JP2005191301A - Model parameter sampling method, and circuit simulation system - Google Patents

Abstract

<P>PROBLEM TO BE SOLVED: To provide a model parameter sampling method and circuit simulation system, wherein the correlations between characteristics that test elements come to assume due to their assigned locations are reflected in model parameters. <P>SOLUTION: In this model parameter sampling method, a more realistic distribution of characteristics is calculated, by accurately incorporating into a test element model the correlations between characteristics, dependent on the positional relations between the devices in the object system comprising a plurality of devices. In a circuit simulator using this method, accuracy is improved in the prediction of changes in the circuit characteristics distribution or the like, due to manufacturing process modifications, and this reduces the need for prototyping for eventual reduction in cost and in time for development. <P>COPYRIGHT: (C)2005,JPO&NCIPI

Description

  The present invention relates to a model parameter extraction method and a circuit simulation system, and more particularly to a technique for extracting a statistical model parameter such as a semiconductor chip and a circuit simulation system using the model parameter.

  Conventionally, when designing an integrated circuit to be built in a semiconductor chip, generally in the semiconductor field, a circuit simulator that simulates the electrical characteristics of the integrated circuit on a computer is used. The designer of the integrated circuit manufactures an actual circuit after designing various circuits on a computer using a circuit simulator and predicting electrical characteristics to obtain desired characteristics. In recent years, the manufacturing process of semiconductors has become complicated, the manufacturing process has become longer, and the manufacturing cost has increased, and the method of obtaining a desired characteristic by repeating trial manufacture of an integrated circuit increases the time and cost for product development. It is a factor. Therefore, the use of a circuit simulator has increased in importance because the number of trial productions can be reduced and the time and cost required for product development can be reduced.

  In the circuit simulator, elements are arranged on a drawing displayed on a CRT or the like, and the design is performed by a circuit diagram by electrically connecting these elements. The elements used at that time are actual electric characteristics. Is modeled so that For example, the modeled element is generated by combining nonlinear elements such as resistors, capacitors, inductors, and diodes, and mathematical expressions describing characteristics. For example, for devices such as bipolar transistors, device models such as Gummel-Poon and VBIC are proposed and used in the circuit simulator.

  On the other hand, an element that is actually manufactured is manufactured after setting a standard for electrical characteristics and the like, and designing the structure of the element cross section and element plane so as to satisfy the standard. Therefore, if the standard defined for the electrical characteristics and the like is different, the characteristics of the elements are also different. If the default values set in advance in the element model on the circuit simulator are used as they are, the element characteristics do not match reality. Therefore, the device model is fitted using some built-in model parameters so that the characteristics of the device model on the circuit simulator match the actual device characteristics, and circuit design can be performed using this device model. Do. This is called model parameter extraction.

  In the example of the element pattern for measurement for extracting the model parameter, wiring is drawn from each terminal of the element so that a series of electrical characteristics for fitting the model parameter can be measured. The wiring extending from each terminal is terminated with an aluminum pad that can be contacted with the electrical probe on the measuring instrument side. Such an element pattern for model parameter extraction is arranged at a certain position on the semiconductor chip as one pattern in TEG (Test Element Group) or the like. The set of model parameters extracted thereby is used for the element model from which the circuit model parameters on the circuit simulator are extracted.

  An example of the simplest model parameter extraction method is to select and extract one element, which is considered to be most representative of electrical characteristics, from a wafer or the like after the completion of manufacture. However, the manufacturing process always varies, and process parameters such as the diffusion layer depth of the elements in the integrated circuit and the size of the opening of the contact window on the actually manufactured semiconductor chip are used to manufacture the wafer on which the semiconductor chip is mounted. It varies depending on the timing of the wafer, the lot of the wafer, the position occupied by the semiconductor chip in the wafer, and the like. As a result, the element characteristics and the characteristics of the integrated circuit using the element characteristics also vary. Therefore, if the manufacturing process varies greatly, the electrical characteristics of the circuit may deviate from the product standard. In other words, a semiconductor chip that is out of the standard cannot be made into a product and is discarded. As the manufacturing process fluctuates, more semiconductor chips are discarded and the loss increases. In order to reduce such loss, circuit characteristics and circuit constants should be examined in advance on the circuit simulator before making an actual product, so that the amount of fluctuation in circuit characteristics due to fluctuations in the manufacturing process on the circuit simulator. Can be predicted. In the circuit simulator, it is possible to control the fluctuation range and the center value of the fluctuation of the circuit characteristics, and when the circuit characteristics deviate from the product standards in subsequent prototypes, it is possible to prevent trial and trial repetition. When the electrical characteristics deviate from the standard, it gives a guideline for adjusting the circuit constants.

  However, if one model is selected and model parameters are extracted as described above, variations in the manufacturing process cannot be reflected in the model parameters. Therefore, various extraction methods are considered so as to reflect the entire variation of the manufacturing process. The extraction method includes a worst case model and a corner model. These are models that select and extract elements that correspond to the maximum and minimum values of fluctuations with respect to the elements manufactured within the fluctuation range within the standard of the manufacturing process managed by PCM (Process Control Module). A circuit simulation is performed by using parameters, or by using the maximum and minimum values of model parameters obtained by calculation by predicting the fluctuation range from the manufacturing process fluctuation. However, in general, the manufacturing process of a semiconductor manufacturing process is long and complicated, and there are many process parameters to be controlled. How such a complex process affects which characteristics of a completed device. Often there is no telling about the causal relationship. Therefore, it is often unclear whether the process model can be created by sufficiently reflecting the process variation if model parameters are prepared by focusing on the process variation. Further, when performing a circuit simulation using the worst case model or the corner model, it is assumed that various model parameters can be freely and independently varied, and a combination of the maximum value and the minimum value is arbitrarily selected. However, the actual model parameters generally vary in correlation with each other from the analysis result of data extracted by increasing the number of elements. Therefore, if the combination of the maximum and minimum values is arbitrarily used, such as the worst case model or corner model, the combination of values that cannot occur by fluctuations with the correlation of the actual model parameters is inevitably used. Become. In other words, predicting the fluctuation range of circuit characteristics using such model parameters would be an overestimation, and the fluctuation range of circuit characteristics could be estimated to be considerably wider than the actual range. There is a possibility that an erroneous determination is made that even a circuit that satisfies the above characteristics has characteristics outside the standard range.

  In order to solve such disadvantages of the worst case model and the corner model, there is a method using an element model called a statistical model (for example, refer to Non-Patent Document 1). The model parameter extraction method described in Non-Patent Document 1 will be described below with reference to the drawings. 10 is an example of a measurement pattern for applying the conventional statistical model extraction method, and FIG. 11 is a block diagram of a circuit simulation system for applying the conventional statistical model extraction method. Shows the procedure of a method for extracting model parameters of a conventional statistical model.

  In FIG. 10, transistors A to C are arranged in the boundary of the semiconductor chip S1. The transistor A is a transistor from which model parameters are to be extracted. The transistor B is arranged at the nearest distance from the transistor A based on the arrangement rule in the semiconductor chip, and has the same manufacturing process sectional structure and manufacturing process planar structure as the transistor A. . The transistor C is a transistor which is disposed at a distance from the transistor A away from the transistor B and has the same manufacturing process sectional structure and manufacturing process planar structure as the transistor A. Then, wirings (not shown) are drawn from the respective terminals of the transistors A to C so that a series of electrical characteristics for fitting the model parameters of the element model can be measured, and the wirings extending from the respective terminals are measured. It is terminated with an aluminum pad that can be contacted with the electrical probe of the instrument.

  In FIG. 11, the circuit simulation system for applying the statistical model extraction method includes a measured system 111, a measurement system 112, an extraction system 113, a data processing system 114, and a circuit simulator 115. The measured system 111 indicates an element itself from which a model parameter of an element model is to be extracted, a semiconductor chip on which such an element is mounted, or a semiconductor wafer on which such a semiconductor chip is mounted. The measurement system 112 performs a series of electrical measurements and the like necessary for extracting model parameters on the system to be measured 111. The extraction system 113 uses the result of a series of electrical measurements and the like to extract the model parameters obtained from the measurement system 112, and the same measurement as the actual measurement on the characteristic curve obtained from them and the circuit diagram of the simulator A characteristic curve obtained from a simulation result of the element characteristics performed by changing the model parameters in various conditions after setting the conditions is compared. Then, the extraction system 113 determines, that is, extracts model parameters so as to be within a certain set error range by using a fitting method using a least square method or the like. The data processing system 114 statistically processes the model parameter extracted by the extraction system 113, and changes the model parameter representing the statistical model to a mathematical expression. The circuit simulator 115 incorporates a mathematical expression of the model parameter representing the statistical model given from the data processing system 114 into the model parameter of the element model. Then, the circuit simulator performs a Monte Carlo simulation or the like using these model parameters, and reflects the variation in the manufacturing process in the variation in the electrical characteristics of the element and the circuit. The circuit simulator 115 also performs simulation of element characteristics when the extraction system 113 fits the above characteristic curves with model parameters.

  In FIG. 12, the measurement system 112 measures the electrical characteristics of the transistor A (see FIG. 10) formed on the semiconductor chip S1 (step S121). At this time, the sample is selected so as to sufficiently reflect statistical fluctuations due to the time when the wafer on which the semiconductor chip S1 is mounted is manufactured, the difference in the manufacturing lot of the wafer, the position occupied by the semiconductor chip S1 in the manufactured wafer, and the like. Measure each. In step S121, the system 111 to be measured is selected as the transistor A, the semiconductor chip S1 on which the transistor A is mounted, or the wafer on which the semiconductor chip S1 is mounted, and a number of transistors A are measured by the measuring system 112.

  Next, the extraction system 113 extracts model parameters from the measurement result of the transistor A obtained in step S121 (step S122). In step S122, the extraction system 113 sets the electrical characteristic curve of the transistor A obtained from the measurement system 112 to be equal to the actual measurement condition on the circuit diagram of the circuit simulator 115, and is used in the circuit simulator 115. This is equivalent to determining the model parameters so that they are compared within a specified error range by comparing with a characteristic curve obtained from a circuit simulation result obtained by changing the values of model parameters in various element models. In addition, a large number of transistors A measured by the measurement system 112 have statistical fluctuations due to the time of manufacturing a wafer on which the transistors A are mounted, the difference in the lot of manufactured wafers, the position of semiconductor chips in the manufactured wafer, and the like. Therefore, the electrical characteristics that differ from measurement to measurement are shown in a manner that reflects variations in the manufacturing process. Therefore, the model parameters of the element model extracted in step S122 have a distribution reflecting the variation in the manufacturing process.

