JP2007108843A - Semiconductor device design support method, semiconductor device design support system and semiconductor device design support program - Google Patents

Abstract

<P>PROBLEM TO BE SOLVED: To identify a statistical model related to a failure occurrence mechanism. <P>SOLUTION: A plurality of hypotheses on the failure occurrence mechanism of each failure mode are assumed, and function types of data distribution from a plurality of state variables to a plurality of output variables are assumed on each of the hypotheses in consideration of a predetermined failure physical model corresponding to each failure mode. A probability variable data set on the plurality of state variables, discretized connection probability distribution and a probability variable data set whose probability density is not zero, are computed using a response surface model and Monte Carlo simulation based on CAE analysis result data given beforehand or under expansion of numerical experiment points, and a parameter corresponding to the function type of the assumed data distribution is estimated from an observation data set obtained beforehand or under monitoring on a plurality of input variables and the plurality of output variables, and from the connection probability distribution of the plurality of state variables computed in a second step. <P>COPYRIGHT: (C)2007,JPO&INPIT

Description

  The present invention relates to a semiconductor device design support method, a semiconductor device design support system, and a semiconductor device design support program, and more particularly to a semiconductor device design support method and a semiconductor device design support system that are useful for performing hypothesis verification on a failure occurrence mechanism. And a semiconductor device design support program.

  In order to strengthen the design review related to reliability at the design stage, it uses CAE (Computer Aided Engineering) based on a physical model that can reproduce the phenomenon and database on materials, manufacturing, inspection / testing, and quality, and is inherent in the design. It is necessary to design reliability after taking uncertainty into account by statistical and probabilistic methods.

  Semiconductor devices have been developed using simulation technology due to advances in element analysis technology, inspection / evaluation technology, and accumulation of knowledge about failure modes in materials and manufacturing processes. Mainly to clarify the failure mechanism by reproducing complex phenomena in the manufacturing process, test / inspection and market conditions, or to optimize the material / structure / process by comparing the simulation result and failure occurrence criteria It is used for the purpose of doing.

  In semiconductor device development, there are many cases where a plurality of failure modes exist while many factors are intricately intertwined, and it is not easy to faithfully reproduce all combinations by experiments. For example, in a minute region such as the inside of a semiconductor package element or a fine junction, it is often difficult to actually measure and evaluate physical quantities such as temperature, stress, and strain. Therefore, there are high expectations for problem solving based on simulation technology. Recently, simulation technology is being applied to reliability design such as to perform robust design against variations inherent in the design, or to set a safety factor (design margin) rationally.

  However, the CAE model has uncertainties inherent in constitutive equations in modeling such as anisotropy characteristics and inelastic characteristics of materials, and in spatio-temporal boundary conditions of a target numerical calculation model. It is not possible to clarify the mechanism of phenomenon occurrence for each failure mode within a limited cost and period only by individual experimental data from bulk materials and TEG (Test Element Group), and judgment based on expertise and experience. In many cases, the uncertainty of the analytical model cannot be identified.

  In addition, in the globalization of procurement / manufacturing, failure modes are diversified, and the variation in material characteristics and manufacturing processes tends to increase. Furthermore, innovations in technology for miniaturization of semiconductor devices, higher speed of signal transmission, and larger capacity, and the appearance of semiconductor devices based on new concepts occur in a discontinuous manner. For example, the development of new materials and structures, or the use in new environments or under different load assumptions. At this time, it becomes difficult to determine a necessary and sufficient reliability specification by extrapolating past experience, and a large uncertainty remains.

In addition, the literature which described the technique relevant to this application includes the following. In the disclosure of this document, there is no concept of a hierarchical structure in which a plurality of potential state variables intervene between a plurality of design variables and a plurality of output variables. There is no mention of either.
JP-A-6-195402

  From the situation described above, in semiconductor device design, in addition to the deep cultivation and accumulation of failure phenomenon analysis / evaluation technology based on physical phenomenon models, many hypotheses regarding failure mechanisms are included in the presence of uncertainty regarding failure factors. It is required to build a rational verification mechanism. In addition, a new method for supporting the identification of a failure mechanism is required by using a statistical model or CAE based on a failure physical model from overall data on materials, manufacturing, and quality.

  The present invention relates to statistics relating to a failure occurrence mechanism using CAE data and data accumulated in actual measurement as total data in a semiconductor device design support method, a semiconductor device design support system, and a semiconductor device design support program. It is an object to provide a semiconductor device design support method, a semiconductor device design support system, and a semiconductor device design support program capable of identifying a model.

