Characteristic variation evaluation method of semiconductor device

Specific parameters used for calculation of a worst-case parameter can be readily selected. Parameters not suitable for characteristic-value expression are eliminated from principal parameters (step ST31), and model equations are extracted (step ST32). Then, parameter values extracted from the same chip are inserted into the model equations to calculate the characteristic value (step ST32). A correlation coefficient between the calculated and measured characteristic values are obtained (step ST33) to select specific parameters (step ST34).

BACKGROUND OF THE INVENTION
 1. Field of the Invention
 The present invention relates to a characteristic variation evaluation
 method for generating a worst-case parameter to evaluate characteristic
 variations of semiconductor devices; a characteristic variation evaluation
 unit; and a recording medium for storing a characteristic variation
 evaluation program, all of semiconductor devices.
 2. Description of the Background Art
 Fluctuations in semiconductor process conditions stochastically cause
 circuit performance variations. Such variations have been relatively
 widened with the shrinking of minimum device size, so that it is getting
 necessary to estimate performance variations, along with standard
 performance, in the device development phase.
 For instance, there are a method for detecting independent model parameters
 by a factor analysis or a principal component analysis in order to
 represent variations of every parameter by the model parameters; and a
 method for classifying parameters into groups with consideration for their
 correlation and analyzing sensitivity of each classified group to circuit
 performance in order to represent circuit performance variations by the
 sensitivity and the parameter variations.
 These methods are both intended to represent variations of characteristic
 values by the parameter variations, but the use of simulations therein
 costs much labor. Hence, Technical Report SDM96-122 (1996-11) of the
 Institute of Electronics, Information and Communication Engineers, for
 example, provides a method for representing performance variations of
 basic circuits due to fluctuations in process conditions by circuit model
 parameters which are obtained by a principal component analysis and a
 multiple regression analysis, and determining parameters having the
 highest probability of being the worst case under a certain condition of
 circuit performance, as parameters representing the worst case (this
 method is hereinafter referred to as "basic method.")
 The features of this conventional basic method are as follows:
 (1) To determine parameter values corresponding to the worst case with
 consideration for a correlation between parameters;
 (2) To increase accuracy in probability estimation as compared with a
 method using corner models;
 (3) To determine parameter values by specifying values corresponding to the
 worst case of circuit performance; and
 (4) To reduce the required number of simulations for parameter
 verification.
 Outline of Basic Method
 A general flow of the basic method is shown in FIG. 13. First, performance
 (drain current, circuit delay etc.) of a large number of chips is
 measured, and a small number of principal parameters (effective channel
 length, threshold voltage etc.) of model parameters in circuit simulation
 are extracted from a plurality of chips (step ST1). After eliminating
 outliers (step ST2), each parameter is applied to distribution (step ST3).
 Then, a typical chip (typical semiconductor device) giving a standard
 performance is selected (step ST4) and its principal parameters are
 extracted as typical parameters (step ST5). At the same time, parameters
 necessary to represent variations of a characteristic value in the worst
 case are selected by a partial regression analysis (step ST6) and a
 multiple regression analysis (step ST7). By adapting the basic method to
 those selected parameters, a parameter value corresponding to the worst
 case (worst-case parameter) is determined (step ST8). Finally, the
 parameters representing the variations of the characteristic value, out of
 the typical parameters extracted from the typical chip, are replaced by
 the worst-case parameter to conduct simulation and verification of the
 worst-case parameter (step ST9).
 We will give a further detailed description of each step on the flow chart
 in FIG. 13. First, parameter extraction at step ST1 is described. The
 parameters to be extracted at step ST1 are, for example, effective channel
 length Leff; effective channel width Weff; external resistance RSH; oxide
 film thickness Tox; threshold voltage Vth; junction capacitance of plane
 components Cj; and junction capacitance of line components Cjsw. Of the
 model parameters in circuit simulation, the above parameters are
 considered as principal parameters having physical connotations and
 influence on circuit performance such as circuit delay Tpd and drain
 current Idmax.
 Next, outlier elimination at step ST2 is described. At step ST2, parameter
 sets of chips at least having one outlier are eliminated from the
 principal parameters in circuit simulation extracted from a plurality of
 chips. Under present circumstances, there are a method for converting data
 of each parameter including outliers into an independent variable by a
 principal component analysis and eliminating outliers until no correlation
 is found in data of converted variables; and a method for eliminating
 outliers by referring to conversion and distribution of each parameter.
 The former method by the principal component analysis is, for example,
 shown in FIGS. 14 and 15. The vertical axis denotes a second principal
 component and the horizontal axis denotes a first principal component. In
 FIG. 14, those principal components are correlated because of outliers
 mixed therein. By eliminating the outliers, a distribution of plots is
 rounded as shown in FIG. 15, in which the principal components are
 uncorrected. Such operation is performed on certain principal components
 for each parameter to eliminate outliers.
 Next, application to distribution at step ST3 is described. At step ST3,
 characteristic values and parameters are applied to suitable probability
 distribution. Assuming in general that a random variable influenced by
 various non-specific factors approximately has a normal distribution, a
 characteristic value is applied to normal distribution and parameters are
 applied to multivariate normal distribution. FIG. 16 shows a
 characteristic value and principal parameters applied to normal
 distribution. As shown, the parameters and the characteristic value almost
 have a normal distribution, and there is a correlation between the
 parameters as shown in FIG. 17.
 Next, chip selection for typical parameter selection at step ST4 is
 described. First, alternatives are narrowed down to chips whose
 characteristic values are about average in distribution. As necessary, the
 principal parameters may also be used to narrow down the alternatives.
 Then, a chip whose principal model parameters have the highest probability
 density is selected as a typical chip from which all model parameters with
 typical values in circuit simulation are extracted. At step ST5,
 predetermined parameters (Leff, Vth etc.) of the typical chip
 (semiconductor device) are extracted as typical parameters.
 Next, parameter selection by the partial regression analysis at step ST6 is
 described. At step ST6, it is checked whether one certain parameter which
 is considered to contribute to the variations of the characteristic value
 includes a component other than the components representing the
 characteristic value. That is, the partial regression analysis reveals a
 contribution of a certain parameter to the characteristic values.
 Next, multiple regression analysis at step ST7 is described by which a
 contribution of a parameter set to the characteristic value is confirmed.
 At step ST7, the degree of correlation between measured values and values
 of linear regression equations calculated from the parameter set selected
 by the partial regression analysis, is checked. This analysis confirms to
 what degree the selected parameters can represent the variations of the
 characteristic value.
 Next, a method for determining the worst-case MOSFET model parameter at
 step ST8 is described. A variable characteristic value p (circuit delay
 time, current driving capability etc.) is considered as a random variable
 having a certain probability distribution. Using a model parameter x in
 circuit simulation having a large contribution to variations of the
 characteristic value p, the variations of the characteristic value p are
 generally expressed as p=F(x)+e, where p is the random variable vector of
 the characteristic value; x is the random variable vector of the
 parameter; and e is the error vector. In the present specification, a
 constant is also included in the concepts of vectors as a vector with one
 element, so the description in the case of a constant will be omitted.
