example, as shown in FIG. 1, the sensitivity coefficient is simply calculated as the slope of a straight line connecting the nominal point P1 and the variation point P2. Such a calculation method lacks a statistical viewpoint, and an error occurs in the statistics. In other words, it cannot be said that the delay model reproduces the actual response with sufficient accuracy.

  The second reason is that the type of manufacturing variation of the element is not taken into account in calculating the sensitivity coefficient. Device manufacturing variations are roughly divided into “random variations” that occur regardless of position and pattern, and “D2D (Die-to-Die) variations” that are variations in the average value of chips, that is, variations between chips. It is done. D2D variation is a constant within the chip, and random variation behaves as a random variable within the chip. Since the D2D variation is constant in the chip, the effect appears as an algebraic sum in series cells. On the other hand, since random variation becomes a random variable within the chip, its effect appears as a statistical sum, that is, the sum of variances. Thus, since random variation and D2D variation have different characteristics, it is not preferable to handle them together. However, in the conventional characterization, the sensitivity coefficient is calculated without distinguishing between random variation and D2D variation. In the conventional cell delay library, the same value is used for the sensitivity coefficient for random variations and the sensitivity coefficient for D2D variations. Therefore, it cannot be said that the delay model sufficiently reproduces the actual response.

  Thus, the accuracy of the delay model provided by the conventional cell delay library is not sufficient. Therefore, the statistical accuracy of statistical STAs using the delay model is not sufficient. It is desired to further improve the accuracy of the delay model provided by the cell delay library.

  In a first aspect of the present invention, a method for creating a cell delay library for statistical STAs is provided. The creation method includes (A) providing a delay model that provides a cell delay value depending on model parameters of transistors constituting the cell. Here, the delay model is expressed by a linear function of model parameter variation, and the variation coefficient is a sensitivity coefficient. Further, the creation method includes (B) calculating a plurality of cell delay values corresponding to each of the plurality of simulation points by performing a circuit simulation using a plurality of simulation points of the model parameter, and (C) a plurality of cells. Determining a constant term and a sensitivity coefficient of the delay model by moment matching between the delay value and the delay model; and (D) creating a cell delay library that provides at least the determined sensitivity coefficient.

  In a second aspect of the invention, a cell delay library for statistical STAs is provided. The cell delay library includes a constant, a first sensitivity coefficient, and a second sensitivity coefficient different from the first sensitivity coefficient. The cell delay value is expressed by a linear function of random variation of model parameters of transistors constituting the cell and variation between chips. The linear function has the above constant as a constant term and includes random variation and chip-to-chip variation as variables. Coefficients of random variation and chip-to-chip variation regarding the same model parameter are the first sensitivity coefficient and the second sensitivity coefficient, respectively.

  In a third aspect of the present invention, a delay analysis method for performing statistical STA using the cell delay library is provided.

  In a fourth aspect of the present invention, a method for manufacturing a semiconductor integrated circuit is provided. The manufacturing method includes (a) designing a semiconductor integrated circuit, (b) performing statistical STA using the cell delay library, and performing delay verification of the designed semiconductor integrated circuit, (c) And a step of manufacturing a semiconductor integrated circuit after the result of the delay verification is a pass.

  According to the present invention, the accuracy of the delay model (sensitivity coefficient) provided by the cell delay library is improved. Therefore, the analysis accuracy of the statistical STA using the cell delay library is also improved. As a result, the yield of the manufactured semiconductor integrated circuit is also improved.

  With reference to the accompanying drawings, a cell delay library for statistical STAs according to an embodiment of the present invention, a method for producing the library, and a method for designing and manufacturing a semiconductor integrated circuit using the library will be described.

1. Manufacturing Variation and Cell Delay Value Manufacturing variation of the element is roughly divided into two, “random variation” and “D2D (Die-to-Die) variation” shown in FIG. Random variations are variations that occur regardless of position, pattern, etc. in one chip. As the random variation, for example, variation in threshold voltage caused by variation in impurity concentration can be cited. On the other hand, D2D variation is variation between chips, and is defined as a difference between an average value in one chip and an average value in another chip. D2D variation is caused by the tendency of process characteristics to change within the plane of a certain wafer, and the difference in process characteristics between different wafers. Random variation behaves as a random variable within the chip, while D2D variation is a constant within the chip. Since the D2D variation is constant in the chip, the effect appears as an algebraic sum in series cells. On the other hand, since random variation becomes a random variable within the chip, its effect appears as a statistical sum, that is, the sum of variances. Thus, the random variation and the D2D variation have different characteristics.

  The delay value D of a certain logic cell varies depending on variations in the model parameters (gate length, gate width, mobility, etc.) of the transistors constituting the logic cell. Hereinafter, two types of model parameters X and Y will be described as examples. The same argument applies as the number of model parameters increases. When the random variation and the D2D variation are considered separately, the model parameters X and Y are expressed as the following equation (3).