  Next, the data processing system 114 statistically processes the sample values of the model parameters of the transistor A extracted in step S122 (step S123).

For example, if it is statistically estimated from the distribution of sample values of model parameters that the distribution of the population of each model parameter throughout the manufacturing process follows a normal distribution, the distribution of the population that follows the normal distribution from the distribution of sample values The average value and standard deviation are statistically estimated, and the distribution of the population of model parameters statistically estimated using these values can be expressed as Equation (1).
P = P_m + P_σ · N (1, 0) (1)
Where P: model parameter, P_m: average value of the distribution of model parameter P, P_σ: standard deviation of the distribution of model parameter P, N (1, 0): normal distribution of average value 0, standard deviation 1 Indicates a variable.

On the other hand, even if there is a model parameter P0 that is statistically estimated from the distribution of the sample values of the model parameter, that the distribution of the population in the entire manufacturing process does not follow the normal distribution among the model parameters. If the population distribution statistically estimated from the distribution of the value f (P0) obtained by converting the sample value of the model parameter P0 by an appropriate function f follows a normal distribution, the distribution of f (P0) is expressed by the equation It can be expressed as (2).
f (P0) = f_m (P0) + f_σ (P0) · N (1, 0) (2)
Here, f (P0): function of model parameter P0, f_m (P0): average value of distribution of f (P0), and standard deviation of distribution of f_σ (P0): f (P0) are shown. As for the above equation (2), by solving the function f with respect to the model parameter P0, the distribution due to process variation can be reflected in the model parameter P0 as in the equation (3).
P0 = f ⁻¹ {f_m (P0) + f_σ (P0) · N (1, 0)} (3)
Here, f ⁻¹ is an inverse function of the function f.

  Also in the following description, when the statistically estimated population distribution of model parameters does not follow the normal distribution, the model parameters are converted by converting the model parameters using the function f as in the above equation (2). The detailed description is omitted assuming that the population of the value distributions of (2) follows a normal distribution and model parameters are expressed in the form of the above equation (3). The model parameters of the above formulas (1) and (3) are incorporated into the model parameters of the element model, and a random number according to a normal distribution with an average value of 0 and a standard deviation of 1 is generated in the circuit simulator 115 to generate N (1, 0). Substitute for. As a result, the distribution of the model parameter values calculated by the equations (1) and (3) reproduces a normal distribution having the average value and standard deviation of the model parameters extracted in step S122. .

  The data processing system 114 statistically estimates the correlation coefficient between the distributions of the model parameters over the entire manufacturing process using the distribution data for the sample values of the model parameters extracted in step S122. (Step S124). For example, in the Gummel-Poon model of an NPN type bipolar transistor as an element model, model parameters RB, IRB, and RBM that are deeply related to the base resistance, and model parameters BF that are deeply related to the static current amplification factor hFE of the transistor, Consider the correlation. The model parameter RB is the base resistance at zero bias, the model parameter IRB is the base current when the base resistance is halved, and the model parameter RBM indicates the minimum value of the base resistance at high injection. Considering the manufacturing process, when the base diffusion depth does not change much, the base acceptor doping concentration generally decreases as the base resistance increases. If the concentration profile in the diffusion depth direction of the emitter does not change much, the current amplification factor hFE increases, so that the distribution of the model parameters RB, IRB, RBM and the distribution of the model parameters BF have a correlation with each other. Is expected. Further, from the actual data, the distribution of the model parameters RB, IRB, RBM and the distribution of the model parameters BF are generally correlated. The actual manufacturing process is complicated, and the diffusion depth of the base may change even in the above case, and it is unlikely that both have a completely negative correlation coefficient “−1”, but it is obtained from measured data. There is some correlation between the model parameters depending on the manufacturing process.

  FIG. 13 is an example of a correlation coefficient statistically estimated as a normal distribution of a population of model parameters statistically estimated over the entire manufacturing process obtained by the process of step S124. . In FIG. 13, a set of model parameters of an element model corresponding to the transistor A is indicated by P1_A to Pn_A. Note that the model parameters do not have to be all necessary for the transistor A, and may be a part of the model parameters. The value at the intersection of the matrix formed by the model parameters P1_A to Pn_A indicates the correlation coefficient of the combination of the vertical model parameter and the horizontal model parameter. The correlation coefficient of the combination of model parameters takes real values in the range of “−1” to “1”, and these are expressed as α11 to αnn. These correlation coefficients α are closer to “1” as the positive correlation is larger, closer to “−1” as the negative correlation is larger, and “0” when there is no correlation with each other. For example, since the correlation coefficient for the same combination of model parameters is “1”, the correlation coefficients α11, α22, and αnn are represented by “1” in FIG. Further, the matrix formed by these correlation coefficients α is always a symmetric matrix due to the basic property of correlation.

  Next, the data processing system 114 performs principal component analysis on the correlation coefficient obtained in step S124 to obtain a principal component (step S125). Here, even if the correlation coefficient α shown in FIG. 13 is obtained, a normal circuit simulator cannot generate random numbers according to a normal distribution having a correlation with each other. On the other hand, since it is easy to generate random numbers according to the normal distribution independent of each other with a normal circuit simulator, using such random numbers, random numbers according to the normal distribution having a correlation coefficient α are generated by the circuit simulator. Generate in. For this purpose, statistical principal component analysis is used. In general, principal component analysis used in statistics is to derive principal components, which are independent quantitative variables, from quantitative variables that are correlated with each other, and to contribute to the statistical variation of the original quantitative variables. By ignoring small principal components, it is regarded as a method of multivariate analysis that classifies and characterizes data with fewer variables or dimensions. However, in statistical models, the purpose is not to classify and characterize data in this way, but to derive principal components that are independent quantitative variables and to convert the original quantitative variables by such principal components. There is to ask. In addition, the principal components can be arranged in order from the largest contribution to the standard deviation of the quantitative variable. Therefore, if the distribution can be reproduced up to the specified percentage of the fluctuation range indicated by the standard deviation of the quantitative variable, it is not necessary to use the same number of principal components as the quantitative variable. Since it is sufficient to use the main components up to the specified ratio, the number of quantitative variables sufficient to represent the variation is reduced, and the load on the circuit simulator is reduced, while the variation in the electrical characteristics of the circuit is accurately performed. It also has the advantage of good circuit simulation.

In order to explain how to determine the principal component, consider n quantitative variables xj (j = 1, 2,..., N) according to a normal distribution. Because of the mathematical ease of handling, the quantitative variable xj is obtained by subtracting the average value mj (j = 1, 2,..., N) of the quantitative variable xj as shown in Equation (4). By dividing by the standard deviation σj (j = 1, 2,..., N), it is converted into a quantitative variable uj (j = 1, 2,..., N). Such conversion is usually called standardization.
uj = (xj−mj) / σj (4)
Here, since the quantitative variable xj follows a normal distribution having an average value mj and a standard deviation σj, the quantitative variable uj follows a normal distribution having an average value 0 and a standard deviation 1.

According to principal component analysis, a linear combination of these quantitative variables uj can be generated and converted into quantitative variables zi (j = 1, 2,..., N) according to n independent normal distributions. Are known. To execute it, first, a correlation coefficient matrix between the quantitative variables uj (j = 1, 2,..., N) is statistically obtained. However, it is statistically derived that the correlation coefficient matrix does not change before and after the standardization of quantitative variables. For example, in the example of FIG. 13, the numerical array is a component of a correlation coefficient matrix between model parameters P1_A to Pn_A that are quantitative variables, and these are standardized model parameters P1_A to Pn_A. It is equal to the component of the later correlation coefficient matrix. A linear combination of the above-described quantitative variable uj is expressed as in Equation (5).
zj = aj1 · u1 + aj2 · u2 + ... + ajn · un (5)
Here, j = 1, 2,..., N, and coefficients aj1, aj2,..., Ajn are coefficients that have not yet been determined. Therefore, the linear combination zj of the quantitative variable uj is also a quantitative variable whose value is not yet determined.

The coefficients of the n expressions represented by the above expression (5) are the n vectors (a11, a12) in which the variance of z1, z2,..., Zn is large in this order and the coefficient of the expression (5) is generated. , ..., a1n), (a21, a22, ..., a2n), ..., (an1, an2, ..., ann) are determined to be 1 and orthogonal to each other. I know mathematically that In order to do that, first, a mathematical eigenvalue problem is solved for the correlation coefficient matrix obtained from n quantitative variables uj. When the correlation coefficient matrix is an n-by-n matrix, there are n eigenvalues, which can be obtained by solving the secular equation | A−λI | = 0. Here, A is an n-by-n correlation coefficient matrix obtained from the quantitative variable uj, and I is an n-order unit matrix. The eigenvalues obtained in this way are λ1, λ2,..., Λj,..., Λn, and the eigenvectors belonging to the eigenvalue λj are (aj1, aj2,..., Ajn). The eigenvectors (aj1, aj2,..., Ajn) obtained by solving the eigenvalue problem in this way are n vectors (a11, a12,..., A1n) generated by the coefficients of the equation (5). , (A21, a22,..., A2n),..., (An1, an2,..., Ann), and the linear combination zj of equation (5) is given by equation (6) Is required statistically.
zj = {λj ^ (1/2)} · Nj (1, 0) (6)
Here, j = 1, 2,..., N is an integer value, and Nj (1, 0) is an average value of 0 and a standard deviation of 1, and n quantitative variables that follow normal distributions independent of each other. Represent. Thus obtained zj in the equation (6) is the main component. As described above, the eigenvalues λ1, λ2,..., Λj,..., Λn are assumed to be larger in this order, so that the principal components z1, z2,. .., Zj,..., Zn have a large variance in this order. Usually, in statistics, the objective is to find such principal components zj and to classify and characterize statistical variables with these principal components zj instead of the original quantitative variables. The contents to be described later are specific to a method necessary for creating a statistical model for a circuit simulator, and are not principal component analysis itself.

  Next, the data processing system 114 uses the principal component obtained in the above step S125, the average value and standard deviation obtained in step S123, and the correlation coefficient obtained in step S124. Is obtained (step S126). Hereinafter, a specific method for obtaining a mathematical expression representing a conventional statistical model will be described.