 FIGS. 18A to 18D show that current driving capability Idmax as the
 characteristic value p can be well represented by the effective channel
 length Leff, the effective channel width Weff, the external resistance
 RSH, the oxide film thickness Tox, and the threshold voltage Vth. FIG. 18A
 shows a correlation between the parameters Leff, Vth, Tox and the
 characteristic value Idmax; FIG. 18B shows a correlation between the
 parameter Tox and the characteristic value Idmax; FIG. 18C shows a
 correlation between the parameters Leff, Weff, RSH, Vth, Tox and the
 characteristic value Idmax; and FIG. 18D shows a correlation between the
 parameters Weff, Vth, Tox and the characteristic value Idmax. Assuming
 that the variations of the characteristic value are considerably small,
 the characteristic value p can be expressed by a linear combination as
 follows:
EQU p=.beta..sup.t x+.beta..sub.0 +e (1)
 where .beta. is the partial regression coefficient vector or matrix in the
 multiple regression analysis; and t is the transposition symbol.
 FIG. 19 shows that the current driving capability Idmax as the
 characteristic value p can be well represented by a linear combination of
 the effective channel length Leff, the effective channel width Weff, the
 external resistance RSH, the oxide film thickness Tox, and the threshold
 voltage Vth. Suppose that the model parameters in circuit simulation which
 can well represent the variations of the characteristic value have a
 certain multidimensional distribution. For example, multivariate normal
 distribution of x.apprxeq.N(.mu..sub.x, .SIGMA..sub.x) is assumed in this
 case. Then, the probability density function can be expressed as:
 ##EQU1##
 where .mu..sub.x is the mean vector of x; and .SIGMA..sub.x is the
 variance-covariance matrix of x.
 Since there is a correlation between the parameters as previously
 described, the parameters are applied to multivariate normal distribution
 which can consider a linear correlation between the parameters. In
 general, a linear combination of random variables having a normal
 distribution shows a normal distribution, so p.apprxeq.N(.mu..sub.p,
 .SIGMA..sub.p). By applying a suitable normal direct matrix A to x, Z can
 have an independent normal distribution:
EQU Z=A.sup.t X (2)
 From Equation (1), the characteristic value p can be expressed by a linear
 combination of Z:
EQU C=b.sup.t Z+b.sub.0 +e (3)
 Further, the probability density function of Z is given by:
 ##EQU2##
 where
 .SIGMA..sub.z =A.sup.t.SIGMA..sub.x A/2;
 .SIGMA..sub.z is the diagonal matrix.
 Ignoring the error term e, consider C as a constant value corresponding to
 the worst/best-case characteristic value. Then, Equation (3) can be
 considered as an equation restricted on Z:
EQU b.sup.t.multidot.Z=C-b.sub.O (5)
 The principle component Z that satisfies Equation (5) and maximizes the
 logarithmic probability density function of Equation (4), is given by:
EQU z=.mu..sub.x +(.SIGMA..sub.z.multidot.b).sup.-1.multidot.(p-.mu..sub.p)
 (6)
 where
 .mu..sub.p =(b.sup.t.multidot..mu..sub.z +b.sub.0): mean vector of the
 characteristic value.
 Further, from Equation (2), a worst-case parameter x.sub.w can be expressed
 as:
 x.sub.w =.mu..sub.x
 +(.SIGMA..sub.x.multidot.B).multidot.(B.sup.t.multidot..SIGMA..sub.x
 B).sup.-1.multidot.(p-.mu..sub.p) (7)
 where
EQU B=(Ab) (8)
 If a characteristic value with desired yield is inserted into Equation (7),
 model parameters in circuit simulation corresponding to the
 worst/best-case characteristic value can be uniquely obtained.
 Next, verification results of the worst-case parameter at step ST9 is
 described. In a 0.35-.mu.m CMOS process, verification of the basic method
 is conducted. To obtain parameters, firstly with a delay time Tpd of an
 inverter chain as a characteristic value, the basis method is applied to
 the current driving capabilities Idmax, the junction capacitances of plane
 components Cj, and the junction capacitances of line components Cjsw of
 both N- and P-channels, to obtain respective worst/best-case values.
 Secondly with the current driving capabilities Idmax as a characteristic
 value, the other parameter values are obtained.
 According to the partial regression analysis with the current driving
 capabilities Idmax, the junction capacitances of plane components Cj, and
 the junction capacitances of line components Cjsw of both NMOS and PMOS
 transistors as a parameter set and the basic circuit as a characteristic
 value, it is found that the partial regression coefficient of the junction
 capacitance Cj of the PMOS transistor is considerably low. That junction
 capacitance Cj is, however, left as one of parameters representing the
 worst/best case in order to maintain a balance of parameters between the
 channels. As a result, the junction capacitance Cj of the PMOS transistor
 becomes about a mean value.
 FIG. 20 shows the results of the multiple regression analysis with the
 delay time Tpd of an inverter chain as a characteristic value and the
 current driving capabilities Idmax, the junction capacitances of plane
 components Cj, the junction capacitances of line components Cjsw of both
 N- and P-channels as explanatory variables.
 The analysis shows that in the basic circuit, the inverter chain has a
 correlation coefficient of 0.90 and the other circuits have a correlation
 coefficient of 0.87 or more. This indicates that the variations in the
 basic circuit can be expressed in a linear form of the current driving
 capabilities Idmax, the junction capacitances of plane components Cj, and
 the junction capacitances of line components Cjsw of both N- and
 P-channels.
 Next, the partial regression analysis is conducted with the effective
 channel length Leff, the effective channel width Weff, the oxide film
 thickness Tox, the external resistance RSH, and the threshold voltage Vth
 as a parameter set, and the current driving capabilities Idmax of both
 NMOS and PMOS transistors, Leff of 0.35, and Weff of 10 .mu.m as a
 characteristic value.
 The effective channel width Weff is excluded from the parameters
 representing the worst/best case because of its considerably low partial
 regression coefficient. The oxide film thickness Tox is also excluded
 because of its small variations and its correlation with the threshold
 voltage Vth. The other parameters have a high partial regression
 coefficient with respect to the current driving capability Idmax.
 FIG. 21 shows the results of the multiple regression analysis with the
 current driving capability Idmax of the NMOS transistor as a
 characteristic value and the effective channel length Leff, the external
 resistance RSH, and the threshold voltage Vth as explanatory variables.
 Based on the above result, in both NMOS and PMOS transistors, the
 effective channel length Leff, the external resistance RSH, and the
 threshold voltage Vth are used as parameters representing the
 worst/best-case current driving capability Idmax.
 FIG. 22 shows a comparison between measured values and simulated values of
 the worst/best-case current driving capability Idmax (-3.sigma.,
 -2.sigma., -.sigma., .sigma., 2.sigma., 3.sigma.). The simulations are
 performed by using the parameter value calculated by the basic method. In
 both NMOS and PMOS transistors, the effective channel length Leff, the
 external resistance RSH, the threshold voltage Vth, the junction
 capacitance of plane components Cj, and the junction capacitance of line
 components Cjsw are used as parameters representing the worst/best-case
 circuit performance. The other parameters are typical parameters extracted
 at step ST5. FIG. 23 shows errors of variations between measured values
 and simulated values of the basic circuit. The variations of the simulated
 values obtained from the above parameters almost agree with the variations
 of the measured values.
 To make the conventional problems easy to understand, we will recapitulate
 step ST8 of calculating the worst-case parameter after step ST10 in FIG.
 13. Step ST8 roughly includes: step ST11 of calculating sensitivity; and
 step ST12 of calculating the worst-case parameter as shown in FIG. 24. In
 the foregoing description, sensitivity B is obtained by the principal
 component analysis and the worst-case parameter x.sub.w is obtained from
 Equation (7). Alternatively, the sensitivity can be calculated as follows:
 First, a product unit having standard characteristics is determined by the
 parameter x and the characteristic value p, and parameters representing
 the characteristics of the product unit are extracted by a structural
 formula showing product characteristics. Then, sensitivity of these
 parameters to the characteristic value is analyzed to calculate a partial
 differential coefficient B[.differential.pi/.differential.xj] (i=1, 2, . .