Formula (3):
X = Xnom + X _RAND + X _D2D
Y = Ynom + Y _RAND + Y _D2D

Xnom is a nominal value (design value) of the model parameter X. X _RAND is a random variable indicating a random variation from the nominal value Xnom of the model parameter X. X _D2D is a random variable indicating the D2D variation from the nominal value Xnom of the model parameter X. Ynom is a nominal value (design value) of the model parameter Y. Y _RAND is a random variable indicating random variation from the nominal value Ynom of the model parameter Y. Y _D2D is a random variable indicating the D2D variation from the nominal value Ynom of the model parameter Y. At this time, the delay value D of the cell is approximately expressed by a linear function of each variation variable as expressed by the following equation (4).

Formula (4):
D = D0 + D _RAND + D _D2D
D _RAND = α _X・ X _RAND + α _Y・ Y _RAND
D _D2D = β _X · X _D2D + β _Y · Y _D2D

In Expression (4), D0 is a constant term of a linear function. D _RAND is a variation in the cell delay value D caused by random variations X _RAND and Y _RAND . α _X is a sensitivity coefficient for the random variation X _RAND of the model parameter X, and α _Y is a sensitivity coefficient for the random variation Y _RAND of the model parameter Y. The sensitivity coefficients α _X and α _Y related to random variations may be collectively referred to as “sensitivity coefficient α”. The delay variation D _RAND is given by a linear combination of random variations X _RAND and Y _RAND . On the other hand, D _D2D is a variation in the cell delay value D caused by the D2D variation X _D2D , Y _D2D . β _X is a sensitivity coefficient for the D2D variation X _D2D of the model parameter X, and β _Y is a sensitivity coefficient for the D2D variation Y _D2D of the model parameter Y. The sensitivity coefficients β _X and β _Y related to the D2D variation may be collectively referred to as “sensitivity coefficient β”. The delay variation D _D2D is given by a linear linear combination of D2D variations X _D2D and Y _D2D .

As shown in Expression (4), the cell delay value D is given as a linear function of the model parameter (X, Y) variation (X _RAND , X _D2D , Y _RAND , Y _D2D ). The linear function expressed by the equation (4) is a “delay model” for the cell. In characterization of the cell, appropriate sensitivity coefficients α and β and a constant term D0 are determined, thereby determining a delay model.

2. Cell Characterization FIG. 3 conceptually shows a characterization process flow according to the present embodiment. In characterization, the constant term D0 and sensitivity coefficients α and β of the delay model are calculated by using various types of information. As a result, the cell delay library DLIB providing the constant term D0 and the sensitivity coefficients α and β is created.

  Since random variation and D2D variation have different characteristics, it is not preferable to handle them together. Therefore, in the characterization of the present embodiment, random variation and D2D variation are distinguished from each other. Therefore, the sensitivity coefficient α for random variation and the sensitivity coefficient β for D2D variation are calculated by different methods. Hereinafter, the characterization method in the present embodiment will be described in detail with reference to FIG.

2-1. Step S10
First, various information necessary for the characterization process is provided. The information includes cell information 1, model parameter information 2, simulation point information 3, delay model information 4, and the like.

(Cell information 1)
Cell information 1 gives information related to a cell necessary for SPICE simulation. For example, the cell information 1 includes a cell net list (SPICE net list) indicating the circuit configuration of the cell. The cell information 1 may include a table of load capacity (Cload) and slew rate (Slew). The cell information 1 is general information used in normal characterization.

(Model parameter information 2)
Model parameter information 2 gives information on model parameters (gate length, gate width, mobility, etc.) of the transistors included in the cell. The model parameter information 2 includes nominal values (Xnom, Ynom, etc.) of each model parameter. Further, the model parameter information 2 includes probability distribution information (variation information) regarding the model parameters X and Y taking into account variation.

For example, FIG. 4 conceptually shows the probability distribution f (X _RAND ) of the random variation X _RAND of the model parameter X. As shown in FIG. 4, the random variation X _RAND has a certain distribution. Typically, the probability distribution f (X _RAND ) is given by a normal distribution N (0, σX _RAND ) with an average value of 0 and a standard deviation σX _RAND . In this case, the average value 0 and the standard deviation σX _RAND are given as the probability distribution information. Similarly, the average value 0 and the standard deviation σX _D2D are given to the D2D variation X _D2D . Similarly, the average value 0 and standard deviations σY _RAND and σY _D2D are given for the other model parameters Y. Such probability distribution information is determined based on actual measurements and specifications.