First, when the above equation (5) is substituted into equation (6), equation (7) is obtained.
{Λj ^ (1/2)} · Nj (1, 0)
= Aj1 · u1 + aj2 · u2 + ... + ajn · un (7)
Here, j = 1, 2,..., N is an integer value. Each of the vectors (a11, a12,..., A1n), (a21, a22,..., A2n),..., (An1, an2,. The size is 1 and they are orthogonal to each other. Therefore, the matrix created by the coefficients is an orthogonal matrix, and its transposed matrix is equal to the inverse of the original matrix. That is, u1, u2,..., Un can be solved by multiplying both sides of Equation (7) by a transposed matrix of a matrix formed by coefficients. The result is as shown in equation (8).
uj = a1j · {λ1 ^ (1/2)} · N1 (1, 0)
+ A2j · {λ2 ^ (1/2)} · N2 (1, 0)
+ ...
+ Anj · {λn ^ (1/2)} · Nn (1, 0) (8)

From the above equation (8), the quantitative variables uj (j = 1, 2,..., N) correlated with each other are n independent variables Nj having a normal distribution with mean value 0 and standard deviation 1. It can be seen that it is represented by a linear combination of (1, 0). In the equation (8), when the fluctuation amount of a predetermined target value (for example, 90% or more) of the fluctuation amount of the quantitative variable uj should be reproduced in the circuit simulation, the integer subscript j of the eigenvalue λj The smaller the value, the larger the value of the eigenvalue, so that the contribution to the standard deviation of zj, that is, the standard deviation of the quantitative variable uj increases as shown in Equation (6). By adopting up to the point of entry and ignoring the remaining principal components, the number of random numbers generated in the circuit simulation can be reduced, and the load on the computer can be reduced. In addition, since uj (j = 1, 2,..., N) in equation (8) is standardized as in equation (4), quantitative variable xj (j = 1, 2,..., If n) represents equation (8), equation (9) is obtained.
xj = mj + σj · [a1j · {λ1 ^ (1/2)} · N1 (1, 0)
+ A2j · {λ2 ^ (1/2)} · N2 (1, 0)
+ ...
+ Anj · {λn ^ (1/2)} · Nn (1, 0)] (9)
Here, since the average value of the quantitative variable uj is 0 and the standard deviation is 1, the average value of the quantitative variable xj is mj and the standard deviation is σj. Further, the correlation coefficient between the quantitative variables xj is the same as the correlation coefficient between the quantitative variables uj from the basis of statistics. In order to simplify the coefficients aij · {λi ^ (1/2)} (i, j = 1, 2,..., N) in equation (9), bij (i, j = 1, 2 respectively). ,..., N), equation (9) can be written as equation (10).
xj = mj + σj · {b1j · N1 (1, 0)
+ B2j · N2 (2, 0)
+ ...
+ Bnj · Nn (n, 0)} (10)

If n model parameters Pj_A (j = 1, 2,..., N) that are desired to incorporate variations in the manufacturing process of the transistor A are applied to the quantitative variable xj in Expression (10), Expression (11) It becomes like this.
Pj_A = Pj_Am + Pj_Aσ · {b1j · N1 (1, 0)
+ B2j · N2 (2, 0)
+ ...
+ Bnj · Nn (n, 0)} (11)
Here, Pj_A (j = 1, 2,..., N): model parameter, Pj_Am (j = 1, 2,..., N): average value of Pj_A, Pj_Aσ (j = 1, 2,. .., N): Pj_A standard deviation, Nj (1, 0) (j = 1, 2,..., N): a quantitative variable according to an independent normal distribution of mean value 0 and standard deviation 1. .

  The above equation (11) is an equation that gives n model parameters Pj_A for which a variation in the manufacturing process of the transistor A is to be taken in by a conventional statistical model. The n model parameters Pj_A may be all model parameters necessary for the element model used for the transistor A or may be a part of them. The equation (11) thus obtained is directly written as text data in the circuit model 115 where the model parameters of the element model on the circuit diagram corresponding to the transistor A of the circuit simulator 115 are described. Model parameters of conventional statistical models can be incorporated.

When the circuit simulation is actually performed, the circuit simulator 115 generates a random number according to a normal distribution having an average value of 0 and a standard deviation of 1 independently at Nj (1, 0) in Expression (11). By substituting the value of the random number every time, the circuit simulation can be performed by a normal Monte Carlo simulation or the like. The distribution of model parameters Pj_A obtained by generating a large number of random numbers has an average value Pj_Am, a standard deviation Pj_Aσ, and a correlation coefficient between the distributions of Pj_A is a correlation coefficient obtained from an actual model parameter extraction value. It becomes equal to α (see FIG. 13).
J. et al. A. Power, D.C. Barry, A.M. Mathewson, and W.M. A. Lane, "ACCURATE AND EFFICENT PREDICTIONS OF STATIONIC ALCIRCUIT PERFORMANCE SPREADS", IEEE 1992 CUSTOM INTEGRATED CIRCUIT CONFERENCE, May 1992, p. 3.3.1-3.4.

  The model parameters extracted by the above-described conventional statistical model model parameter extraction method are given to the entire circuit to be designed. For example, when the mask layout on the semiconductor chip S1 (see FIG. 10) is realized, N1 (1, 0) to Nn in the above equation (11) for all transistors on the semiconductor chip S1 corresponding to the element model. Model parameters calculated by independently generating random numbers according to a normal distribution with a mean value of 0 and a standard deviation of 1 are applied to n variables of (1, 0). In this case, every time n independent random numbers are generated, the value of the model parameter set changes, and after the random numbers are generated many times, the distribution of the model parameters reflects the process variation.

  However, since the relative variation between the model parameters in all the transistors arranged on the semiconductor chip S1 corresponding to the element model is not modeled, no relative variation occurs. Actually, there is always a processing variation in each step of the semiconductor process in the semiconductor chip S1, and for example, no matter how uniform the size of the emitter opening window of the bipolar transistor is processed, the processing variation always occurs in the chip. . For example, consider the transistor B arranged at the closest distance from the transistor A shown in FIG. 10 and the transistor C arranged at a distance more than the closest distance. Between the combinations of the transistors A and B and A and C in the semiconductor chip S1, there is always a processing variation at each stage of the semiconductor process, which means that there is no completely identical transistor.

  Here, when the degree of processing variation between the transistors A and B and A and C in the semiconductor chip S1 is compared, the processing variation varies because the distance between the transistors A and B is closer. Is generally known to be small. Although it is said that the absolute accuracy with respect to the electrical characteristics of the elements arranged on the semiconductor chip S1 is generally worse than that produced as a single component, the relative accuracy is better than that produced as a single component. . Further, when the distance between each other in the same chip is close as in the combination of the transistors A and B, the processing variation becomes smaller and the relative variation of the element characteristics becomes smaller. In the semiconductor field, in order to cover the point where the absolute accuracy of the electrical characteristics is worse than that of the single component as described above, a differential circuit configuration utilizing the fact that the relative fluctuation of the element characteristics in the semiconductor chip is small. It has been used as an indispensable component, and as a result, it has succeeded in manufacturing a stable circuit with small characteristic fluctuation for each chip.

  The operation principle of the above-mentioned differential circuit is based on the premise that the characteristics of transistors of the same type are ideally perfectly matched, and there is a difference in characteristics between combinations of transistors using differential. In this case, the output of the circuit is sensitively affected as if due to the difference in the differential input applied to the differential circuit or part of the circuit. For this reason, the greater the difference in transistor characteristics between the differential parts actually manufactured, the greater the deviation from the ideal circuit characteristics that should be. Therefore, in the case of a differential circuit, how close the characteristics of the transistors used for the differential are also an important factor for achieving the target circuit characteristics, but it is extracted by the conventional extraction method. The statistical model parameters using the principal components did not take into account such variations in the relative characteristics of the transistors.

  For example, in the case of the arrangement example shown in FIG. 10, transistor pairs or the like that are required for differential in the differential circuit are located as close as the arrangement rules allow, such as the transistors A and B in the semiconductor chip S1. Arranging the elements in the circuit approaches the ideal circuit characteristics. However, the number of transistors in the transistor pair that is the key to differential is not always two, and many combinations are usually required, or there are many pairs of points to be considered in the circuit. . Further, when the designed circuit is arranged on the semiconductor chip S1 as a mask layout, a pair which is a key of the differential is arranged at the nearest distance like the relationship between the transistors A and B due to layout restrictions. It is not always possible. In addition, based on circuit theory and experience related to differential circuits, there is knowledge about the relative fluctuations in the characteristics of which pairs in the circuit from the stage before circuit design has a major impact on the fluctuations in circuit characteristics. If it is obtained to some extent, it is possible to obtain a quantitatively accurate result, consider a circuit for which knowledge is not obtained in advance, or whether a portion that is not a differential pair relatively affects the fluctuation of characteristics. I need to think about it.

  In order to obtain these answers, the larger and more complex the circuit becomes, the more difficult it becomes by hand calculation based on theory and intuition based on experience, and it cannot be realized without a circuit simulator. However, as described above, the statistical model using the model parameters extracted by the conventional extraction method does not include any information on the relative characteristic variation in the chip, and thus cannot obtain these answers. . When such an answer is unknown, it is possible that a large amount of non-standard circuit characteristics may be generated due to a relative variation in element characteristics that does not match the actual distribution. Furthermore, when the cause of the fluctuation of the circuit characteristics is the relative fluctuation of the element characteristics, in order to make the circuit characteristics within the standard, which element combination has a quantitative influence, how much of each element Since it is not clear whether the relative positional relationship should be optimized, it is not possible to devise an occurrence countermeasure. In addition, it has been necessary to repeatedly make a prototype by correcting the circuit configuration and the circuit mask layout in a thought-and-error manner, which may increase the cost and time for development.

  Therefore, an object of the present invention is to provide a model parameter extraction method and a circuit simulation system in which a correlation of characteristics depending on a position where an element is arranged is reflected in a model parameter.

  In order to achieve the above object, the present invention employs the following configuration. Note that the reference numerals in parentheses, step numbers, and the like indicate correspondence with the embodiments described later in order to help understanding of the present invention, and do not limit the scope of the present invention.