 . , k, j=1, 2, . . . , n).
 The parameters used for the generation of the worst-case parameter at step
 ST8 is empirically selected at step ST10 through the partial regression
 analysis at step ST6 and the multiple regression analysis at step ST7.
 In the aforementioned conventional characteristic variation evaluation
 method of semiconductor devices, the parameters used for the generation of
 the worst-case parameter are selected by repeating much trial and error
 based on experiences. This complicates the procedure for generating the
 worst-case parameter and consumes much time.
 Another problem is that the calculation of sensitivity used for the
 generation of the worst-case parameter requires the principal component
 analysis. This complicates calculation.
 Another problem is that it is difficult to increase accuracy of the
 worst-case parameter.
 Another problem is that calculation becomes complicated when the number of
 parameters used for the generation of the worst-case parameter is
 different from the number of characteristic values.
 Another problem is that the calculation of the worst-case parameter is
 complicated and time consuming.
 SUMMARY OF THE INVENTION
 A first aspect of the present invention is directed to a characteristic
 variation evaluation method of semiconductor devices. The method comprises
 the steps of: (a) extracting predetermined parameters concerning a
 characteristic value of semiconductor devices and measuring the
 characteristic value; (b) eliminating outliers from data measured at the
 step (a); (c) selecting a typical semiconductor device having data that is
 typical among the data with outliers eliminated therefrom, from
 semiconductor devices used for the measurement of the step (a), and
 extracting the predetermined parameters of the typical semiconductor
 device as typical parameters; (d) selecting specific parameters to be used
 for the generation of a worst-case parameter, from the predetermined
 parameters; (e) generating a worst-case parameter from the specific
 parameters on the basis of the data with outliers eliminated therefrom;
 and (f) verifying a worst-case parameter on the basis of the
 characteristic value calculated from the typical parameters excepting the
 specific parameters and the worst-case parameter generated at the step
 (e). The step (d) includes the steps of: (d-1) obtaining a coefficient
 concerning the specific parameters; and (d-2) selecting the specific
 parameters on the basis of the coefficient.
 According to a second aspect of the present invention, the step (d-1) of
 the characteristic variation evaluation method of the first aspect
 includes the steps of: determining an equation that expresses the
 characteristic value by some of the predetermined parameters; and
 obtaining a correlation coefficient between the characteristic value
 measured at the step (a) and a characteristic value calculated from the
 equation, as the coefficient.
 According to a third aspect of the present invention, the step (d-1) of the
 characteristic variation evaluation method of the first aspect includes
 the steps of: obtaining sensitivity from some of the predetermined
 parameters; and obtaining respective inner products of the sensitivity and
 those of the predetermined parameters to obtain respective correlation
 coefficients between the characteristic value measured at the step (a) and
 the inner products, as the coefficient.
 According to a fourth aspect of the present invention, the step (d-1) of
 the characteristic variation evaluation method of the first aspect
 includes the steps of: obtaining sensitivity from some of the
 predetermined parameters; and obtaining respective standard partial
 regression coefficients of those of the predetermined parameters from the
 sensitivity, as the coefficient.
 According to a fifth aspect of the present invention, the step (d-1) of the
 characteristic variation evaluation method of the first aspect includes
 the step of: obtaining respective random coefficients of some of the
 predetermined parameters, as the coefficient.
 A sixth aspect of the present invention is directed to a characteristic
 variation evaluation method of semiconductor devices. The method comprises
 the steps of: (a) extracting predetermined parameters concerning a
 characteristic value of semiconductor devices and at the same time
 measuring the characteristic value; (b) eliminating outliers from data
 measured at the step (a); (c) selecting a typical semiconductor device
 having data that is typical among the data with outliers eliminated
 therefrom, from semiconductor devices used for the measurement of the step
 (a), and extracting the predetermined parameters of the typical
 semiconductor device as typical parameters; (d) selecting specific
 parameters to be used for the generation of a worst-case parameter from
 the predetermined parameters; (e) generating a worst-case parameter from
 the specific parameters on the basis of the data with outliers eliminated
 therefrom; and (f) verifying a worst-case parameter on the basis of the
 characteristic value calculated from the typical parameters excepting the
 specific parameters and the worst-case parameter generated at the step
 (e). The step (e) includes the steps of: calculating sensitivity, on the
 basis of the data with outliers eliminated therefrom, from:
EQU B=.SIGMA..sub.x.sup.-1 (x-.mu..sub.x)/(p-.mu..sub.p).sup.t.SIGMA..sub.p
 where B is a sensitivity vector or matrix, x is a parameter vector or
 matrix, p is a characteristic-value vector or matrix, .mu..sub.x and
 .mu..sub.p are mean vectors or matrices of x and p, respectively,
 .SIGMA..sub.x and .SIGMA..sub.p are variance-covariance matrices of x and
 p, respectively, -1 indicates an inverse matrix, and t is a transposed
 matrix; and calculating a worst-case parameter from the sensitivity.
 A seventh aspect of the present invention is directed to a characteristic
 variation evaluation method of semiconductor devices. The method comprises
 the steps of: (a) extracting predetermined parameters concerning a
 characteristic value of semiconductor devices and at the same time
 measuring the characteristic value; (b) eliminating outliers from data
 measured at the step (a); (c) selecting a typical semiconductor device
 having data that is typical among the data with outliers eliminated
 therefrom, from semiconductor devices used for the measurement of the step
 (a), and extracting the predetermined parameters of the typical
 semiconductor device as typical parameters; (d) selecting specific
 parameters to be used for the generation of a worst-case parameter, from
 the predetermined parameters; (e) generating a worst-case parameter from
 the specific parameters on the basis of the data with outliers eliminated
 therefrom; and (f) verifying a worst-case parameter on the basis of the
 characteristic value calculated from the typical parameters excepting the
 specific parameters and the worst-case parameter generated at the step
 (e). The step (e) includes the steps of: preparing a table showing a
 correlation between the sensitivity and the specific parameters; and
 calculating a worst-case parameter, using the table, from:
EQU x.sub.w =.mu..sub.x +B.sub.Table.SIGMA..sub.x
 where x.sub.w is a worst-case parameter vector or matrix; .mu..sub.x is a
 mean vector or matrix of a parameter x; B.sub.Table is a sensitivity
 vector or matrix; and .SIGMA..sub.x is a variance vector or matrix.
 According to an eighth aspect of the present invention, the step (e) of the
 characteristic variation evaluation method includes, instead of the steps
 of the sixth aspect, the step of calculating sensitivity, on the basis of
 the data with outliers eliminated therefrom, from:
EQU B=EXP.left brkt-bot.(p-.mu..sub.p)/(x-.mu..sub.x).sup.t.right brkt-bot.
 where B is a sensitivity vector or matrix, x is a parameter vector or
 matrix, p is a characteristic-value vector or matrix, .mu..sub.x and
 .mu..sub.p are mean vectors or matrices of x and p, respectively, EXP is
 an expected value, and t is a transposed matrix; and the step of
 calculating a worst-case parameter from the sensitivity.