(Simulation point information 3)
The simulation point information 3 gives a plurality of simulation points SP deviated from the nominal values Xnom and Ynom with respect to the model parameters X and Y. FIG. 5 conceptually shows an example of a plurality of simulation points SP. The horizontal axis represents the model parameter X, the vertical axis represents the model parameter Y, and the origin represents the nominal condition. For example, the simulation points SP include SP1 (x ₁ , 0), SP3 (0, y ₁ ), SP5 (−x ₁ , 0), and SP7 (0, −y ₁ ). The simulation points SP are SP2 (x ₂ , y ₂ ), SP4 (−x ₂ , y ₂ ), SP6 (−x ₂ , −y ₂ ), and both of which the plural types of model parameters X and Y vary simultaneously. SP8 _{(x 2,} -y 2) may be contained. Note that the number and position of the simulation points SP are determined based on the order of a higher-order approximation formula used in moment matching described later and the required accuracy.

(Delay model information 4)
The delay model information 4 gives the function form of the delay model expressed by the above equation (4). The purpose of characterization is to obtain a constant term D0 and sensitivity coefficients α and β in the delay model and determine the delay model.

2-2. Step S20
In step S20, a nominal delay value Dnom when the model parameter is a nominal value is calculated through circuit simulation (SPICE). Therefore, cell information 1 and nominal values (Xnom, Ynom, etc.) of each model parameter are input to SPICE. By executing the SPICE simulation using the nominal value, the nominal delay value Dnom is calculated.

2-3. Step S30
In step S30, the cell delay value when the model parameters X and Y vary is calculated through circuit simulation (SPICE). For this purpose, cell information 1 and simulation point information 3 are input to SPICE. By using a plurality of simulation points SP1 to SP8 given by the simulation point information 3, a SPICE simulation is executed. As a result, a plurality of cell delay values DS1 to DS8 corresponding to the plurality of simulation points SP1 to SP8 are calculated. For example, cell delay value DS1 is a cell delay value when the model parameter X varies by x ₁ from the nominal value Xnom. The cell delay value DS2, the model parameter X varies by _{x 2} from the nominal value Xnom, and a cell delay value when the model parameter Y varies by _{y 2} from the nominal value Ynom. Thus, by appropriately setting the simulation point SP, it is possible to calculate a cell delay value when a plurality of types of model parameters vary at the same time. As a result of step S30, simulation result information indicating the correspondence between the plurality of simulation points SP1 to SP8 and the calculated cell delay values DS1 to DS8 is created.

2-4. Step S40
In step S40, a delay model for random variation is determined. Here, only random variations are considered, and D2D variations are not considered. Therefore, the delay model is expressed by the following equation (5).

Formula (5):
D = F1 (X _RAND , Y _RAND )
= DR + α _X • X _RAND + α _Y • Y _RAND

In Equation (5), DR is a constant term of a linear function. According to the present embodiment, the sensitivity coefficient α (α _X , R X) with respect to random variations is obtained by moment matching between the plurality of cell delay values DS1 to DS8 calculated in step S30 and the delay model F1 represented by the equation (5). α _Y ) and the constant term DR are determined. Here, the moment matching means that the moment amount (average value, variance) related to the cell delay values DS1 to DS8 and the moment amount (average value, variance) related to the delay model F1 are matched as much as possible. That is, from a statistical point of view, the simulation result and the delay model are matched so that the error of the statistic becomes small. As a result, a sensitivity coefficient α in which statistical accuracy is considered is obtained.

  FIG. 6 is a flowchart showing details of step S40 in the present embodiment.

(Step S41)
First, the plurality of cell delay values DS1 to DS8 obtained in step S30 are fitted with a predetermined approximate expression. The approximate expression is a polynomial that gives the cell delay value D as a function of random variations X _RAND and Y _RAND . In the present embodiment, for example, a high-order polynomial F2 represented by the following equation (6) is used.

Formula (6):
D = F2 (X _RAND , Y _RAND )
= B _XX · X _RAND ² + b _XY · X _RAND · Y _RAND + b _YY · Y _RAND ² + b _X · X _RAND + b _Y · Y _RAND + b0

Coefficients b _XX , b _XY , b _YY , b _x , b _Y are obtained by the least square method based on the high-order polynomial F2, the plurality of simulation points SP1 to SP8, and the plurality of cell delay values DS1 to DS8 obtained by the simulation. Can be determined. Specifically, the error between the cell delay value obtained by substituting each of the simulation points SP1 to SP8 into the high-order polynomial F2 and the cell delay values DS1 to DS8 obtained by the simulation is minimized. Each parameter of the high-order polynomial F2 is determined.

The higher-order polynomial F2 expressed by the equation (6) is closer to the actual response than the delay model F1 (first-order linear function) expressed by the equation (5). It should be noted that the high-order polynomial F2 represented by the equation (6) includes a component “b _XY · X _RAND · Y _RAND ” that is proportional to the product of a plurality of types of model parameters. Such a component means an “interaction” on the cell delay value D by a plurality of types of model parameters. That is, the high-order polynomial F2 includes an interaction with the cell delay value D.