  In the model parameter extraction method of the present invention, parameters of a model formula are extracted from measurement data measured with a plurality of elements (A1, B1) as a pair. The model parameter extracting method includes a step (S1) of measuring predetermined characteristic data by pairing a plurality of elements, a step (S2) of extracting each model parameter from the characteristic data of the plurality of elements, A step (S3) of estimating an average value and a standard deviation of a population in each model parameter using the sample value, and a correlation between each model parameter between the populations using the sample value of each model parameter Using the mean value, the standard deviation, the correlation coefficient, and the principal component, the step of estimating the number (S4), the principal component analysis of the correlation coefficient to obtain the principal component (S6), and the model parameter A parameter indicating the statistical distribution of the population and a correlation depending on the relative positional relationship between the populations of different elements. And a step (S6) of obtaining a formula which gives a statistical model expressed by the motor.

  The step of obtaining the above mathematical expression further uses a mean value, a standard deviation, a correlation coefficient, and a principal component to simultaneously calculate a parameter indicating a statistical distribution of a population of model parameters and a correlation depending on a relative positional relationship. You may obtain the numerical formula which gives the statistical model represented with the parameter which shows the correlation between the populations of the same element (S14).

  For example, a plurality of element pairs are arranged within a predetermined boundary (S1). In this case, as an example, the measuring step measures characteristic data in a plurality of element pairs at a plurality of positions (A2, B2) within the boundary. The step of obtaining the mathematical formula further shows the statistical distribution of the population of model parameters using the mean value, standard deviation, correlation coefficient, and principal component, and also depends on the position of the pair within the boundary. And a mathematical formula that gives a statistical model expressed by a parameter indicating a correlation depending on a position of a pair in a boundary between populations of the same element simultaneously with a correlation depending on a relative positional relationship.

  As another example, the measuring step measures the characteristic data by changing the position within the boundary and the relative positional relationship between a plurality of pairs of elements (B3, B4). The step of obtaining the mathematical formula further shows the statistical distribution of the population of model parameters using the mean value, standard deviation, correlation coefficient, and principal component, and also depends on the position of the pair within the boundary. And a mathematical formula that gives a statistical model expressed by a parameter indicating a correlation depending on a position of a pair in a boundary between populations of the same element simultaneously with a correlation depending on a relative positional relationship.

  Specifically, the plurality of elements forming the pair are arranged closest to each other based on a predetermined arrangement rule.

  The circuit simulation system of the present invention extracts a parameter of a model formula from measurement data measured with a plurality of elements as a pair, and performs circuit simulation. The circuit simulation system includes a measurement unit (112) that measures predetermined characteristic data with a plurality of elements as a pair (111), an extraction unit (113) that extracts each model parameter from the characteristic data measured by the measurement unit, A data processing unit (114) that calculates a mathematical formula that gives a statistical model using the model parameters extracted by the extraction unit, and a circuit simulator (115) that performs circuit simulation incorporating the statistical model using the mathematical formula calculated by the data processing unit With. The data processing unit uses a sample value of each model parameter extracted by the extraction unit, statistical distribution estimation means (S3) for estimating the average value and standard deviation of the population in each model parameter, and the extraction unit Correlation coefficient estimation means (S4) for estimating a correlation coefficient between populations of each model parameter using the sample values of each extracted model parameter, and a correlation coefficient estimated by the correlation coefficient estimation means Principal component analysis means (S5) for obtaining principal components through principal component analysis, statistical distribution estimation means, correlation coefficient estimation means, and mean values, standard deviations, correlation coefficients, and principal values obtained by principal component analysis means Parameters that indicate the statistical distribution of the model parameter population using components and correlations that depend on the relative positional relationships between the populations of different elements Calculating a formula that gives the statistical model expressed in and a formula calculation means (S6).

  The above mathematical formula calculation means further uses the average value, standard deviation, correlation coefficient, and principal component, and is the same as the parameter indicating the statistical distribution of the population of model parameters simultaneously with the correlation depending on the relative positional relationship. A mathematical expression that gives a statistical model represented by a parameter indicating the correlation between element populations may be calculated.

  For example, a plurality of element pairs are arranged within a predetermined boundary. In this case, as an example, the measurement unit measures characteristic data in a plurality of pairs of elements at a plurality of positions within the boundary. The mathematical formula calculation means further shows a statistical distribution of the population of model parameters using the average value, the standard deviation, the correlation coefficient, and the principal component, and a parameter depending on the position of the pair within the boundary. Then, a mathematical expression that gives a statistical model expressed by a parameter indicating a correlation depending on a position of a pair in a boundary between populations of the same element simultaneously with a correlation depending on a relative positional relationship is calculated.

  As another example, the measurement unit measures the characteristic data by changing the position in the boundary and the relative positional relationship between a plurality of pairs of elements. The mathematical formula calculation means further shows a statistical distribution of the population of model parameters using the average value, the standard deviation, the correlation coefficient, and the principal component, and a parameter depending on the position of the pair within the boundary. Then, a mathematical expression that gives a statistical model expressed by a parameter indicating a correlation depending on a position of a pair in a boundary between populations of the same element simultaneously with a correlation depending on a relative positional relationship is calculated.

  Specifically, the plurality of elements forming the pair are arranged closest to each other based on a predetermined arrangement rule.

  According to the model parameter extraction method of the present invention, the correlation of characteristics depending on the relative positional relationship of each device in the system composed of a plurality of target devices can be accurately incorporated into the element model, It becomes possible to calculate the distribution of characteristics closer to reality. As a result, when predicting the distribution due to process fluctuations in circuit characteristics, it is possible to improve the prediction accuracy and reduce the cost and time required for development by reducing the number of prototypes, etc. It will be.

  In addition, when obtaining a mathematical formula that gives a statistical model including a parameter indicating a correlation between populations of the same element as well as a correlation depending on the relative positional relationship, the correlation between the model parameters in a plurality of elements It is possible to further predict the distribution of characteristics incorporating the correlation between model parameters in the element.

  In addition, when obtaining a mathematical formula that gives a statistical model that includes a parameter that depends on the position of a pair within the boundary, the distribution of characteristics that depend on the position within the boundary, such as the characteristics of elements that change due to the stress that occurs within the boundary. Further predictions can be made.

  Also, when changing the relative positional relationship between multiple elements and obtaining a mathematical formula that gives a statistical model that includes parameters that depend on the relative positional relationship, the distance between multiple elements is fixed due to various constraints. Even if the relative positional relationship changes in a circuit that cannot be used, the distribution of the characteristics can be further predicted.

  Further, even in the case of a differential circuit formed by a pair arranged closest to each other based on a predetermined arrangement rule, a distribution that takes into account variations in their characteristics can be predicted.

  According to the circuit simulation system of the present invention, the same effect as the model parameter extraction method described above can be obtained.

(First embodiment)
Hereinafter, a model parameter extraction method and a circuit simulation system using the method according to a first embodiment of the present invention will be described with reference to the drawings. FIG. 1 is a diagram showing a concept of a system used by the model parameter extraction method, and FIG. 2 is a flowchart showing the model parameter extraction method. Note that the block diagram of the circuit simulation system is the same as FIG. 11 described in the background art, and thus detailed description thereof is omitted. However, as will be described later, when the model parameter extraction method is used, the method of selecting a sample in the measured system 111 and the data processing method in the data processing system 114 are different.

  In FIG. 1, devices A1 and B1 are arranged based on a relative positional relationship within a system boundary S1 composed of a plurality of devices. For example, in the semiconductor integrated circuit field, the boundary S1 is a boundary of a semiconductor chip or a boundary of a circuit block. The devices A1 and B1 are elements such as transistors, for example. The devices A1 and B1 are arranged close to each other at the closest distance allowed by the element arrangement rule, and are arranged near the center in the boundary S1. The devices A1 and B1 are assumed to be an essential part of a circuit that utilizes the fact that relative fluctuations in mutual element characteristics are small in a differential circuit or the like, and TEG (Test Element Group) Are arranged on the measurement pattern. The devices A1 and B1 are provided with electrical wiring that can perform a series of measurements in order to extract model parameters.

  Hereinafter, for the sake of specific explanation, the boundary S1 is a semiconductor chip boundary (hereinafter referred to as a semiconductor chip S1), and the devices A1 and B1 are both bipolar transistors (hereinafter referred to as transistors A1 and B1). Let the element model be Gummel-Poon. In addition, variations in the characteristics of the transistors A1 and B1 due to differences in the manufacturing time of the wafer on which the semiconductor chip S1 is mounted, the difference in the lot of the manufacturing wafer, the position occupied by the semiconductor chip S1 in the manufacturing wafer, and the like. Consider the case where the electrical characteristics of the differential circuit are particularly strongly affected by the relative fluctuation of the current amplification factor hFE of B1 (that is, the relative fluctuation of the model parameter BF that is closely related to the current amplification factor hFE). .

  FIG. 2 shows a flowchart of a model parameter extraction technique for a statistical model that can incorporate relative variations in the characteristics of transistors A1 and B1. In FIG. 2, the measurement system 112 measures the electrical characteristics of the transistors A1 and B1 formed on the semiconductor chip S1 (step S1). At this time, as described above, in the transistor pair A1 and B1, the change in the relative characteristics on the differential circuit formed on the semiconductor chip S1 is necessary to affect the change in the electrical characteristics of the circuit. In addition, the measurement system 112 sufficiently reflects statistical fluctuations due to differences in timing of manufacturing a wafer on which the semiconductor chip S1 is mounted, differences in the lot of manufactured wafers, positions occupied by the semiconductor chips S1 in the manufactured wafer, and the like. Select a sample and measure many samples. The measurement system 112 always collects data of the paired transistors A1 and B1 from the same chip, and clearly records the correspondence relationship of the data collected from the same chip. Here, in step S1, the system 111 to be measured is selected from the transistors A1 and B1, the semiconductor chip S1 on which the transistors A1 and B1 are mounted, or the wafer on which the semiconductor chip S1 is mounted. Measuring.