 According to a ninth aspect of the present invention, the step (e) of the
 characteristic variation evaluation method includes, instead of the steps
 of the sixth aspect, the step of calculating sensitivity on the basis of
 the data with outliers eliminated therefrom; and the step of calculating a
 worst-case parameter, using a calculation result of the sensitivity, from:
EQU x.sub.w =.mu..sub.x +.SIGMA..sub.x B.SIGMA..sub.p.sup.-1.delta..sub.p
 where x.sub.w is a worst-case parameter vector or matrix, .mu..sub.x is a
 mean vector or matrix of the parameter x, .SIGMA..sub.x and .SIGMA..sub.p
 are variance-covariance matrices of x and p, respectively, .delta..sub.p
 is a vector obtained by subtracting the mean vector or matrix .mu..sub.p
 from the characteristic-value vector or matrix p, B is a sensitivity
 vector or matrix, and -1 indicates an inverse matrix.
 According to a tenth aspect of the present invention, the step (e) of the
 characteristic variation evaluation method includes, instead of the steps
 of the sixth aspect, the step of calculating sensitivity on the basis of
 the data with outliers eliminated therefrom; and the step of calculating a
 worst-case parameter, using a calculation result of the sensitivity, from:
EQU x.sub.w =.mu..sub.x +(B.sup.t).sup.+.delta..sub.p
 where x.sub.w is a worst-case parameter vector or matrix, .mu..sub.x is a
 mean vector or matrix of the parameter x, B is a sensitivity vector or
 matrix, t is a transposed matrix, (B.sup.t).sup.+ is a generalized inverse
 matrix of B.sup.t, .delta..sub.p is a vector or matrix obtained by
 subtracting the mean vector or matrix .mu..sub.p from the
 characteristic-value vector or matrix p.
 An eleventh aspect of the present invention is directed to a characteristic
 variation evaluation unit of semiconductor devices. The unit comprises an
 outlier elimination unit eliminating outliers from measured data of a
 characteristic value of semiconductor devices; a typical parameter
 extraction unit selecting a typical semiconductor device having data that
 is typical among the measured data with outliers eliminated therefrom from
 semiconductor devices used for the measurement and extracting
 predetermined parameters concerning the characteristic value of the
 typical semiconductor device as typical parameters; a selection unit
 selecting specific parameters used for the generation of a worst-case
 parameter from the predetermined parameters; a worst-case parameter
 generation unit generating a worst-case parameter from the specific
 parameters on the basis of the measured data with outliers eliminated
 therefrom; and a worst-case parameter verification unit verifying a
 worst-case parameter by utilizing the worst-case parameter generated by
 the worst-case parameter generation unit and the typical parameters
 excepting the worst-case parameter. The parameter selection unit includes
 means for extracting an equation that expresses the characteristic value
 by some of the predetermined parameters; means for obtaining a correlation
 coefficient between the measured characteristic value and a characteristic
 value calculated from the equation; and means for selecting specific
 parameters from those of the predetermined parameters, using the
 correlation coefficient.
 According to a twelfth aspect of the present invention, the parameter
 selection unit of the characteristic variation evaluation unit includes,
 instead of the means of the eleventh aspect, means for obtaining
 sensitivity from some of the predetermined parameters; means for obtaining
 respective inner products of the sensitivity and those of the
 predetermined parameters to obtain respective correlation coefficients
 between the inner products and the measured characteristic value; and
 means for selecting specific parameters from the correlation coefficients.
 According to a thirteenth aspect of the present invention, the parameter
 selection unit of the characteristic variation evaluation unit includes,
 instead of the means of the eleventh aspect, means for obtaining
 sensitivity from some of the predetermined parameters; means for obtaining
 respective standard partial regression coefficients of those of the
 predetermined parameters from the sensitivity; and means for selecting
 specific parameters from the standard partial regression coefficients.
 According to a fourteenth aspect of the present invention, the parameter
 selection unit of the characteristic variation evaluation unit includes,
 instead of the means of the eleventh aspect, means for obtaining
 respective variable coefficients of some of the predetermined parameters;
 and means for selecting specific parameters from the variable
 coefficients.
 A fifteenth aspect of the present invention is directed to a characteristic
 variation evaluation unit of semiconductor devices. The unit comprises: an
 outlier elimination unit eliminating outliers from measured data of a
 characteristic value of semiconductor devices; a typical-parameter
 extraction unit selecting a typical semiconductor device having data that
 is typical among the measured data with outliers eliminated therefrom from
 semiconductor devices used for the measurement and extracting
 predetermined parameters concerning the characteristic value of the
 typical semiconductor device as typical parameters; a selection unit
 selecting specific parameters used for the generation of a worst-case
 parameter from the predetermined parameters; a worst-case parameter
 generation unit generating a worst-case parameter from specific parameters
 on the basis of the measured data with outliers eliminated therefrom; and
 a worst-case parameter verification unit verifying a worst-case parameter
 by utilizing the worst case-parameter generated by the worst-case
 parameter generation unit and the typical parameters excepting the
 parameters used for the generation of the worst-case parameter. The
 worst-case parameter generation unit includes means for calculating
 sensitivity, on the basis of the data with outliers eliminated therefrom,
 from:
EQU B=.SIGMA..sub.x.sup.-1 (x-.mu..sub.x)/(p-.mu..sub.p).sup.t.SIGMA..sub.p
 where B is a sensitivity vector or matrix, x is a parameter vector or
 matrix, p is a characteristic-value vector or matrix, .mu..sub.x and
 .mu..sub.p are mean vectors or matrices of x and p, respectively,
 .SIGMA..sub.x and .SIGMA..sub.p are variance-covariance matrices of x and
 p, respectively, -1 indicates an inverse matrix, and t is a transposed
 matrix; and means for calculating a worst-case parameter from the
 sensitivity.
 According to a sixteen aspect of the present invention, the worst-case
 parameter generation unit of the characteristic variation evaluation unit
 includes, instead of the means of the fifteenth aspect, means for
 calculating sensitivity, on the basis of the data with outliers eliminated
 therefrom, from:
EQU B=EXP.left brkt-bot.(p-.mu..sub.p)/(x-.mu..sub.x).sup.t.right brkt-bot.
 where B is a sensitivity vector or matrix, x is a parameter vector or
 matrix, p is a characteristic-value vector or matrix, .mu..sub.x and
 .mu..sub.p are mean vectors or matrices of x and p, respectively, EXP is
 an expected value, and t is a transposed matrix; and means for calculating
 a worst-case parameter from the sensitivity.
 According to a seventeenth aspect of the present invention, the worst-case
 parameter generation unit of the characteristic variation evaluation unit
 includes, instead of the means of the fifteenth aspect, means for
 calculating sensitivity on the basis of the data with outliers eliminated
 therefrom; and means for calculating a worst-case parameter, using a
 calculation result of the sensitivity, from:
EQU x.sub.w =.mu..sub.x +.SIGMA..sub.x B.SIGMA..sub.p.sup.-1.delta..sub.p
 where x.sub.w is a worst-case parameter vector or matrix, .mu..sub.x is a
 mean vector or matrix of the parameter x, .SIGMA..sub.x and .SIGMA..sub.p
 are variance-covariance matrices of x and p, respectively, .delta..sub.p
 is a vector obtained by subtracting the mean vector or matrix .mu..sub.p
 from the characteristic-value vector or matrix p, B is a sensitivity
 vector or matrix, and -1 indicates an inverse matrix.