(Step S42)
In step S42, probability distribution information given by the high-order polynomial F2 and model parameter information 2 determined in step S41 is used. The moment amount (average value E1, variance V1) of the distribution of the cell delay value D can be calculated by a variance analysis based on the higher-order polynomial F2 and the probability distribution information.

FIG. 7 conceptually shows the process in step S42. Here, for simplicity, only the one-dimensional model parameter X is considered. A graph (a) in FIG. 7 shows the high-order polynomial F2 (X _RAND ) determined in step S41. The horizontal axis is the random variation X _RAND and the vertical axis is the cell delay value D. The graph (b) is the same as FIG. 4 and shows the probability distribution f (X _RAND ) of the random variation X _RAND . The horizontal axis is random variation X _RAND , and the vertical axis is probability density. The probability distribution G2 (D) of the cell delay value D is obtained by combining the high-order polynomial F2 (X _RAND ) that gives the cell delay value D and the probability distribution f (X _RAND ) of the random variation X _RAND . A graph (c) in FIG. 7 shows the probability distribution G2 (D). The horizontal axis is the cell delay value D, and the vertical axis is the probability density. Since a high-order polynomial is used, the shape of the probability distribution G2 (D) is asymmetric. The average value of the probability distribution G2 (D) is “first average value E1”, and the variance is “first variance V1”.

In the present embodiment, average value 0 and standard deviations σX _RAND and σY _RAND are given as probability distribution information of random variations X _RAND and Y _RAND (E (X _RAND ) = 0, E (Y _RAND ) = 0, V (X _RAND ) = σX _RAND ² , V (Y _RAND ) = σY _RAND ² ). In this case, the first average value E1 and the first variance V1 are given by the following equations (7) and (8), respectively. The first average value E1 and the first variance V1 can be easily calculated, which is preferable.

Formula (7):
E1 = E (G2 (D))
= E (F2 (X _RAND : N (0, σX _RAND ), Y _RAND : N (0, σY _RAND )))
= B _XX · E (X _RAND ² ) + b _XY · E (X _RAND Y _RAND ) + b _YY · E (Y _RAND ² ) + b _X · E (X _RAND ) + b _Y · E (Y _RAND ) + b 0
= B _XX · V (X _RAND ) + b _YY · V (Y _RAND ) + b0
= B _XX · σX _RAND ² + b _YY · σY _RAND ² + b0

Formula (8):
V1 = V (G2 (D))
= B _XX ² · V (X _RAND ² ) + b _XY ² · V (X _RAND Y _RAND ) + b _YY ² · V (Y _RAND ² ) + b _X ² · V (X _RAND ) + b _Y ² · V (Y _RAND ) + B0

(Step S43)
In step S43, the probability distribution information given by the delay model (primary linear function) F1 and the model parameter information 2 expressed by the above equation (5) is used. By the analysis of variance based on the delay model F1 and the probability distribution information, the moment amount (average value E2, variance V2) of the distribution of the cell delay value D can be calculated.

FIG. 8 conceptually shows the process in step S43. Here, for simplicity, only the one-dimensional model parameter X is considered. The graph (a) in FIG. 8 shows the delay model F1 (X _RAND ). The horizontal axis is the random variation X _RAND and the vertical axis is the cell delay value D. The graph (b) is the same as FIG. 4 and shows the probability distribution f (X _RAND ) of the random variation X _RAND . The horizontal axis is random variation X _RAND , and the vertical axis is probability density. The probability distribution G1 (D) of the cell delay value D is obtained by combining the delay model F1 (X _RAND ) that gives the cell delay value D and the probability distribution f (X _RAND ) of the random variation X _RAND . A graph (c) in FIG. 8 shows the probability distribution G1 (D). The horizontal axis is the cell delay value D, and the vertical axis is the probability density. Since the delay model is a linear function, the probability distribution G1 (D) has a symmetrical shape. The average value of the probability distribution G1 (D) is “second average value E2”, and the variance thereof is “second variance V2”.

At this stage, the constant term DR and the sensitivity coefficients α _X and α _Y are not yet determined. Accordingly, the second average value E2 and the second variance V2 are functions of the constant term DR and the sensitivity coefficients α _X and α _Y. If the constant term DR and the sensitivity coefficients α _X and α _Y change, the second average value E2 and the second variance V2 can also change.

In the present embodiment, average value 0 and standard deviations σX _RAND and σY _RAND are given as probability distribution information of random variations X _RAND and Y _RAND (E (X _RAND ) = 0, E (Y _RAND ) = 0, V (X _RAND ) = σX _RAND ² , V (Y _RAND ) = σY _RAND ² ). In this case, the second average value E2 and the second variance V2 are given by the following equations (9) and (10), respectively. The second average value E2 and the second variance V2 can be easily calculated, which is preferable.