  Next, the extraction system 113 extracts model parameters from the measurement results of the transistors A1 and B1 obtained in step S1 (step S2). In step S2, the extraction system 113 sets the simulation conditions on the circuit diagram of the circuit simulator 115 so as to be equal to the measurement conditions of the characteristic curve obtained from the measurement system 112. Then, the extraction system 113 compares the characteristic curve obtained from the simulation result performed by changing the value of the model parameter in the element model used in the circuit simulator 115 with the characteristic curve obtained from the measurement system 112, The model parameters are determined so that they match within the specified error range. A number of transistors A1 and B1 measured by the measurement system 112 are statistically based on the timing of manufacturing the wafer on which they are mounted, the difference in the lot of the manufactured wafer, the position of the semiconductor chip S1 in the manufactured wafer, and the like. Since it is selected to reflect the fluctuation sufficiently, it shows different electrical characteristics for each measurement in a way that reflects the fluctuation of the manufacturing process. Therefore, the model parameter of the element model extracted in step S2 has a distribution reflecting the variation of the manufacturing process, and information on the correlation between the model parameters BF obtained from the transistors A1 and B1 is also the manufacturing process. It is included in the extracted value in a way that reflects the fluctuation.

  Next, the data processing system 114 statistically processes the sample values of the model parameters of the transistors A1 and B1 extracted in step S2 (step S3).

  For example, if it is statistically estimated from the sample value distribution of measured model parameters that the population distribution of each model parameter in the entire manufacturing process follows a normal distribution, the population that follows the normal distribution from the sample value distribution The mean and standard deviation of are statistically estimated. The sample values for obtaining these average values and standard deviations are statistics based on differences in the manufacturing time of the wafer on which the semiconductor chip S1 is mounted, the manufacturing wafer lot, the position occupied by the semiconductor chip S1 in the manufacturing wafer, and the like. Therefore, a slight positional shift in the chips of the transistors A1 and B1 does not affect the average value and the standard deviation. Therefore, one of the transistors A1 and B1 may be represented. The distribution of model parameters using these can be expressed by the above equation (1) similar to the background art.

  Here, the model parameter P of the above equation (1) is incorporated into the model parameter of the element model, and random numbers normally distributed with an average value of 0 and a standard deviation of 1 are generated in the circuit simulator 115 a number of times. 0), the distribution of the model parameter P values reproduces the distribution of the model parameter population extracted in step S2. However, this stage models the fact that the characteristics of the element affect the circuit characteristics by changing together in the entire circuit in the chip due to the influence of the fluctuations throughout the manufacturing process. The relative variation of the device characteristics is not incorporated in the device model. In the case of the differential circuit described above, it is important to estimate this relative variation.

  Next, the data processing system 114 uses the distribution data for the sample values of the model parameters extracted in step S2 to statistically calculate the correlation coefficient between the distributions of the model parameters for the transistors A1 and B1. (Step S4).

  FIG. 3 is an example of a correlation coefficient between the model parameter BF of the transistor A1 and the model parameter BF of the transistor B1 obtained by the process of step S4. In FIG. 3, model parameters BF of the transistors A1 and B1 are indicated by BF_A1 and BF_B1, respectively. The value at the intersection of the matrix formed by the model parameters BF_A1 and BF_B1 indicates the correlation coefficient of the combination of the vertical model parameter, the horizontal model parameter, and the data. The correlation coefficient of the combination of the model parameters BF takes real values ranging from “−1” to “1”, and these are expressed by αaa, αab, αba, and αbb. These correlation coefficients α are closer to “1” as the positive correlation is larger, closer to “−1” as the negative correlation is larger, and “0” when there is no correlation with each other. For example, since the correlation coefficient for the same combination of model parameters is “1”, the correlation coefficients αaa and αbb are indicated by “1” in FIG. Even in the case of a combination of different model parameters BF, as described above, the transistors A1 and B1 are bipolar transistors, and the model parameters BF fluctuate with correlation. Since the transistors A1 and B1 are the same type of elements having a minimum proximity distance, the characteristics of the respective transistors are usually close to each other, and the correlation coefficient (that is, αab and αba) between them is “1”. Has a close value. Further, the matrix formed by these correlation coefficients α is always a symmetric matrix due to the basic property of correlation. The off-diagonal elements in FIG. 3 (ie αab and αba) are equal.

  Next, the data processing system 114 performs principal component analysis on the correlation coefficient matrix formed by the correlation coefficient obtained in step S4 to obtain a principal component (step S5). Here, even if the correlation coefficient α between the model parameters BF of the transistors A1 and B1 shown in FIG. 3 is obtained, a normal circuit simulator may generate random numbers according to a normal distribution having a correlation with each other. Can not. On the other hand, since it is easy to generate random numbers according to the normal distribution independent of each other with a normal circuit simulator, using such random numbers, random numbers according to the normal distribution having a correlation coefficient α are generated by the circuit simulator. Generate in. For this purpose, statistical principal component analysis is used.

In the principal component analysis performed in step S5, standardization is performed using the above formula (4) described in the background art for ease of mathematical handling. Then, the linear combination of the quantitative variables derived by the above equation (4) is calculated by the above equation (5). Since these methods are the same as those in the background art, detailed description thereof is omitted. When the correlation coefficient matrix shown in FIG. 3 has 2 rows and 2 columns, the principal component is given by Equation (12) where n = 2 in Equation (5) above.
zj = aj1 · u1 + aj2 · u2 (12)
Here, j = 1 and 2 and u1 and u2 are quantitative variables obtained by standardizing model parameters BF_A1 and BF_B1, respectively.

The variances of z1 and z2 shown in the above equation (12) are large in this order, and the two vectors (a11, a12) and (a21, a22) created by the coefficients of the equation (12) Eigenvectors obtained by solving a mathematical eigenvalue problem for a column correlation coefficient matrix, each having a magnitude of 1 and orthogonal to each other. If eigenvalues obtained by solving this eigenvalue problem are λ1 and λ2 in descending order, eigenvectors belonging to eigenvalues λ1 and λ2 are (a11, a12) and (a21, a22), respectively. Further, as described in the background art, the principal components z1 and z2 in the above equation (12) have a relationship as in the equation (13) between the eigenvalues λ1 and λ2.
zj = {λj ^ (1/2)} · Nj (1, 0) (13)
Here, j is an integer value of 1 and 2, and Nj (1, 0) is an average value of 0 and a standard deviation of 1 and represents two quantitative variables according to a normal distribution independent of each other. Normally, statistics end up to the point where such principal components z1 and z2 are obtained, and the contents described later are specific for creating a statistical model.

  Next, the data processing system 114 gives a statistical model using the principal component obtained in step S5, the average value and standard deviation obtained in step S3, and the correlation coefficient obtained in step S4. A mathematical formula is obtained (step S6). Here, the correlation coefficient used in the statistical model described in the background art is for model parameters in the same element model that gives the characteristics of the transistor A (see FIG. 10). On the other hand, in the case of the statistical model of the present invention, the correlation coefficient depending on the relative positional relationship between the element model parameters in the semiconductor chip S1 corresponding to each of the transistors A1 and B1 is considered. The same correlation is used, but it is completely different. Hereinafter, a specific method for obtaining a mathematical expression representing the statistical model of the present invention will be described.

First, when the formula (12) is substituted into the formula (13), the formula (14) is obtained.
{Λj ^ (1/2)} · Nj (1, 0) = aj1 · u1 + aj2 · u2 (14)
Here, the magnitudes of the vectors (a11, a12) and (a21, a22) formed by the coefficients on the right side are 1 and are orthogonal to each other. Therefore, the matrix created by the coefficients is an orthogonal matrix, and its transposed matrix is equal to the inverse of the original matrix. That is, u1 and u2 can be solved by multiplying both sides of the equation (14) by a transposed matrix of a matrix formed by coefficients. The result is as shown in equation (15).
uj = a1j · {λ1 ^ (1/2)} · N1 (1, 0)
+ A2j · {λ2 ^ (1/2)} · N2 (1, 0) (15)

From the above equation (15), the quantitative variables uj (j = 1, 2) that are correlated with each other are variables N1 (1, 0) and N2 that follow a normal distribution with two independent mean values 0 and standard deviation 1. It can be seen that it is represented by a linear combination of (1, 0). In Expression (14), since the quantitative variable uj (j = 1, 2) is standardized, Expression (16) and Expression (15) are respectively expressed by Expression (15) with the original quantitative variables BF_A1 and BF_B1. (17)
BF_A1 = BF_A1m
+ BF_A1σ × [a11 · {λ1 ^ (1/2)} · N1 (1, 0)
+ A21 · {λ2 ^ (1/2)} · N2 (1, 0)]
... (16)
BF_B1 = BF_B1m
+ BF_B1σ × [a12 · {λ1 ^ (1/2)} · N1 (1, 0)
+ A22 · {λ2 ^ (1/2)} · N2 (1, 0)]
... (17)
Here, BF_A1m and BF_B1m are average values of the distribution of the model parameters BF_A1 and BF_B1, respectively. BF_A1σ and BF_B1σ are standard deviations of the distribution of the model parameters BF_A1 and BF_B1, respectively. N1 (1, 0) and N2 (1, 0) are quantitative variables according to normal distributions having an average value of 0 and a standard deviation of 1, respectively, and independent from each other.

Further, in the above equations (16) and (17), the coefficients aij · {λi ^ (1/2)} (i, j = 1, 2) are respectively expressed as bij (i, j = 1, 2).
bij = aij · {λi ^ (1/2)} (18)
Then, from the above equation (18), the above equations (16) and (17) become equations (19) and (20), respectively.
BF_A1 = BF_A1m
+ BF_A1σ × {b11 · N1 (1, 0) + b21 · N2 (1, 0)}
... (19)
BF_B1 = BF_B1m
+ BF_B1σ × {b12 · N1 (1, 0) + b22 · N2 (1, 0)}
... (20)
Here, since N1 (1, 0) and N2 (1, 0) in the above formulas (19) and (20) are respectively independent quantitative variables, the phase between the quantitative variables BF_A1 and BF_B1 When the relation number is α_A1B1 and is calculated using the above formulas (19) and (20), formula (21) is obtained.
α_A1B1 = b11 · b12 + b21 · b22 (21)

The correlation coefficient α_A1B1 between the quantitative variables BF_A1 and BF_B1 depends on the relative positional relationship of the transistors A1 and B1 in the semiconductor chip S1 as described above. Therefore, b11, b12, b21, and b22 constituting the right side of the above equation (21) also depend on the relative positional relationship between the transistors A1 and B1. This dependency is explicitly expressed using equation (22).
bij = bij (POS_B1-POS_A1) (22)
Here, POS_A1 and POS_B1 represent the positions of the transistors A1 and B1 in the semiconductor chip S1, respectively. These positions are indicated by positions of specific points defined in the respective transistors A1 and B1 (for example, the center point of the planar structure in the transistor manufacturing process). And (POS_B1-POS_A1) represents an arrow line vector from the position POS_A1 to the position POS_B1. In addition, as described above, the transistors A1 and B1 are the same type, and even if the positions of the transistors A1 and B1 are interchanged, the result does not change, and therefore Equation (23) holds.
bij = bij (POS_A1-POS_B1) (23)

  The above equations (19) and (20) are equations that give the model parameters of the statistical model reflecting the correlation coefficient depending on the relative positional relationship in the semiconductor chip S1 between the transistors A1 and B1. However, bij (i, j = 1, 2) in the above formulas (19) and (20) is represented by aij and λi (i, j = 1, 2) as in the above formula (18). Further, bij depends on the relative positional relationship in the semiconductor chip S1 of the transistors A1 and B1 as in the above formula (22) or formula (23).