 According to an eighteenth aspect of the present invention, the worst-case
 parameter generation unit of the characteristic variation evaluation unit
 includes, instead of the means of the fifteenth aspect, means for
 calculating sensitivity on the basis of the data with outliers eliminated
 therefrom; and means for calculating a worst-case parameter, using a
 calculation result of the sensitivity, from:
EQU x.sub.w =.mu..sub.x +(B.sup.t).sup.+.delta..sub.p
 where x.sub.w is a worst-case parameter vector or matrix, .mu..sub.x is a
 mean vector or matrix of the parameter x, B is a sensitivity vector or
 matrix, t is a transposed matrix, (B.sup.t).sup.+ is generalized inverse
 matrix of B.sup.t, .delta..sub.p is a vector or matrix obtained by
 subtracting the mean vector or matrix .mu..sub.p from the
 characteristic-value vector or matrix p.
 According to a nineteenth aspect of the present invention, the worst-case
 parameter generation unit of the characteristic variation evaluation unit
 includes, instead of the means of the fifteenth aspect, means for storing
 a table showing a correlation between the sensitivity and the specific
 parameters; and means for calculating a worst-case parameter, using the
 table, from:
EQU x.sub.w =.mu..sub.x +B.sub.Table.SIGMA..sub.x
 where x.sub.w is a worst-case parameter vector or matrix; .mu..sub.x is a
 mean vector or matrix of the parameter x; B.sub.Table is a sensitivity
 vector or matrix; and .SIGMA..sub.x is a variance vector or matrix.
 A twentieth aspect of the present invention is directed to a storage medium
 for storing a program for having a computer execute at least the steps,
 excepting the step (a), of the characteristic variation evaluation method
 of either of the first to tenth aspects.
 A twenty-first aspect of the present invention is directed to a storage
 medium for storing a program for having a computer function as the
 characteristic variation evaluation unit of either of the eleventh to
 nineteenth aspects.
 In the characteristic variation evaluation method of semiconductor devices
 according to the first aspect, the specific parameters used for the
 generation of the worst-case parameter are selected on the basis of the
 coefficient concerning the specific parameters. This facilitates selection
 of the specific parameters.
 In the characteristic variation evaluation method of semiconductor devices
 according to the second aspect, the characteristic variation evaluation
 unit of semiconductor devices according to the eleventh aspect, and the
 storage medium according to the twentieth or twenty-first aspect, the
 specific parameters used for the generation of the worst-case parameter
 are selected on the basis of the correlation coefficient between the
 calculated characteristic value and measured characteristic value. This
 allows unique selection of the specific parameters without great
 difficulty.
 In the characteristic variation evaluation method of semiconductor devices
 according to the third aspect, the characteristic variation evaluation
 unit of semiconductor devices according to the twelfth aspect, and the
 storage medium according to the twentieth or twenty-first aspect, the
 specific parameters used for the generation of the worst-case parameter
 are selected on the basis of the correlation coefficients between the
 characteristic value and the inner produces calculated from several
 parameters and sensitivity. This allows unique selection of the specific
 parameters without great difficulty and shortens calculation time.
 In the characteristic variation evaluation method of semiconductor devices
 according to the fourth aspect, the characteristic variation evaluation
 unit of semiconductor devices according to the thirteenth aspect, and the
 storage medium according to the twentieth or twenty-first aspect, the
 specific parameters used for the generation of the worst-case parameter
 are selected on the basis of the standard partial regression coefficients
 of several parameters. This allows unique selection of the specific
 parameters without great difficulty and facilitates calculation of each
 parameter.
 In the characteristic variation evaluation method of semiconductor devices
 according to the fifth aspect, the characteristic variation evaluation
 unit of semiconductor devices according to the fourteenth aspect, and the
 storage medium according to the twentieth or twenty-first aspect, the
 specific parameters used for the generation of the worst-case parameter
 are selected on the basis of the variable coefficients of several
 parameters. This allows unique selection of the specific parameters
 without great difficulty and facilitates calculation of each parameter.
 In the characteristic variation evaluation method of semiconductor devices
 according to the sixth aspect, the characteristic variation evaluation
 unit of semiconductor devices according to the fifteenth aspect, and the
 storage medium according to the twentieth or twenty-first aspect,
 sensitivity is obtained from the mean values and the variances in the step
 of generating the worst-case parameter. This facilitates calculation.
 In the characteristic variation evaluation method of semiconductor devices
 according to the seventh aspect, the characteristic variation evaluation
 unit of semiconductor devices according to the nineteenth aspect, and the
 storage medium according to the twentieth or twenty-first aspect, the
 worst-case parameter can be calculated only from the mean values and the
 standard deviation in the step of generating the worst-case parameter.
 In the characteristic variation evaluation method of semiconductor devices
 according to the eighth aspect, the characteristic variation evaluation
 unit of semiconductor devices according to the sixteenth aspect, and the
 storage medium according to the twentieth or twenty-first aspect,
 statistical sensitivity can be readily calculated in the step of
 generating the worst-case parameter.
 In the characteristic variation evaluation method of semiconductor devices
 according to the ninth aspect, the characteristic variation evaluation
 unit of semiconductor devices according to the seventeenth aspect, and the
 storage medium according to the twentieth or twenty-first aspect, the step
 of generating the worst-case parameter increases accuracy in the
 calculation of the variations of the characteristic value as compared in
 the conventional method.
 In the characteristic variation evaluation method of semiconductor devices
 according to the tenth aspect, the characteristic variation evaluation
 unit of semiconductor devices according to the eighteenth aspect, and the
 storage medium according to the twentieth or twenty-first aspect, the step
 of generating the worst-case parameter enables calculation even when the
 number of parameters is different from the number of characteristic
 values.
 Thus, an objective of the present invention is to facilitate the selection
 of parameters used for the generation of the worst-case parameter by doing
 it mechanically. Another objective is to facilitate the calculation of
 sensitivity in the generation of the worst-case parameter. Another
 objective is to increase accuracy in the calculation of the worst-case
 parameter as compared with the conventional method. Another objective is
 to facilitate the calculation of the worst-case parameter when the number
 of parameters is different from the number of characteristic values.
 Another objective is to simplify the calculation of the worst-case
 parameter to shorten calculation time.
 These and other objects, features, aspects and advantages of the present
 invention will become more apparent from the following detailed
 description of the present invention when taken in conjunction with the
 accompanying drawings.

DESCRIPTION OF THE PREFERRED EMBODIMENTS
 1. First Preferred Embodiment
 Characteristic variation evaluation method and unit of semiconductor
 devices according to a first preferred embodiment of the present invention
 are obtained with improvements in parameter selection by the conventional
 basic method. That is, step ST10 of the conventional method in FIG. 13 is
 substituted by step ST20 as shown in FIG. 1.
 FIG. 2 is a flow chart of step ST20 in FIG. 1 according to the first
 preferred embodiment. First at step ST31, parameters having little
 possibility of being the worst-case parameter are eliminated from
 principal parameters. Here, the principal parameters correspond to the
 predetermined parameters extracted at step ST1. FIG. 3 is a conceptual
 view of a wafer with a plurality of integrated circuit formed thereon. For
 instance, a characteristic value such as circuit delay time Tpd and
 current driving capability Idmax, is measured from each chip 101 on a
 wafer 100. The characteristic value is measured by the remaining
 parameters not eliminated at step ST31, but it is also possible to use all
 principal parameters for the measurement. Sufficient measuring points are
 used for the measurement but not so many that adjustment in flexibility is
 required. In general, parameter values are different in each chip 101 with
 an integrated circuit, so that the measured characteristic value varies
 from chip to chip. FIG. 4 is a histogram showing an example of the
 measured characteristic value (in this case, delay time of an inverter
 chain). As shown in FIG. 4, the characteristic value usually has a normal
 distribution.