Formula (9):
E2 = E (G1 (D))
= E (F1 (X _RAND : N (0, σX _RAND ), Y _RAND : N (0, σY _RAND )))
= DR + α _X · E (X _RAND ) + α _Y · E (Y _RAND )
= DR

Formula (10):
V2 = V (G1 (D))
= Α _X ² · V (X _RAND ) + α _Y ² · V (Y _RAND )
= Α _X ² · σX _RAND ² + α _Y ² · σY _RAND ²

As is clear from the equations (9) and (10), the second average value E2 and the second variance V2 are functions of the constant term DR and the sensitivity coefficients α _X and α _Y.

(Step S44)
In step S44, moment matching is performed. That is, matching is performed between the moment amount (E1, V1) calculated in step S42 and the moment amount (E2, V2) calculated in step S43. Since the second average value E2 and the second variance V2 are functions of the constant term DR and the sensitivity coefficients α _X and α _Y , the sensitivity coefficients α _X and α _Y can be determined by this moment matching.

FIG. 9 conceptually shows the process in step S44. Here, for simplicity, only the one-dimensional model parameter X is considered. The graph (a) in FIG. 9 shows the high-order polynomial F2 (X _RAND ) and the delay model F1 (X _RAND ). The horizontal axis is the random variation X _RAND and the vertical axis is the cell delay value D. On the other hand, the graph (b) shows the probability distributions G2 (D) and G1 (D) of the cell delay value D described above. The horizontal axis is the cell delay value D, and the vertical axis is the probability density. When the constant term DR and sensitivity coefficient (slope) alpha _X delay model F1 is changed, the second average value E2 and the second distributed V2 of the probability distribution G1 (D) varies accordingly. Therefore, it is possible to search for a constant term DR and a sensitivity coefficient α _X that reduce an error between the second average value E2 and the first average value E1 and an error between the second variance V2 and the first variance V1. .

  For example, in the case of the above formulas (7) to (10), the moment matching condition is given by the following formula (11).

Formula (11):
E1 = E2 = DR
V1 = V2 = α _X ² · σX _RAND ² + α _Y ² · σY _RAND ²

The constant term DR can be determined under the condition that the first average value E1 and the second average value E2 match. Further, the sensitivity coefficients α _X and α _Y can be determined under the condition where the first variance V1 and the second variance V2 are the same. Note that the sensitivity coefficients α _X and α _Y may not be uniquely determined only by moment matching. As is clear from the above equation (11), for example, the moment matching condition is also satisfied when the sensitivity coefficient is “−α _X ”. FIG. 9 also shows a linear function F1 ′ (X _RAND ) having a slope of “−α _X ”. When the second variance V2 of the linear linear function F1 (X _RAND ) matches the first variance V1, the variance of the linear linear function F1 ′ (X _RAND ) also matches the first variance V1.

Therefore, according to the present embodiment, sensitivity coefficients α _X and α _Y are uniquely determined by using the least square method together with moment matching. Specifically, by using the delay model F1 (X _RAND , Y _RAND ), cell delay values DF1 to DF8 corresponding to the plurality of simulation points SP1 to SP8 are calculated. A condition that minimizes the error between the calculated cell delay values DF1 to DF8 and the cell delay values DS1 to DS8 calculated in the above-described step S30 is expressed by the following equation (12).

Formula (12):
Σ _i (DFi-DSi) ² : minimum

Then, in addition to the moment matching condition given by Expression (11), sensitivity coefficients α _X and α _Y satisfying the condition given by Expression (12) are obtained. The sensitivity coefficients α _X and α _Y satisfying such conditions can be determined based on, for example, Lagrange's undetermined multiplier method. In this way, sensitivity coefficients α _X and α _Y that satisfy the moment matching condition and are closest to the simulation result are uniquely determined.

2-5. Step S50
In step S50, a delay model for D2D variation is determined. Here, only D2D variation is considered, and random variation is not considered. Therefore, the delay model is expressed by the following equation (13).

Formula (13):
D = DD + β _X · X _D2D + β _Y · Y _D2D

In Expression (13), DD is a constant term of a linear function. According to the present embodiment, the simulation result obtained in step S30 and the delay model represented by the equation (13) are used, and the constant term DD and the sensitivity coefficients β _X and β _Y are determined by the least square method. . Specifically, cell delay values DD1 to DD8 corresponding to the plurality of simulation points SP1 to SP8 are calculated by using the delay model of Expression (13). Then, the constant term DD and the sensitivity coefficients β _X and β _Y are determined so that the error between the calculated cell delay values DD1 to DD8 and the cell delay values DS1 to DS8 calculated in step S30 is minimized. Is done. As described above, the constant term DD and the sensitivity coefficients β _X and β _Y are determined by using the least square method so that the delay model matches the SPICE simulation result as much as possible.