  According to another aspect of the present invention, there is provided a semiconductor device design support method in which a plurality of design variables (defect factors) related to occurrence of defects are input variables, and measurable amounts related to a plurality of failure modes are output variables. In a statistical model of a hierarchical structure in which a plurality of potential state variables are interposed between a plurality of input variables and a plurality of output variables, a plurality of hypotheses relating to failure occurrence mechanisms in each failure mode are assumed, and the hypothesis A first step of assuming a function type of data distribution from the plurality of state variables to the plurality of output variables in consideration of a predetermined failure physical model corresponding to each failure mode; Given a random variable data set for state variables and a randomized variable probability data set with a discretized joint probability distribution and probability density of zero. Or a second step of calculating using a response surface model and Monte Carlo simulation based on the CAE analysis result data during the expansion of numerical experimental points, and the plurality of input variables and the plurality of output variables obtained in advance or A third step of estimating a parameter according to a function form of the assumed data distribution from the observation data set being monitored and the joint probability distribution of the plurality of state variables calculated in the second step; It has.

  Further, the semiconductor device design support apparatus according to one aspect of the present invention has a plurality of design variables (defect factors) related to occurrence of defects as input variables, and measurable amounts related to a plurality of failure modes as output variables, and In a statistical model of a hierarchical structure in which a plurality of potential state variables are interposed between the plurality of input variables and the plurality of output variables, assuming a plurality of hypotheses regarding the failure occurrence mechanism of each failure mode, For each of the hypotheses, a first means for assuming a function type of data distribution from the plurality of state variables to the plurality of output variables in consideration of a predetermined failure physical model corresponding to each failure mode; A random variable data set for a plurality of state variables and a discrete variable probability distribution and a random variable data set with a probability density not 0 are given in advance. Or a second means for calculating using the response surface model and Monte Carlo simulation based on the CAE analysis result data in which numerical experimental points are being expanded, and the plurality of input variables and the plurality of output variables obtained or monitored in advance A third means for estimating a parameter corresponding to the function form of the assumed data distribution from the observed data set in the medium and the joint probability distribution of the plurality of state variables calculated in the second step. To do.

  Further, the semiconductor device design support program according to one aspect of the present invention has a plurality of design variables (defect factors) related to occurrence of defects as input variables, and measurable amounts related to a plurality of failure modes as output variables, and In a statistical model of a hierarchical structure in which a plurality of potential state variables are interposed between the plurality of input variables and the plurality of output variables, assuming a plurality of hypotheses regarding the failure occurrence mechanism of each failure mode, A first step of assuming, for each of the hypotheses, a function type of data distribution from the plurality of state variables to the plurality of output variables in consideration of a predetermined failure physical model corresponding to each failure mode; A random variable data set for a plurality of state variables and a discretized joint probability distribution and a random variable data set with a probability density other than 0 are set in advance. A second step of calculating using a response surface model and a Monte Carlo simulation based on the CAE analysis result data for which a given or numerical experimental point is being expanded, and a plurality of input variables and a plurality of output variables obtained in advance; Or a third step of estimating a parameter corresponding to the assumed functional form of the data distribution from the observed data set being monitored or the joint probability distribution of the plurality of state variables calculated in the second step. And let the computer run.

  According to the present invention, in the semiconductor device design support method, the semiconductor device design support system, and the semiconductor device design support program, the CAE data and the data accumulated in the actual measurement are used as the total data, and the defect occurrence mechanism is used. The statistical model involved can be identified.

  Embodiments of the present invention will be described below with reference to the drawings. First, FIG. 1 shows a hierarchical statistical model in which a plurality of design variables are input variables, a measurable amount related to a plurality of failure modes is an output variable, and a plurality of potential state variables are interposed therebetween. It is a general-view figure for demonstrating. The embodiment described below is based on the assumption of the illustrated hierarchical structure.

  As shown in FIG. 1, in the semiconductor device to be designed, the amount that can be observed (measurable amount) corresponds to each failure mode y1. y1... is also an output variable 13. There are a plurality of factors x1... For each failure mode y1..., And each has a variation due to a variation factor such as a load variation. Factor x1... Is also an input variable (design variable) 11. The output variable 13 and the input variable 11 can be given specific values by the observation data 15.

  It is assumed that an amount (state quantity z1...) That is desired to be known in order to suppress each failure mode y1... But is not actually measurable (state quantity z1...) Is interposed between the input variable 11 and the output variable 13. The state quantity z1... Is a state variable 12. The state variable 12 can be given a specific value by CAE data 14 obtained by a simulation technique (so-called CAE) for design, production, and the like. The failure physical model 16 can be applied to the relationship from the state quantity z1 to each failure mode y1.