 At the next step ST32, model equations are extracted or linear equations
 are calculated as follows:
EQU Id max.sub.sim =f(Leff)
EQU Id max.sub.sim =f(Weff)
EQU .
EQU .
EQU .
EQU Id mmax.sub.sim =f(Leff, Weff)
EQU .
EQU .
EQU .
EQU Id max.sub.sim =f(Leff, Weff, RSH, Vth, Tox, Vo)
 As shown, various parameter combinations are set in the model or linear
 equations. Then at step ST33, parameter values extracted from the same
 chip whose characteristic value is measured are inserted into the above
 equations to obtain a calculated characteristic value.
 After that, at step ST34, a correlation coefficient between the measured
 characteristic value and the calculated characteristic value is obtained
 for each parameter combination expressed by the model or linear equations.
 For example, the model equation, Idmax=f(Leff, Vth, RSH) has a correlation
 coefficient of 0.958. In this way, the correlation coefficient of each
 parameter combination is individually obtained. Then, a parameter
 combination having the highest correlation coefficient is selected as
 specific parameters (step ST35). This combination of specific parameters
 is utilized for generation of the worst-case parameter at step ST8 in FIG.
 1. The other steps in FIG. 1 are identical with those denoted by the same
 reference characters in FIG. 13.
 In this fashion, the correlation coefficients of various combinations are
 obtained. This makes it possible to mechanically and uniquely obtain a
 combination of specific parameters suitable for generation of the
 worst-case parameter, thereby facilitating the parameter selection.
 Next, we will describe the characteristic variation evaluation unit of
 semiconductor devices according to the first preferred embodiment. FIG. 5
 is a block diagram of the structure of this characteristic variation
 evaluation unit. A characteristic variation evaluation unit 1 measures
 characteristics of semiconductor devices, e.g., characteristics of chips
 formed on the wafer 100. The characteristics to be measured and principal
 parameters necessary for the measurement are inputted, for example, from
 an input unit 10. The information from the input unit 10 is stored in a
 data storage unit 11, through which the other units obtain the
 information. A parameter extraction unit 12 extracts principal parameters
 having influences on semiconductor device performance from the measured
 data stored in the data storage unit 11. An outlier elimination unit 13
 performs the outlier elimination of step ST2 in FIG. 1, with the measured
 data in the data storage unit 11. A fitting unit 14 performs the
 application to distribution of step ST3 in FIG. 1, with the measured data
 in the data storage unit 11. A parameter selection unit 15 selects
 specific parameters used for the generation of the worst-case parameter,
 from several parameters extracted by the parameter extraction unit 12.
 Prior to the selection, the parameter selection unit 15 eliminates
 outliers indicated by the outlier elimination unit 13. The parameter
 selection unit 15 further can stop the parameter selection when the
 fitting unit 14 fails in the application to distribution. This is because
 a failure in the application to distribution is often caused by
 imperfections in measurement, so that the worst-case parameter generated
 from such data is likely to be wrong. The generation of the worst-case
 parameter is thus practicable without the fitting unit 14. A worst-case
 parameter generation unit 16 generates the worst-case parameter, with the
 measured data in the data storage unit 11 and the specific parameters from
 the parameter selection unit 15. The generation of the worst-case
 parameter by the worst-case parameter generation unit 16 is performed as
 step ST8 in FIG. 1. At the same time, a typical parameter extraction unit
 18 extracts typical parameters through the same processing as steps ST4
 and ST5 in FIG. 1 from the measured data in the data storage unit 11. A
 worst-case parameter verification unit 17 conducts verification of the
 worst-case parameter, with the worst-case parameter extracted by the
 worst-case parameter generation unit 16, the typical parameters extracted
 by the typical parameter extraction unit 18, and the measured data in the
 data storage unit 11.
 2. Second Preferred Embodiment
 Characteristic variation evaluation method and unit of semiconductor
 devices according to a second preferred embodiment of the present
 invention are obtained with improvements in parameter selection by the
 conventional basic method. That is, step ST10 of the conventional method
 in FIG. 13 is substituted by step ST20 as shown in FIG. 1.
 FIG. 6 is a flow chart of step ST20 in FIG. 1 according to the second
 preferred embodiment. Step ST41 in FIG. 6 is identical with step ST31 in
 FIG. 2. More specifically, at step ST41, alternates for specific
 parameters used for the generation of the worst-case parameter are
 narrowed down to some of the principal parameters. Then, at step ST42,
 sensitivity B is obtained from those of the principal parameters, for
 example, using Equation (8), i.e. B=(A.multidot.b), as in the conventional
 method.
 Next, at step ST43, inner products of the sensitivity B and respective
 those of the principal parameters are obtained to calculate respective
 correlation coefficients between the inner products and the measured
 values. Of these parameters, a predetermined number of parameters in
 descending order of their correlation coefficients calculated at step
 ST43; or parameters whose correlation coefficients are equal to or larger
 than a predetermined value are selected as specific parameters (step
 ST44). The subsequent processing after step ST20 in FIG. 1 is performed by
 utilizing a combination of these specific parameters.
 In this fashion, step ST20 of selecting the specific parameters facilitates
 calculation and reduces the calculation time.
 Further, since the sensitivity B calculated at step ST42 is obtained for
 each of the parameters not eliminated at step ST41, it is usually
 different from the sensitivity calculated at step ST8.
 The characteristic variation evaluation unit of the second preferred
 embodiment is different from that of the first preferred embodiment only
 in the function of the parameter selection unit 15 in FIG. 5. The other
 part of the structure is identical with that of the first preferred
 embodiment. The parameter selection unit 15 of the second preferred
 embodiment has a function of performing the processing illustrated in FIG.
 6.
 3. Third Preferred Embodiment
 Characteristic variation evaluation method and unit of semiconductor
 devices according to a third preferred embodiment are obtained with
 improvements in parameter selection by the conventional basic method. That
 is, step ST10 of the conventional method in FIG. 13 is substituted by step
 ST20 as shown in FIG. 1.
 FIG. 7 is a flow chart of step ST20 in FIG. 1 according to the third
 preferred embodiment. Steps ST51 and ST52 in FIG. 7 are identical with
 steps ST41 and ST42 in FIG. 6. More specifically, at step ST51, alternates
 for specific parameters used for the generation of the worst-case
 parameter are narrowed down to some of the principal parameters. After
 steps ST51 and ST52, standard partial regression coefficients of those of
 the principal parameters are calculated from the sensitivity B obtained at
 step ST52 (step ST53). The standard partial regression coefficient is
 obtained by {(standard deviation of the parameter)/(standard deviation of
 the characteristic value).times.sensitivity B}. Of these parameters, a
 predetermined number of parameters in descending order of their standard
 partial regression coefficients; or parameters whose standard partial
 regression coefficients are equal to or larger than a predetermined value,
 are selected as the specific parameters (step ST54). The subsequent
 processing after step ST20 in FIG. 1 is performed by utilizing a
 combination of these specific parameters.
 In this fashion, step ST20 of selecting the specific parameters facilitates
 calculation of each parameter.
 Further, since the sensitivity B calculated at step ST52 is obtained for
 each of the parameters not eliminated at step ST51, it is usually
 different from the sensitivity calculated at step ST8.