2-6. Step S60
The cell delay library DLIB is created by using the nominal delay value Dnom, the constant term DR, the sensitivity coefficient α, the constant term DD, and the sensitivity coefficient β obtained by the above processing. As the random variation sensitivity coefficient α, the sensitivity coefficient α obtained by the process of step S40 is used. As the sensitivity coefficient β of the D2D variation, the one obtained by the process of step S50 is used. As the constant term D0 of the delay model expressed by the equation (4), the following various candidates are conceivable.

(Candidate 1) Constant term DR obtained in step S40
(Candidate 2) Constant term DD obtained in step S50
(Candidate 3) Weighted average of constant term DR and constant term DD (Candidate 4) Nominal delay value Dnom obtained in step S20

  For example, when the random variation is more dominant than the D2D variation, the candidate 1 constant term DR may be used as the constant term D0. On the other hand, when the D2D variation is more dominant than the random variation, it is conceivable to use the candidate 2 constant term DD as the constant term D0. When the random variation and the D2D variation are approximately the same, it is appropriate to use the weighted average of the constant term DR and the constant term DD as the constant term D0. Also, the nominal delay value Dnom can be used as the constant term D0.

FIG. 10 conceptually shows the cell delay library DLIB to be created. As shown in FIG. 10, the cell delay library DLIB provides a constant term D0 and sensitivity coefficients α _X , α _Y , β _X , β _Y for each cell. In addition, for one cell, the constant term D0 and the sensitivity coefficients α _X , α _Y , β _X , and β _Y may be determined for each of various load capacities Cload and slew rates Slew, and recorded in the cell delay library DLIB.

The cell delay library DLIB according to the present embodiment has the same format as the existing cell delay library. The difference from the existing cell delay library is that the sensitivity coefficient α and the sensitivity coefficient β for the same model parameter are different from each other. For example, regarding the model parameter X, the sensitivity coefficient α _{X of} random variation and the sensitivity coefficient β _X of D2D variation are different from each other. This is because random variation and D2D variation are handled separately in cell characterization, and sensitivity coefficient α and sensitivity coefficient β are calculated by different methods.

3. Effect According to the technique shown in FIG. 1, the sensitivity coefficient is simply calculated as the slope of a straight line connecting the nominal point P1 and the variation point P2. Such a calculation method lacks a statistical viewpoint, and an error occurs in the statistics. In other words, it cannot be said that the delay model reproduces the actual response with sufficient accuracy.

  According to the present embodiment, the simulation result and the delay model are matched so as to reduce the error of the statistic from a statistical viewpoint. By this moment matching, a constant term D0 and a sensitivity coefficient α with sufficient statistical accuracy taken into account are calculated. Therefore, the obtained delay model can reproduce the actual response with sufficient accuracy.

  In the present embodiment, the simulation result is approximated using a high-order polynomial F2 (see Expression (6)). The higher order polynomial F2 is closer to the actual response than the delay model which is a linear function. In the present embodiment, the delay model is matched with the high-order polynomial F2 from the viewpoint of the moment amount. As a result, the same average and variation accuracy as the higher-order delay model can be realized by the first-order delay model. This also contributes to improving the accuracy of the delay model.

  Further, when the high-order polynomial F2 is used, the interaction with the cell delay value D by a plurality of types of model parameters can be considered. That is, the sensitivity coefficient α and the constant term D0 can be obtained in consideration of the effect of interaction. This also contributes to improving the accuracy of the delay model.

  Furthermore, according to the present embodiment, when calculating the sensitivity coefficient, random variation and D2D variation are distinguished from each other. Since the random variation and the D2D variation have different characteristics, it is preferable that the sensitivity coefficient α related to the random variation and the sensitivity coefficient β related to the D2D variation are calculated by a method suitable for each. According to the present embodiment, the sensitivity coefficient α related to random variation is determined by moment matching considering the probability distribution. On the other hand, the sensitivity coefficient β related to the D2D variation having some tendency is determined by the least square method that is closer to the simulation result. As a result, the sensitivity coefficient α and the sensitivity coefficient β for the same model parameter are different from each other. Such a delay model including different sensitivity coefficients α and β can better reproduce the actual response.

  As described above, according to the present embodiment, the accuracy of the delay model provided by the cell delay library DLIB is improved. Therefore, the analysis accuracy of the statistical STA using the cell delay library is also improved. As a result, the yield of the manufactured semiconductor integrated circuit is also improved.