  First, a semiconductor device design support method according to an embodiment of the present invention will be schematically described. In order to create a statistical model for explaining observation data and identifying a failure mechanism from observation data 15 in a plurality of failure modes and a parameter survey result (CAE data 14) using CAE relating to failure factors, hierarchical Bayes Utilize models and response surface methodology. Here, a probability distribution obtained by Monte Carlo simulation from a response surface model based on CAE is used as the distribution related to the latent variable (state quantity) in the hierarchical Bayes model. Statistical estimation methods such as maximum posterior probability estimation (MAP (Maximization A Posterior) estimation) and EM (Expectation Maximization) method are used for parameter estimation of the data distribution model, and resampling method is used for robustness evaluation of the model. use. This makes it possible to estimate a statistical model that expresses a failure mechanism in the form of merging observation data and CAE results.

The general flow of the above data processing is as follows.
1) Set data distribution model p (y | z; α) of observation data in consideration of failure physics (see FIG. 2). When using proportional hazard model:
However,
2) Using CAE, response surface model and Monte Carlo simulation, calculate probability distribution p (z | w) for latent variable z. 1. Calculate probability distribution of state quantities using response surface model and Monte Carlo simulation. 2. Discretization of probability distribution after designating division range and number of divisions Extract data set z with probability density not 0 3) Select hyperparameters α and w of prior distribution and statistically estimate model parameter θ of data distribution Here, selection and identification of parameter θ is based on information criterion Take advantage of the method.
The bootstrap method can also be used in this estimation method.
4) Calculate posterior distribution p (z | y; α, w) by Bayesian method
5) Evaluate the stability of the estimated statistical model (evaluate by perturbing the estimation condition)
6) Identifying a defect occurrence hypothesis from the obtained posterior distribution (in this embodiment, Kalman filter and multiple hypothesis test are used)

  FIG. 3 is a somewhat detailed flowchart showing a semiconductor device design support method according to an embodiment of the present invention. Hereinafter, a description will be given with reference to FIG.

  In the target complex or single defect phenomenon, first, it is assumed that a defect occurrence mechanism (hypothesis) is assumed (step 30, defect phenomenon database 21). Possible hypotheses are assumed after each failure mode y1... And failure factor x1. However, in the hypothesis assumption, not only hypotheses that are considered in combination and uncertainties regarding many failure factors x1, but also uncertainties such as physical phenomenon modeling, material modeling, boundary conditions, and load conditions in the CAE analysis model. It also includes hypotheses obtained as a combination of factors considered in gender.

  For each target failure mode y1..., A target failure factor x1 and a latent variable (state quantity z1...) Are determined for each hypothesis (steps 51 and 53). Further, the observation data to be processed is determined for each failure mode y1... (Step 52). The failure mode y1, failure factor x1, and observation data are determined by referring to the observation database 44. Here, the latent variable means an important state quantity (deformation amount, strain range, curvature, stress, etc.) that influences the failure phenomenon, although it is difficult to actually carry out accurate measurements easily. ing.

  Next, the data distribution for each failure mode y1... Is determined in consideration of the failure physical model in the target failure mode y1... (Step 54, data distribution / failure physical model base 71). The statistical model (probability model) of the data distribution includes a plurality or a single observation variable y, latent variable (state quantity) z, and model parameter θ. In general, failure physical models are broadly divided into stress-strength models and probability models. Examples of the former include failure mechanism models, velocity process models, and cumulative damage models. Examples of the latter include failure rate models and multi-states. There are a stochastic process model (Markov model, multi-state model), weakest link model (Weibull model), bundle model (bundle model, Rosen model), and the like (see FIG. 2).

  A case where a proportional hazard model is used as an assumption example of a statistical model of data distribution will be described below. However, since it can be applied not only to the proportional hazard model but also to a statistical model based on other failure physics such as a Markov model, it is desirable to select a model that matches the failure physics of each failure phenomenon. Even when a clear and reproducible causal relationship cannot be found between the failure factor x1 related to many failure phenomena and the failure data, the relationship between the state quantity (latent variable) z1 that governs the failure phenomenon and the failure data Often, certain stochastic dependencies are found.

A proportional hazard model is used as a statistical estimation model to treat this stochastic dependence as a competitive risk problem in which individual variables affect multiple risks, and to determine the relative importance of each variable based on observed data. Exists. The proportional hazard model can quantitatively evaluate the effect of variables that affect the failure phenomenon and determine its importance. The failure rate μ is expressed as follows in order to express that the effect of a plurality of variables is received on the baseline failure rate μ ₀ in a multiple manner. However, if there is an interaction between the variables, the interaction term may be included in the argument of the following exp function.
At this time, the defect occurrence probability is applied as follows.
Here, B is a parameter of the statistical model, and is estimated based on observation data. R ₀ and R (t | z) are the following equations, and the distribution type is determined in consideration of failure physics.