  Of the model parameters of the element model for the transistors A1 and B1, the remaining model parameters not included in the correlation coefficient shown in FIG. 3 are derived by the equations (1) to (3) described in the background art. The model formula is applied.

  Then, the description of the model equation obtained by the above equations (19) and (20) is directly written as text data in the description of the model parameters of the element model on the circuit diagram of the circuit simulator 115 corresponding to the transistors A1 and B1. Thus, the circuit simulator 115 performs circuit simulation incorporating a statistical model (step S7). When the circuit simulator 115 actually performs the circuit simulation, the Nj (1, 0) in the above formula (19) and the formula (20) are each independently supplied with a large number of random numbers according to the normal distribution of the average value 0 and the standard deviation 1. Generate the number of times and assign the value. As a result, the circuit simulator 115 can perform circuit simulation by the Monte Carlo method or the like. The distributions of model parameters obtained by generating random numbers many times with respect to the above equations (19) and (20) are the average values of BF_A1m and BF_B1m, the standard deviations of the distributions of BF_A1σ and BF_B1σ, BF_A1 and BF_B1, respectively. The correlation coefficient (see FIG. 3) obtained from the measured model parameter extraction value can be given.

(Second Embodiment)
Hereinafter, a model parameter extraction method and a circuit simulation system using the method according to a second embodiment of the present invention will be described with reference to the drawings. In the first embodiment described above, correlation between model parameters in a plurality of apparatuses is taken in. However, in reality, correlation between model parameters in one apparatus needs to be considered at the same time. In the second embodiment, the circuit simulation is performed by simultaneously incorporating the correlation between the model parameters in the devices into the correlation between the model parameters in the plurality of devices. FIG. 4 is a flowchart showing the model parameter extraction method. Note that the block diagram of the circuit simulation system is the same as FIG. 11 described in the background art, and thus detailed description thereof is omitted. In FIG. 1, the concept of the system used by the model parameter extraction method is the same as that of FIG. 1 used in the first embodiment.

  In FIG. 4, the processes in steps S11 to S13 are the same as steps S1 to S3 described with reference to FIG. 2 in the first embodiment, and thus detailed description thereof is omitted.

  Next, the data processing system 114 uses the distribution data for the sample values of the model parameters extracted in step S12, and uses the correlation coefficient between the populations of the model parameters in the transistor A1 and the model parameters in the transistor B1. A correlation coefficient between populations and a correlation coefficient between populations of model parameters in the transistors A1 and B1 are statistically estimated (step S14). Specifically, what is considered in the first embodiment is between the model parameter BF of the transistor A1 and the model parameter BF of the transistor B1, whereas in this embodiment, the set of model parameters of the transistor A1 The arbitrary number n of model parameters to be considered from the above and the arbitrary number n of model parameters to be considered from the set of model parameters of the transistor B1 are selected and statistically estimated from each model parameter. Find the correlation coefficient between population distributions.

  FIG. 5 is an example of the correlation coefficient obtained by the process of step S14. In FIG. 5, a set of model parameters of the element model corresponding to the transistor A1 is indicated by P1_A1 to Pn_An. A set of model parameters of the element model corresponding to the transistor B1 is indicated by P1_B1 to Pn_Bn. These model parameters do not have to be all necessary for the transistor A1 or B1, and may be a part of them. The values at the intersections of the 2n rows and 2n columns formed by the model parameters P1_A to Pn_A and the model parameters P1_B1 to Pn_Bn indicate the correlation coefficient of the combination of the vertical model parameter and the horizontal model parameter. The correlation coefficient of the combination of model parameters takes a real value in the range of “−1” to “1”. These correlation coefficients are closer to “1” as the positive correlation is larger, closer to “−1” as the negative correlation is larger, and “0” when there is no correlation. For example, the correlation coefficient for the same combination of model parameters is the correlation coefficient “1”.

  The matrix shown in FIG. 5 is an intersection of model parameters of the transistor A1, an intersection of model parameters of the transistor B1, a vertical alignment of the transistor A1, a horizontal alignment of the model parameters of the transistor B1, and a vertical alignment of the transistor. At B1, the horizontal arrangement is divided into four areas where the model parameters of the transistor A1 intersect. Hereinafter, these regions are referred to as regions E11, E22, E12, and E21, respectively. The correlation coefficient α belongs to the area E11, the correlation coefficient β belongs to the area E22, the correlation coefficient γ belongs to the area E12, and the correlation coefficient δ belongs to the area E21.

  Transistors A1 and B1 are considered to be the same type of elements with a minimum proximity distance allowed by the placement rule. Therefore, the model parameters P1_A1 to Pn_A1 of the transistor A1 to be considered for correlation and the model parameters P1_B1 to Pn_B1 of the transistor B1 to be considered for correlation are the same. Then, when the arrangement order is matched, diagonal elements (correlation coefficients) γ11, γ22,..., Γnn and δ11, δ22,. Since the transistor is close to the minimum proximity distance, the value is close to “1”. Further, the non-diagonal elements of the matrices belonging to the regions E12 and E21 are not necessarily close to “1” because they are combinations of different model parameters. The matrices belonging to the regions E12 and E21 depend on the relative positional relationship between the transistors A1 and B1 in the semiconductor chip S1 as in the first embodiment.

  On the other hand, a region E11 is a correlation coefficient between model parameters of the transistor A1, and a region E22 is a correlation coefficient between model parameters of the transistor B1. Therefore, the regions E11 and E22 correspond to the statistical model described in the background art that does not depend on the relative positional relationship between the transistors A1 and B1 in the semiconductor chip S1. For example, since the correlation coefficient for the same combination of model parameters is the correlation coefficient “1”, the correlation coefficients α11, α22, and αnn and the correlation coefficients β11, β22, and βnn are set to “1” in FIG. ". 5 is a symmetric matrix because of its basic statistical properties, the non-diagonal elements of the entire correlation coefficient matrix are combinations of elements in which rows and columns are interchanged. Are all equal.

  Next, the data processing system 114 performs principal component analysis on the correlation coefficient matrix formed by the correlation coefficient obtained in step S14 to obtain a principal component (step S15). In order to generate in the circuit simulator 115 random numbers according to a normal distribution having a correlation coefficient between model parameters in the transistors A1 and B1 shown in FIG. 5, the above-described statistical principal component analysis is used.

First, in the principal component analysis performed in step S15, standardization is performed using the above-described formula (4) described in the first embodiment and the background art for ease of mathematical handling. As shown in FIG. 5, when the correlation coefficient matrix is 2n rows and 2n columns, the principal component is given by Equation (24) in which n = 2n is substituted in Equation (5) above.
zj = aj1 · u1 + aj2 · u2 + ... + aj [2n] · u [2n] (24)
Here, j = 1, 2,..., 2n. In addition, the coefficient and variable subscripts in the above equation (24) are clarified by enclosing them in parentheses [] to distinguish them from others. Hereinafter, the same notation is used for distinction. U1, u2,..., U [2n] are quantitative variables obtained by standardizing the model parameters P1_A1 to Pn_A1 and P1_B1 to Pn_B1, respectively.

The variance of z1, z2,..., Z2n shown in the above equation (24) is large in this order, and 2n vectors (a11, a12,..., A1 [ 2n]), (a21, a22, ..., a2 [2n]), ..., (a [2n] 1, a [2n] 2, ..., a [2n] [2n]) Eigenvectors obtained by solving a mathematical eigenvalue problem for a 2n × 2n column correlation coefficient matrix, each having a magnitude of 1 and orthogonal to each other. If eigenvalues obtained by solving this eigenvalue problem are λ1, λ2,..., Λ [2n] in descending order, the eigenvectors belonging to the eigenvalues λ1, λ2,. , A12, ..., a1 [2n]), (a21, a22, ..., a2 [2n]), ..., (a [2n] 1, a [2n] 2, ..., a [2n] [2n]). Furthermore, as described in the background art, the principal components z1, z2,..., Z [2n] in the above equation (24) are expressed between the eigenvalues λ1, λ2,. There is a relationship like (25).
zj = {λj ^ (1/2)} · Nj (1, 0) (25)
Here, j = 1, 2,..., 2n are integer values, and Nj (1, 0) represents an average value of 0 and a standard deviation of 1 and represents 2n quantitative variables that follow normal distributions independent of each other. .

Next, the data processing system 114 gives a statistical model using the principal component obtained in step S15, the average value and standard deviation obtained in step S13, and the correlation coefficient obtained in step S14. A mathematical formula is obtained (step S16). First, the data processing system 114 substitutes the above equation (25) into the above equation (24). Then, the data processing system 114 solves uj (j = 1, 2,..., [2n]) in the same manner as in the first embodiment described above, and returns to the variable before standardization, thereby obtaining the equation (26). ) And (27).
Pi_A1 = Pi_A1m
+ Pi_A1σ × {b1i · N1 (1, 0)
+ B2i · N2 (1, 0)
+ ...
+ B [2n] i · N [2n] (1, 0)} (26)
Pi_B1 = Pi_B1m
+ Pi_B1σ × {b1i + n · N1 (1, 0)
+ B1i + n · N2 (1, 0)
+ ...
+ B [2n] i + n · N [2n] (1, 0)}
... (27)
Here, in the above formula (26) and formula (27), i = 1, 2,..., N. Also, bij is the eigenvalue λi (i = 1, 2,..., [2n]) obtained by solving the eigenvalue problem of the correlation coefficient matrix shown in FIG. j = 1, 2,... [2n]).
bij = aij · {λi ^ (1/2)} (28)
Here, in formula (28), i = 1, 2,..., 2n.