4). LSI Design and Manufacturing Method LSI design is performed by using a computer. FIG. 11 is a block diagram schematically showing a configuration of a design system 10 (computer system) for designing an LSI. The design system 10 includes a storage device 20, a processor 30, an input device 40, and an output device 50. Examples of the storage device 20 include RAM and HDD. Examples of the input device 40 include a mouse and a keyboard. An example of the output device 50 is a display.

  The storage device 20 stores the cell information 1, model parameter information 2, simulation point information 3, delay model information 4, and cell delay library DLIB described above. Furthermore, the storage device 20 stores a netlist NET and layout data LAY. The netlist NET is design data indicating connections between cells of the design circuit. The layout data LAY is design data indicating the layout of the design circuit.

  The characterization program 100, the simulation program 110, the design program 200, and the statistical STA program 300 are software programs executed by a computer. These programs may be recorded on a computer-readable recording medium. The processor 30 implements LSI design processing by executing these programs.

  FIG. 12 is a flowchart schematically showing an LSI design / manufacturing method according to the present embodiment. An LSI design / manufacturing method according to the present embodiment will be described with reference to FIGS.

(Step S100)
The characterization program 100 executes the characterization process described above and creates the cell delay library DLIB. At this time, the characterization program 100 reads necessary information (cell information 1, model parameter information 2, simulation point information 3, delay model information 4) from the storage device 20, and uses the simulation program 110 as appropriate. The simulation program 110 is SPICE. The cell delay library DLIB that provides a highly accurate delay model is created by the characterization method according to the present embodiment.

(Step S200)
The design program 200 designs a desired circuit and creates a netlist NET and layout data LAY. This circuit design method is a general method.

(Step S300)
The statistical STA program 300 performs the statistical STA by using the cell delay library DLIB, the netlist NET, the layout data LAY, and the like. The cell delay library DLIB according to the present embodiment has the same format as the existing cell delay library. Therefore, the statistical STA program 300 may also be an existing statistical STA program. In the statistical STA, delay analysis / delay verification of a design circuit is performed by statistically considering manufacturing variation. Information relating to manufacturing variation of model parameters is obtained from model parameter information 2. The cell delay library DLIB according to the present embodiment provides a highly accurate delay model (sensitivity coefficient). Therefore, the analysis accuracy of the statistical STA using the cell delay library is also improved.

(Step S400)
When the result of the delay verification by the statistical STA is failure (step S400; No), the process returns to step S200. Then, the layout of the design circuit is corrected. When the result of the delay verification becomes a pass (step S400; Yes), the layout of the design circuit is determined.

(Step S500)
A design circuit is manufactured based on the determined layout. According to the present embodiment, delay analysis and delay verification are performed with high accuracy. Therefore, the yield of manufactured semiconductor integrated circuits is also improved.

  The embodiments of the present invention have been described above with reference to the accompanying drawings. However, the present invention is not limited to the above-described embodiments, and can be appropriately changed by those skilled in the art without departing from the scope of the invention.

FIG. 1 is a conceptual diagram for explaining how to obtain a general delay model and sensitivity coefficient. FIG. 2 is a conceptual diagram showing random variation and D2D variation. FIG. 3 is a conceptual diagram showing a processing flow of characterization according to the embodiment of the present invention. FIG. 4 is a graph showing an example of the probability distribution of model parameters in the present embodiment. FIG. 5 is a conceptual diagram showing an example of simulation points in the present embodiment. FIG. 6 is a flowchart showing a method for determining the sensitivity coefficient α relating to random variation in the present embodiment. FIG. 7 is a conceptual diagram showing the processing in step S42 in the present embodiment. FIG. 8 is a conceptual diagram showing the processing in step S43 in the present embodiment. FIG. 9 is a conceptual diagram showing the processing in step S44 in the present embodiment. FIG. 10 is a conceptual diagram showing the cell delay library according to the present embodiment. FIG. 11 is a block diagram schematically showing the configuration of the design system according to the present embodiment. FIG. 12 is a flowchart schematically showing a method for designing and manufacturing a semiconductor integrated circuit according to the present embodiment.

Explanation of symbols

1 cell information 2 model parameter information 3 simulation point information 4 delay model information 10 design system 20 storage device 30 processor 40 input device 50 output device 100 characterization program 110 simulation program 200 design program 300 statistical STA program DLIB cell delay library NET net List LAY Layout data

Claims (12)