 The characteristic variation evaluation units of the third preferred
 embodiment is different from that of the first preferred embodiment only
 in the function of the parameter selection unit 15 in FIG. 5. The other
 part of the structure is identical with that of the first preferred
 embodiment. The parameter selection unit 15 of the third preferred
 embodiment has a function to perform the processing illustrated in FIG. 7.
 4. Fourth Preferred Embodiment
 Characteristic variation evaluation method and unit of semiconductor
 devices according to a fourth preferred embodiment are obtained with
 improvements in parameter selection by the conventional basic method. That
 is, step ST10 of the conventional method in FIG. 13 is substituted by step
 ST20 as shown in FIG. 1.
 FIG. 8 is a flow chart of step ST20 in FIG. 1 according to the third
 preferred embodiment. Step ST61 in FIG. 8 is identical with step ST31 in
 FIG. 2. More specifically, at step ST61, alternates for specific
 parameters used for the generation of the worst-case parameter are
 narrowed down to some of the principal parameters. After step ST61, random
 coefficients of those of the principal parameters are calculated at step
 ST62. The random coefficient is obtained by dividing the standard
 deviation of the parameter by the mean value of the same. Of these
 parameters, a predetermined number of parameters in descending order of
 their random coefficients; or parameters whose random coefficients are
 equal to or larger than a predetermined value, are selected as the
 specific parameters (step ST63). This is because in general a parameter
 with a higher random coefficient has a greater influence on the
 characteristics. The subsequent processing after Step ST20 in FIG. 1 is
 performed by utilizing a combination of these specific parameters.
 In this fashion, step ST20 of selecting the specific parameters facilitates
 calculation of each parameter.
 The characteristic variation evaluation unit of the fourth preferred
 embodiment is different from that of the first preferred embodiment only
 in the function of the parameter selection unit 15 in FIG. 5. The other
 part of the structure is identical with that of the first preferred
 embodiment. The parameter selection unit 15 of the fourth preferred
 embodiment has a function to perform the processing illustrated in FIG. 8.
 5. Fifth Preferred Embodiment
 Characteristic variation evaluation method and unit of semiconductor
 devices according to a fifth preferred embodiment are obtained with
 improvements in the step or means of calculating sensitivity of the
 conventional basic method. That is, step ST8 of the conventional method in
 FIG. 13 is substituted by step ST21 as shown in FIG. 9. More specifically,
 step ST11 (sensitivity calculation) of the conventional method in FIG. 24
 is substituted by step ST70 as shown in FIG. 10 to achieve the
 characteristic variation evaluation method and unit of the fifth preferred
 embodiment.
 In the conventional method, the sensitivity B is calculated from Equation
 (8) through the principal component analysis and the regression analysis
 on the characteristic value. In the method of the fifth preferred
 embodiment, on the other hand, the sensitivity B is obtained at step ST70
 by:
EQU B=.SIGMA..sub.x.sup.-1 (x-.mu..sub.x)/(p-.mu..sub.p).sup.t.SIGMA..sub.p
 (9)
 where .mu.x and .mu.p are the mean values of the parameter x and the
 characteristic value p, respectively; .SIGMA.x is the variance-covariance
 matrix of the parameter x; -1 is the inverse matrix; B is the sensitivity
 vector or matrix; .SIGMA.p is the variance-covariance matrix of the
 characteristic value p. The mean values .mu.x and .mu.p can be substituted
 by measured values close to the mean values. Further, the variance and the
 covariance in the variance-covariance matrices .SIGMA.x and .mu.p can be
 substituted by measured values close to the variance and the covariance,
 respectively. The measured value close to the variance S.sup.2 of the
 parameter x is, for example, a value close to the variance S.sup.2 in the
 set (xi-.mu.x).sup.2, where xi is a member in the set of the parameter x.
 The measured value close to the covariance S.sub.x1, x2 of the parameter x
 is, for example, a value close to the covariance S.sub.x1, x2 in the set
 (x.sub.1i -.mu.x.sub.1) (x.sub.2i -.mu.x.sub.2), where the parameter x is
 a vector; x.sub.1i and x.sub.2i are a set of elements of the vector; and
 .mu.x.sub.1 and .mu.x.sub.2 are the mean values of those elements,
 respectively.
 For instance, we will describe calculation of Equation (9) with the
 effective channel length Leff, the threshold voltage Vth, and the external
 terminal RSH as a parameter. At this time, the vector p is a vector with
 one row and two columns, having characteristic values p1 and p2 as its
 elements; and the vector x is a vector with three rows and one column,
 having the parameters Leff, Vth, and RSH as its elements. In this case,
 Equation (9) can be rewritten as follows:
 ##EQU3##
 where .delta. is the distance from the mean value, e.g.,
 .delta.Leff=Leff-.mu..sub.Leff ; S.sub.i is the standard deviation of i;
 S.sub.i,j is the covariance of i and j; and
 .differential.p/.differential.i is the sensitivity of p to i.
 In Equation (9) at step ST21 of the fifth preferred embodiment, the
 sensitivity B can be calculated from the mean values and the variances.
 This facilitates calculation.
 Further, while step ST20 of the first to fourth preferred embodiments uses
 the conventional method to obtain sensitivity, it is also possible to use
 step ST21 of the fifth preferred embodiment for the calculation of
 sensitivity in the first to fourth preferred embodiments.
 The characteristic variation evaluation unit of the fifth preferred
 embodiment in FIG. 11 is different from those of the first to fourth
 preferred embodiments in FIG. 5 only in the function of a worst-case
 parameter generation unit 20. The other part of the structure is identical
 with that of the first to fourth preferred embodiments. The worst-case
 parameter generation unit 20 of the fifth preferred embodiment has a
 function to perform the processing of step ST21 illustrated in FIG. 9.
 6. Sixth Preferred Embodiment
 Characteristic variation evaluation method and unit of semiconductor
 devices according to a sixth preferred embodiment of the present invention
 are obtained with improvements in the step or means of calculating
 sensitivity of the conventional basic method. That is, step ST8 of the
 conventional method in FIG. 13 is substituted by step ST21 as shown in
 FIG. 9. More specifically, step ST11 (sensitivity calculation) of the
 conventional method in FIG. 24 is substituted by step ST70 as shown in
 FIG. 10 to achieve the characteristic variation evaluation method and unit
 of the fifth preferred embodiment.
 In the conventional method, the sensitivity B is calculated from Equation
 (8) through the principal component analysis and the regression analysis
 on the characteristic value. In the method of the sixth preferred
 embodiment, on the other hand, the sensitivity B is obtained at step ST70
 by:
EQU B=EXP.left brkt-bot.(p-.mu..sub.p)/(x-.mu..sub.x).sup.t.right brkt-bot.
 (10)
 To obtain an expected value of a plurality of chips C1 to Cn, for example,
 the value (p-.mu.p)/(x-.mu.x).sup.t of each chip is calculated. That is,
 the characteristic value p and the parameter x of each chip are first
 extracted for Equation (10). Then, the average of the values
 (p-.mu.p)/(x-.mu.x).sup.t of the chips C1 to Cn is taken to obtain the
 expected value EXP [(p-.mu.p)/(x-.mu.x).sup.t ].
 In Equation (10) at step ST21 of the sixth preferred embodiment, the
 sensitivity B can be calculated from the mean values and the variances.
 This facilitates calculation.
 Further, while step ST20 of the first to fourth preferred embodiments uses
 the conventional method to obtain sensitivity, it is also possible to use
 step ST21 of the sixth preferred embodiment for the calculation of
 sensitivity in the first to fourth preferred embodiments.