The quantitative variables of 2n N1 (1, 0), N2 (1, 0),..., N2n (1, 0) in the above formulas (26) and (27) are independent from each other and Although the number of relations is “0”, each model parameter Pi_A1, Pi_B1 (i = 1, 2,..., N) can have a correlation through the coefficient bij, and has the correlation coefficient shown in FIG. It will be. Further, the coefficient bij (i = 1, 2,..., [2n]) includes the dependency on the relative positional relationship of the transistors A1 and B1 in the semiconductor chip S1, and the dependency is expressed by the formula ( 29).
bij = bij (POS_B1-POS_A1) (29)
Here, in the formula (29), i, j = 1, 2,..., 2n.

  Then, the description of the model formula obtained by the above formulas (26) and (27) is directly written as text data in the description of the model parameters of the element model on the circuit diagram of the circuit simulator 115 corresponding to the transistors A1 and B1. For example, the circuit simulator 115 performs circuit simulation incorporating a statistical model (step S17). Normally, the circuit simulator 115 can easily generate random numbers according to normal distributions independent of each other. Therefore, by substituting 2n independent random numbers into the above equations (26) and (27), the model parameter Pi_A1 , Pi_B1 (i = 1, 2,..., N) can satisfy the correlation coefficient shown in FIG.

  In the above description, the two transistors A1 and B1 are arranged at a distance of the closest distance. However, such arrangement is not necessary. For example, as shown in FIG. 6, the same type of four transistors A1, B1, C1, and D1 that are separated by the closest distance can be similarly implemented. In this case, in the same statistical model as in the first and second embodiments described above, it is possible to create a model parameter that incorporates a correlation depending on the positional relationship between the transistors A1, B1, C1, and D1. It goes without saying that the present invention can be similarly implemented with three transistors or five or more transistors separated by the closest distance.

  The transistors A1 and B1 have been described on the assumption that the element manufacturing process has the same plane and cross-sectional structure. However, the combinations in which the relative distribution and correlation of the element characteristics influence the circuit characteristics are not limited to the transistors, but include resistance and the like. Needless to say, the present invention can be applied to combinations of other devices. Further, when the transistors A1 and B1 are bipolar transistors of the same type, the device models used for the bipolar transistors may be different device models by applying Gumel-Poon to A1 and VBIC to B1. For example, the base resistance and the like are used in both element models, and usually have a correlation between the model parameters. Therefore, the model parameter extraction method of the first and second embodiments can be applied.

  Further, as shown in FIG. 7, a semiconductor chip S1 is considered in which an arbitrary element Y1 having a planar structure or a cross-sectional structure that is different from the element A1 is arranged with respect to the element A1. Even in such an element Y1, there is a possibility that the relative distribution or correlation of the element characteristics may be effective on the circuit characteristics depending on the target circuit. In this case, the model parameters of the first and second embodiments 1 are used. Extraction methods can be applied.

(Third embodiment)
Hereinafter, a model parameter extraction method and a circuit simulation system using the method according to a third embodiment of the present invention will be described with reference to the drawings. FIG. 8 is a diagram illustrating a concept of a system used by the model parameter extraction method.

  In FIG. 8, devices A1 and B1, A2 and B2 are arranged in a system boundary S1 composed of a plurality of devices, forming a pair based on a certain relative positional relationship. For example, in the semiconductor integrated circuit field, the boundary S1 is a boundary of a semiconductor chip or a boundary of a circuit block. The devices A1, B1, A2, and B2 are elements such as transistors, for example. The devices A1 and B1 are a pair arranged at the closest distance allowed by the element arrangement rule, and are arranged near the center in the boundary S1. The devices A2 and B2 are a pair arranged at the closest distance allowed by the element arrangement rule, and are arranged at a position farther from the position where the devices A1 and B1 are arranged than the closest distance. Yes. The devices A1 and B1, A2, and B2 are assumed to be the main parts of a circuit that utilizes the fact that relative fluctuations in mutual element characteristics are small in a differential circuit or the like. And arranged on a measurement pattern such as TEG. The devices A1 and B1, A2, and B2 are provided with electrical wiring that can perform a series of measurements in order to extract model parameters. Hereinafter, for the sake of specific explanation, the boundary S1 is defined as a semiconductor chip boundary (hereinafter referred to as a semiconductor chip S1), and the devices A1, B1, A2, and B2 are all bipolar transistors (hereinafter referred to as transistors A1, B1). , A2 and B2. Transistors A1 and B1 are transistor pairs having the same planar structure and cross-sectional structure in the manufacturing process. Transistors A2 and B2 are also transistor pairs having the same planar structure and cross-sectional structure in the manufacturing process.

  The semiconductor chip S1 is separated from the wafer state into a chip state by dicing in the course of the manufacturing process, and packaged with plastic or the like. When the semiconductor chip S1 is packaged, the internal stress generated in the chip varies greatly depending on the position in the chip surface. On the other hand, since the characteristics of the element arranged on the semiconductor chip S1 vary due to the stress, the characteristic variation of the element after packaging has a position dependency on the chip surface. Therefore, when a circuit simulation is performed on the assumption that circuit characteristics vary due to such packaging, the correlation between the model parameters of the transistor varies depending on the transistor arrangement position in the surface of the semiconductor chip S1.

  Such a change is caused by the fact that the coefficients bij in the equations (19) and (20) in the first embodiment are equal to those in the equation (22) or in the equations (26) and (27) in the second embodiment. It cannot be expressed simply that the coefficient bij depends on the relative positional relationship in the chip of the transistors A1 and B1 as shown in the equation (29). For example, in the above formula (22) and formula (29), the position dependency of the coefficient bij of the transistors A1 and B1 is represented by the formula bij = bij (POS_B1-POS_A1). In this embodiment, the dependency of the transistors A1 and B1 on the average position (POS_A1 + POS_B1) / 2 in the semiconductor chip S1, the dependency on the position POS_A1 of the transistor A1 in the semiconductor chip S1, and the semiconductor of the transistor B1 It is necessary to add dependency depending on the position POS_B1 in the chip S1. Here, the dependency due to the average position of the transistors A1 and B1 is added in order to express the position dependency in the correlation between the model parameter of the transistor A1 and the model parameter of the transistor B1. This is because an average position index representing the entire position is necessary.

  By the way, the combination of the respective positions POS_A1 and POS_B1 of the transistors A1 and B1 is designated, and the combination of the difference vector POS_B1−POS_A1 of the positions of the transistors A1 and B1 and the average position vector (POS_A1 + POS_B1) / 2 is designated. Are mathematically equivalent, and if one is known, the other can be derived. However, the dependency of the coefficient bij on the position POS_A1 of the transistor A1 and the dependency on the position POS_B1 of the transistor B1 are added because the correlation between the model parameters of the transistor A1 and the correlation between the model parameters of the transistor B1 are relative to each other. This is to express explicitly that it depends on the absolute position in the semiconductor chip S1 without depending on the specific positional relationship.

Therefore, the dependency of the coefficient bij on the positions of the transistors A1 and B1 in this embodiment is expressed by Expression (30).
bij = bij ((POS_A1 + POS_B1) / 2,
POS_B1-POS_A1, POS_A1, POS_B1)
... (30)
Here, i, j = 1, 2,..., 2n, where n is the number of model parameters of the transistors A1 and B1.

  With respect to the coefficient bij representing the correlation between the model parameters using the above equation (30) instead of using the above equations (22) and (29), the positions of the transistor pairs shown in FIG. 8 are systematically arranged in the semiconductor chip S1. Instead, the absolute position dependency in the semiconductor chip S1 in the transistor pair of Expression (30) is obtained. Thus, depending on the arrangement position of the transistor pair in the semiconductor chip S1, the correlation coefficient between the model parameters of the transistor A1, the correlation coefficient between the model parameters of the transistor B1, and the phase relationship between the model parameters of the transistors A1 and B1. The number can be calculated using the above equations (19), (20), (26), and (27).

Here, when the position of the transistor pair is systematically changed in the semiconductor chip S1, the symmetry of the semiconductor chip S1 is used when an actual measurement pattern or the like cannot be arranged only at a limited position due to a practical restriction. By examining one area (for example, ¼ area) in the chip or using an interpolation method or the like, an area that cannot be actually measured can be compensated. In addition, if there is no need to consider the effect of the above-mentioned distribution of package stress on the chip surface, the relative distribution of characteristics between transistor pairs or the dependence of the correlation on the chip surface can be ignored. (POS_A1 + POS_B1) / 2 dependency, POS_A1 dependency, and POS_B1 dependency are not required, and Equation (31) is obtained. In this case, the coefficient bij depends only on the relative positional relationship of the transistors A1 and B1 in the semiconductor chip S1, and again results in the results of the above equations (22) and (29).
bij = bij (POS_A1-POS_B1) (31)

On the other hand, when there is a package stress as described above, when the positions of the transistor pairs are largely separated from each other in the semiconductor chip, such as the transistor pairs A1 and B1 and A2 and B2 shown in FIG. The mean and standard deviation of the population distribution of model parameters in the transistor can also change. In such a case, taking the transistor A1 as an example, the average value P_A1m and the standard deviation P_A1σ of the model parameters in the equation that gives the model parameters have the dependency on the position POS_A1 of the transistor A1, and this effect is statistically modeled. (32) and (33).
P_A1m = P_A1m (POS_A1) (32)
P_A1σ = P_A1σ (POS_A1) (33)

(Fourth embodiment)
Hereinafter, a model parameter extraction method and a circuit simulation system using the method according to a third embodiment of the present invention will be described with reference to the drawings. FIG. 9 is a diagram illustrating a concept of a system used by the model parameter extraction method.

  In FIG. 9, devices A1 and B1 are arranged in a system boundary S1 composed of a plurality of devices in a pair based on a certain relative positional relationship. For example, in the semiconductor integrated circuit field, the boundary S1 is a boundary of a semiconductor chip or a boundary of a circuit block. The devices A1 and B1 are elements such as transistors, for example. The devices A1 and B1 are a pair arranged at the closest distance allowed by the element arrangement rule, and are arranged near the center in the boundary S1. Hereinafter, for the sake of concrete explanation, the boundary S1 is defined as a semiconductor chip boundary (hereinafter referred to as a semiconductor chip S1), and the devices A1 and B1 are both bipolar transistors (hereinafter referred to as transistors A1 and B1). To do. Transistors A1 and B1 are transistor pairs having the same planar structure and cross-sectional structure in the manufacturing process.