A method for creating a cell delay library for statistical STAs, comprising:
(A) providing a first delay model that provides a cell delay value depending on a model parameter of a transistor constituting the cell;
Here, the first delay model is represented by a linear function of the variation of the model parameter, the constant term of the first delay model is a first constant term, and the coefficient of variation is a first sensitivity coefficient. ,
(B) performing a circuit simulation using a plurality of simulation points of the model parameter, and calculating a plurality of cell delay values corresponding to each of the plurality of simulation points;
(C) determining the first constant term and the first sensitivity coefficient by moment matching between the plurality of cell delay values and the first delay model;
(D) creating a cell delay library that provides at least the determined first sensitivity coefficient. A method for creating a cell delay library. A method for creating a cell delay library according to claim 1,
The step (C) includes:
(C1) determining an approximate expression that gives a cell delay value as a function of the variation by a least square method by using the plurality of simulation points and the plurality of cell delay values;
(C2) calculating an average value and a variance of cell delay value distribution obtained from the approximate expression and the probability distribution information of the model parameter, respectively, as a first average value and a first variance;
(C3) calculating an average value and a variance of cell delay value distribution obtained from the first delay model and the probability distribution information as a second average value and a second variance, respectively;
(C4) determining the first constant term under the condition that the second average value matches the first average value;
(C5) A method of creating a cell delay library, comprising: determining the first sensitivity coefficient under a condition that at least the second variance matches the first variance. A method for creating a cell delay library according to claim 2,
In the step (C5), the error between the cell delay values calculated from the plurality of simulation points and the first delay model and the cell delay values calculated in the step (B) is minimized. A method for creating a cell delay library, wherein the first sensitivity coefficient is determined under conditions. A method for creating a cell delay library according to claim 2 or 3,
The approximation formula is a high-order polynomial method for creating a cell delay library. A method for creating a cell delay library according to claim 4,
The plurality of simulation points include a point where a plurality of types of model parameters are simultaneously deviated from a nominal value,
The high-order polynomial includes a component proportional to a product of the plurality of types of model parameters. A method for creating a cell delay library according to any one of claims 1 to 5,
The cell delay value depends on the random variation of the model parameter and the variation between chips,
The method of creating a cell delay library, wherein the first delay model is a delay model related to the random variation. A method for creating a cell delay library according to claim 6,
Furthermore,
(X) providing a second delay model for the inter-chip variation;
Here, the second delay model is represented by a linear function of the model parameter variation between chips, the constant term of the second delay model is a second constant term, and the coefficient of variation between chips is a second coefficient. Sensitivity coefficient,
(Y) so that an error between the cell delay values calculated from the plurality of simulation points and the second delay model and the cell delay values calculated in the step (B) is minimized. Determining two sensitivity coefficients and the second constant term;
The cell delay library provides at least the determined first sensitivity coefficient and second sensitivity coefficient. A method of creating a cell delay library. A method for creating a cell delay library according to claim 7,
Furthermore,
(Z) determining a constant term of a delay model giving a cell delay value based on the determined first constant term and second constant term;
The cell delay library provides the determined constant term, the first sensitivity coefficient, and the second sensitivity coefficient. A method of creating a cell delay library. A method for creating a cell delay library according to claim 8,
The method of creating a cell delay library, wherein the constant term is any one of the first constant term, the second constant term, and a weighted average of the first constant term and the second constant term. A cell delay library for statistical STAs,
Constants,
A first sensitivity coefficient;
A second sensitivity coefficient different from the first sensitivity coefficient,
The cell delay value is expressed by a linear function of random variation and inter-chip variation of model parameters of transistors constituting the cell,
The linear function has the constant as a constant term, and includes the random variation and the inter-chip variation as variables,
The cell delay library, wherein the random variation coefficient and the chip-to-chip variation coefficient related to the same model parameter are the first sensitivity coefficient and the second sensitivity coefficient, respectively. A delay analysis method for performing statistical STA using a cell delay library, comprising:
The cell delay library is:
Constants,
A first sensitivity coefficient;
A second sensitivity coefficient different from the first sensitivity coefficient,
The cell delay value is expressed by a linear function of random variation and inter-chip variation of model parameters of transistors constituting the cell,
The linear function has the constant as a constant term, and includes the random variation and the inter-chip variation as variables,
The delay analysis method, wherein the random variation coefficient and the inter-chip variation coefficient related to the same model parameter are the first sensitivity coefficient and the second sensitivity coefficient, respectively. (A) designing a semiconductor integrated circuit;
(B) performing statistical STA using a cell delay library and performing delay verification of the designed semiconductor integrated circuit;
(C) after the result of the delay verification becomes a pass, manufacturing the semiconductor integrated circuit, and
The cell delay library is:
Constants,
A first sensitivity coefficient;
A second sensitivity coefficient different from the first sensitivity coefficient,
The cell delay value is expressed by a linear function of random variation and inter-chip variation of model parameters of transistors constituting the cell,
The linear function has the constant as a constant term, and includes the random variation and the inter-chip variation as variables,
The method for manufacturing a semiconductor integrated circuit, wherein the random variation coefficient and the inter-chip variation coefficient related to the same model parameter are the first sensitivity coefficient and the second sensitivity coefficient, respectively.

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