 The characteristic variation evaluation unit of the sixth preferred
 embodiment in FIG. 11 is different from those of the first to fourth
 preferred embodiments in FIG. 5 only in the function of the worst-case
 parameter generation unit 20. The other part of the structure is identical
 with that of the first to fourth preferred embodiments. The worst-case
 parameter generation unit 20 of the sixth preferred embodiment has a
 function to perform the processing of step ST21 illustrated in FIG. 9.
 7. Seventh Preferred Embodiment
 Characteristic variation evaluation method and unit of semiconductor
 devices according to a seventh preferred embodiment of the present
 invention are obtained with improvements in the step or means of
 generating the worst-case parameter of the conventional basic method. That
 is, step ST12 of the conventional method in FIG. 24 is substituted by step
 ST80 as shown in FIG. 12.
 In the conventional method, the worst-case parameter x.sub.w is generated
 from Equation (7). In the method of the seventh preferred embodiment, on
 the other hand, the worst-case parameter x.sub.w is found by:
EQU x.sub.w =.mu..sub.x +.SIGMA..sub.x B.SIGMA..sub.p.sup.-1.delta..sub.p (11)
 where p is the characteristic value corresponding to the worst case. The
 use of Equation (11) for the generation of the worst-case parameter
 x.sub.w increases accuracy in the calculation of the variations of the
 characteristic value p as compared in the conventional method.
 For instance, we will describe calculation of Equation (11) with the
 effective channel length Leff, the threshold voltage Vth, and the external
 terminal RSH as a parameter x. At this time, the characteristic value p is
 a constant; and the vector x is a vector with three rows and one column,
 having the parameters Leff, Vth, and RSH as its elements. In this case,
 Equation (11) can be rewritten as follows:
 ##EQU4##
 Alternatively, when the vector p is a vector with two rows and one column,
 having characteristic values p1 and p2 as its elements; and the vector x
 is a vector with three rows and one column, having the parameters Leff,
 Vth, and RSH as its elements, Equation (11) can be rewritten as follows:
 ##EQU5##
 The characteristic variation evaluation unit of the seventh preferred
 embodiment in FIG. 11 is different from those of the first to fourth
 preferred embodiments in FIG. 5 only in the function of the worst-case
 parameter generation unit 20. The other part of the structure is identical
 with that of the first to fourth preferred embodiments. The worst-case
 parameter generation unit 20 of the seventh preferred embodiment has a
 function to perform the processing of step ST21 illustrated in FIG. 12.
 8. Eighth Preferred Embodiment
 Characteristic variation evaluation method and unit of semiconductor
 devices according to an eighth preferred embodiment of the present
 invention are obtained with improvements in the step or means of
 generating the worst-case parameter of the conventional basic method. That
 is, step ST12 of the conventional method in FIG. 24 is substituted by step
 ST80 as shown in FIG. 12.
 In the conventional method, the worst-case parameter x.sub.w is generated
 from Equation (7). In the method of the eighth preferred embodiment, on
 the other hand, the worst-case parameter x.sub.w is found by:
EQU x.sub.w =.mu..sub.x +(B.sup.t).sup.+.delta..sub.p (12)
 where p is a characteristic value corresponding to the worst case; and
 (B.sup.t).sup.+ is the generalized inverse matrix of B.sup.t. The use of
 Equation (12) for the generation of the worst-case parameter x.sub.w
 increases accuracy in the calculation of the variations of the
 characteristic value p.
 For instance, we will describe calculation of Equation (12) with the
 effective channel length Leff, the threshold voltage Vth, and the external
 terminal RSH as the parameters x. At this time, the vector p is a vector
 with two rows and one column, having characteristic values p1 and p2 as
 its elements; and the vector x is a vector with three rows and one column,
 having the parameters Leff, Vth, and RSH as its elements. In this case,
 Equation (12) can be rewritten as follows:
 ##EQU6##
 The characteristic variation evaluation unit of semiconductor devices
 according to the eighth preferred embodiment is as shown in FIG. 11. It is
 different from the units of the first to fourth preferred embodiments in
 FIG. 5 only in the function of the worst-case parameter generation unit
 20. The other part of the structure is identical with that of the first to
 fourth preferred embodiments. The worst-case parameter generation unit 20
 of the eighth preferred embodiment has a function to perform the
 processing of step ST21 illustrated in FIG. 12.
 9. Ninth Preferred Embodiment
 Characteristic variation evaluation method and unit of semiconductor
 devices according to a ninth preferred embodiment of the present invention
 are obtained with improvements in the step or means of generating the
 worst-case parameter of the conventional basic method. That is, step ST12
 of the conventional method in FIG. 24 is substituted by step ST80 as shown
 in FIG. 12.
 In the method of the ninth preferred embodiment, the worst-case parameter
 x.sub.w is found by:
EQU x.sub.w =.mu..sub.x +B.sub.Table.SIGMA..sub.x (13)
 More specifically, the characteristic variation evaluation method and unit
 of the ninth preferred embodiment makes an table of a correlation between
 the worst-case parameter calculated from Equation (7), (11), or (12) and
 the standard deviation of each parameter. Table 1 shows an example of
 B.sub.Table in Equation (13).
 TABLE 1
 NMOS PMOS NMOS PMOS
 Leff/ RSH/ Vth/ Leff/ RSH/ Vth/ Cj/ Cjsw/ Cj/ Cjsw/ Tpd/
 .sigma..sub.Leff .sigma..sub.RSH .sigma..sub.Vth .sigma..sub.Leff
 .sigma..sub.RSH .sigma..sub.Vth .sigma..sub.Cj .sigma..sub.Cjsw
 .sigma..sub.Cj .sigma..sub.Cjsw .sigma..sub.Tpd
 0.59 0.17 0.57 0.95 0.51 0.19 0.80 0.49 0.026 0.22 1
 1.17 0.35 1.13 1.89 1.01 0.38 1.60 0.98 0.052 0.45 2
 1.76 0.52 1.70 2.84 1.52 0.57 2.40 1.47 0.078 0.67 3
 The use of Equation (13) and Table 1 makes it possible to calculate the
 worst-case parameter only from the mean values and the standard
 deviations.
 The characteristic variation evaluation unit of semiconductor devices
 according to the ninth preferred embodiment is as shown in FIG. 11. It is
 different from the units of the first to fourth preferred embodiments in
 FIG. 5 only in the function of the worst-case parameter generation unit
 20. The other part of the structure is identical with that of the first to
 fourth preferred embodiments. The worst-case parameter generation unit 20
 of the ninth preferred embodiment has a function to perform the processing
 of step ST21 illustrated in FIG. 12.
 The flow charts of the aforementioned first to ninth preferred embodiments
 of the present invention are implemented by having a computer execute a
 program stored in a storage unit thereof, but it is also possible to have
 a computer load and execute a program stored in a recording medium such as
 a flexible disc via a flexible disc driver. Further, the program stored in
 the recording medium may be a program for having a computer execute not
 the whole but part of the flow charts. This is because the flow chart can
 be implemented by linking that program with a program already loaded in a
 computer. Further, a program for achieving the characteristic variation
 evaluation method of semiconductor devices may include step ST1 of
 extracting data in FIG. 1 so as to have a computer control a measuring
 device, or it may not include step ST1 by using already-measured/extracted
 data.
 While the invention has been described in detail, the foregoing description
 is in all aspects illustrative and not restrictive. It is understood that
 numerous other modifications and variations can be devised without
 departing from the scope of the invention.