  In the fourth embodiment, transistors B3 and B4 are further arranged in the semiconductor chip S1. The transistor B3 is arranged farther away than the distance from the transistors A1 to B1 and about the closest distance allowed by the arrangement rule. The transistor B3 has the same planar structure and cross-sectional structure as the transistor A1 in terms of the manufacturing process. The transistor B4 is arranged at a position farther from the transistor A1 than the closest distance. The transistor B4 is also formed with the same planar structure and cross-sectional structure in the manufacturing process as the transistor A1.

  In the equations (22), (29), and (30) described in the first to third embodiments, the relative positional relationship between the transistors A1 and B1 in the coefficient bij that represents the correlation between the model parameters is expressed as follows. Regarding the dependency on POS_B1-POS_A1 to be expressed, the absolute value or the size thereof is assumed to be fixed in the range of the closest distance. However, when elements are mask-laid out in an actual circuit, even a transistor pair that dominates circuit characteristics may not be arranged at the closest distance. For example, there are various restrictions, or when the designer does not expect that the transistor pair will dominate the circuit characteristics, and the circuit layout designed for the mask layout needs to be simulated. In this case, the transistor that forms the pair of the transistor A1 shown in FIG. 9 may have to be arranged at the position of the transistors B3 and B4 instead of B1.

  In such a case, it is necessary to consider that POS_B1-POS_A1 in the above formula (22), formula (29), and formula (30) also changes to its absolute value. By the way, unless the chip size of the current semiconductor chip is small, such as 1 mm square, the correlation between the model parameter of the transistor A1 and the model parameter of the paired transistor becomes weaker as the absolute value of POS_B1-POS_A1 increases. Thus, the correlation coefficient approaches “0”. Further, if the control of the process variation of the semiconductor manufacturing process has progressed from the present, and the processing variation in the semiconductor chip is reduced, the model parameter of the transistor A1 and the pair of transistors are paired even when arranged at the chip center and chip end. There may be sufficient correlation between model parameters. When such a long-distance correlation is established, if the direction of the arrow vector POS_A1-POS_B1 changes, the position in the chip changes extremely. Therefore, in the above equation (30), the direction dependence of POS_B1-POS_A1 Sex will also appear prominently.

Further, as the transistors B1, B3, and B4 shown in FIG. 9 are separated from each other, the average value and standard deviation of the population distribution in the model parameters of each transistor may change as the distance from the transistor A1 increases. There is. This can occur remarkably after the semiconductor chip S1 is separated from the wafer by dicing and packaged as described in the third embodiment. In such a case, taking the transistor A1 as an example, the average value P_A1m and the standard deviation P_A1σ of the model parameter in the equation that gives the model parameter depend on the position POS_A1 of the transistor A1 as in the equations (34) and (35). It is necessary to incorporate this effect into the statistical model.
P_A1m = P_A1m (POS_A1) (34)
P_A1σ = P_A1σ (POS_A1) (35)

  As described above, according to the model parameter extraction method and the circuit simulation system of the present invention, the correlation of characteristics depending on the relative positional relationship of each device in a system composed of a plurality of target devices can be obtained. Therefore, it is possible to calculate the distribution of characteristics closer to reality. As a result, when predicting the distribution due to process fluctuations in circuit characteristics, it is possible to improve the prediction accuracy and reduce the cost and time required for development by reducing the number of prototypes, etc. It will be.

  The model parameter extraction method and the circuit simulation system according to the present invention can accurately incorporate the correlation of characteristics depending on the relative positional relationship of each device into an element model, and can be integrated circuits or the like built in a semiconductor chip. It can be applied to applications such as simulating reproduction of characteristics when designing.

The figure showing the concept of the system which the model parameter extraction method concerning the 1st and 2nd embodiment of the present invention uses. The flowchart which shows the model parameter extraction method which concerns on the 1st Embodiment of this invention Example of correlation coefficient obtained by processing of step S4 in FIG. The flowchart which shows the model parameter extraction method which concerns on the 2nd Embodiment of this invention. Example of correlation coefficient obtained by the process of step S14 The figure showing the other example in the concept of the system which the model parameter extraction method of FIG. 1 uses The figure showing the other example in the concept of the system which the model parameter extraction method of FIG. 1 and FIG. 6 uses The figure showing the concept of the system which the model parameter extraction method concerning a 3rd embodiment of the present invention uses. The figure showing the concept of the system which the model parameter extraction method concerning a 4th embodiment of the present invention uses. Examples of measurement patterns for applying conventional statistical model extraction methods Block diagram of a circuit simulation system for applying model parameter extraction techniques The flowchart which shows the procedure of the method of extracting the model parameter of the conventional statistical model Example of correlation coefficient obtained by processing in step S124 of FIG.

Explanation of symbols

111 ... System under test 112 ... Measurement system 113 ... Extraction system 114 ... Data processing system 115 ... Circuit simulator

Claims (10)

A model parameter extraction method for extracting a parameter of a model formula from measurement data measured by pairing a plurality of elements,
Measuring predetermined characteristic data by pairing the plurality of elements;
Extracting each model parameter from characteristic data of the plurality of elements;
Estimating the mean and standard deviation of the population in each model parameter using the sample values of the respective model parameters;
Estimating a correlation coefficient between populations of each model parameter using the sample values of the respective model parameters;
Obtaining a principal component by principal component analysis of the correlation coefficient;
Using the average value, standard deviation, correlation coefficient, and principal component, a parameter indicating a statistical distribution of the population of the model parameters and a correlation depending on a relative positional relationship between the populations of different elements And a step of obtaining a mathematical formula that gives a statistical model represented by the indicated parameter.   The step of obtaining the mathematical expression further depends on a parameter indicating a statistical distribution of a population of the model parameter using the average value, standard deviation, correlation coefficient, and principal component, and the relative positional relationship. 2. The model parameter extraction method according to claim 1, wherein a mathematical formula is obtained which gives a statistical model expressed by a parameter indicating a correlation between populations of the same element simultaneously with the correlation. The plurality of element pairs are disposed within a predetermined boundary;
The measuring step measures characteristic data in the plurality of element pairs respectively at a plurality of positions within the boundary;
The step of obtaining the mathematical formula further shows a statistical distribution of the population of the model parameters using the mean value, standard deviation, correlation coefficient, and principal component, and depends on the position of the pair within the boundary. And a mathematical formula that gives a statistical model expressed by a parameter indicating a correlation depending on the position of a pair in the boundary between populations of the same element as well as a correlation depending on the relative positional relationship. The model parameter extraction method according to claim 1. The plurality of element pairs are disposed within a predetermined boundary;
In the measuring step, characteristic data is measured by changing a position within the boundary and a relative positional relationship between the pair of elements,
The step of obtaining the mathematical formula further shows a statistical distribution of the population of the model parameters using the mean value, standard deviation, correlation coefficient, and principal component, and depends on the position of the pair within the boundary. And a mathematical formula that gives a statistical model expressed by a parameter indicating a correlation depending on the position of a pair in the boundary between populations of the same element as well as a correlation depending on the relative positional relationship. The model parameter extraction method according to claim 1.   The model parameter extraction method according to claim 1, wherein the plurality of elements forming the pair are arranged closest to each other based on a predetermined arrangement rule. A circuit simulation system for extracting a model equation parameter from measurement data obtained by measuring a plurality of elements as a pair and performing circuit simulation,
A measuring unit for measuring predetermined characteristic data by pairing the plurality of elements;
An extraction unit for extracting each model parameter from the characteristic data measured by the measurement unit;
A data processing unit that calculates a mathematical formula that gives a statistical model using the model parameters extracted by the extraction unit;
A circuit simulator for performing a circuit simulation incorporating the statistical model using a mathematical formula calculated by the data processing unit,
The data processing unit
Statistical distribution estimation means for estimating the mean value and standard deviation of the population in each model parameter using the sample values of each model parameter extracted by the extraction unit;
Correlation coefficient estimating means for estimating a correlation coefficient between populations of each model parameter using sample values of each model parameter extracted by the extraction unit;
A principal component analysis means for obtaining a principal component by performing a principal component analysis on the correlation coefficient estimated by the correlation coefficient estimation means;
Using the mean value, standard deviation, correlation coefficient, and principal component obtained by the statistical distribution estimation means, the correlation coefficient estimation means, and the principal component analysis means, the statistical of the population of the model parameters A circuit comprising: a mathematical expression calculating means for calculating a mathematical expression that gives a statistical model represented by a parameter indicating a distribution and a parameter indicating a correlation depending on a relative positional relationship between populations of different elements; Simulation system.   The mathematical formula calculation means further uses the mean value, standard deviation, correlation coefficient, and principal component to calculate a parameter indicating a statistical distribution of a population of the model parameters and a correlation depending on the relative positional relationship. 7. The circuit simulation system according to claim 6, wherein a mathematical expression that gives a statistical model represented by a parameter indicating a correlation between populations of the same element is calculated at the same time. The plurality of element pairs are disposed within a predetermined boundary;
The measurement unit measures characteristic data in the plurality of pairs of elements at a plurality of positions within the boundary,
The mathematical formula calculation means further shows a statistical distribution of the population of the model parameters using the average value, standard deviation, correlation coefficient, and principal component, and depends on the position of the pair within the boundary. Calculating a mathematical formula that gives a statistical model represented by a parameter and a parameter indicating a correlation depending on a position of a pair in the boundary between populations of the same element at the same time as a correlation depending on the relative positional relationship The circuit simulation system according to claim 6. The plurality of element pairs are disposed within a predetermined boundary;
The measurement unit measures characteristic data by changing a position in the boundary and a relative positional relationship between the plurality of elements as a pair,
The mathematical formula calculation means further shows a statistical distribution of the population of the model parameters using the average value, standard deviation, correlation coefficient, and principal component, and depends on the position of the pair within the boundary. Calculating a mathematical formula that gives a statistical model represented by a parameter and a parameter indicating a correlation depending on a position of a pair in the boundary between populations of the same element at the same time as a correlation depending on the relative positional relationship The circuit simulation system according to claim 6. The circuit simulation system according to claim 6, wherein the plurality of elements forming the pair are arranged closest to each other based on a predetermined arrangement rule.

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