This proportional hazards model and unknown parameters included in R _0, the unknown parameters B _i is inherent. In addition, when the defective data is not at the end of life, it is a physical quantity such as intensity data y instead of t.

  Next, the statistical estimation method to be adopted will be described. Hyperparameter α in the probability distribution of parameter θ included in the statistical model of data distribution, hyperparameter w inherent in the probability distribution of state quantity z (probability distribution parameters of each failure factor x1,..., Parameters inherent in response surface model, etc. ) Is determined, the prior distribution of the hyperparameters α, w is determined (steps 55 and 56). For example, in the case of determining the prior distribution of the hyperparameter of the data distribution, the prior distribution is determined based on the expected value of the distribution parameter and the confidence interval obtained from the result of applying the probability distribution to the observed data set in advance. It is possible.

  By considering the prior distribution of hyperparameters, it is possible to reflect subjective information such as knowledge based on experience. Here, only one of α and w is considered, or neither is considered. When the hyper parameter is considered, the estimation of the parameter θ in the statistical model of the data distribution is obtained as a probability distribution, and when it is not considered, the estimation is a point estimation. For estimation of the hyper parameter, a statistical estimation method such as MAP estimation based on an information amount standard such as ABIC or EIC can be used.

  Next, for each hypothesis of each failure mode, a probability distribution of the state quantity z1... Is obtained (step 57, steps 31 to 39, material / shape / load database 41, CAE analysis model base, response surface model base 43, observation. Database 44). A random variable set of state quantities is a causal correlation between defect factors x1... And state quantities z1... Based on a data set of state quantities z1. Is expressed by a response surface model and then calculated by Monte Carlo simulation using the response surface model. Here, for the state value of continuous value, discretization is performed after specifying the number of divisions and each division range, and after the discretization of the probability distribution of the state quantity, a data set whose probability density is not 0 is extracted. To do. This can greatly streamline the number of calculations when estimating a hierarchical statistical model.

  Next, the parameter θ included in the statistical model of the data distribution is estimated based on the target observation data set, the data distribution set above, the probability distribution of the parameter θ, and the probability distribution of the state quantity z (step 58). For the estimation of the parameter θ, a maximum likelihood method or an EM (Expectation Maximization) method is utilized.

  Based on the estimation result of the obtained parameter θ, the fitness (error evaluation) between the data distribution and the observed data is evaluated (step 59). If only one hypothesis satisfies the conformity evaluation specification, the hypothesis can be identified. If there is no case that satisfies the suitability evaluation specification, the determination of the target observation data, the determination of the data distribution, the determination of the prior distribution of the hyperparameters, and the calculation of the probability distribution of the state quantity are corrected and repeated again. If there is no hypothesis that fits even after these repetitions, the hypothesis assumption is corrected and the hypothesis is executed again.

  Next, a posteriori distribution is calculated from the obtained data distribution and the probability distribution of the state quantities z1,... By the Bayesian method (step 60). If the estimated specifications are not satisfied by the evaluation of the posterior distribution (step 61), the determination of the target observation data, the determination of the data distribution, the determination of the prior distribution of the hyperparameters, and the calculation of the probability distribution of the state quantity are corrected. Repeat again. If there is no hypothesis that fits even after these repetitions, the hypothesis assumption is corrected and the hypothesis is executed again.

  Next, the stability of the estimation result is evaluated (step 62). The above estimation result may be greatly influenced by estimation conditions such as variable discretization conditions and sampling selection, and it is important to evaluate the stability against fluctuations in the estimation conditions. It is possible to evaluate the stability (robustness) against the fluctuation of the estimation condition by repeating the estimation again after changing the estimation condition. If the stability can be confirmed, state quantity data for each hypothesis is determined (step 63).

  When only a matching hypothesis is obtained, a combination of failure factor data is identified by searching for a combination of failure factors x satisfying the state quantity data set obtained for the matching hypothesis using a response surface model. . When a plurality of matching hypotheses are obtained, the following defect factor data is identified. The case where the state estimation method and the multiple hypothesis test are used is shown below.

  First, a response surface model for estimation of defect factor data is expressed in a state space (step 81). Based on this state space expression, the state equation and the observation equation are expressed (step 82). The target state quantity data set obtained in the above process is converted into a normal variable and used as state quantity observation data for each hypothesis (step 83).

  Next, sampling data of failure factors for each hypothesis is generated from the probability distribution followed by the target failure factors x1... (Step 84). From the response surface model corresponding to each hypothesis, the sampling data of the target state quantity data is generated by the Monte Carlo method for the sampling data of the failure factor (step 85).

  Based on the observation data corresponding to each hypothesis, the failure factor data is estimated, sequentially updated, and the residual distribution is estimated by a state estimation method such as a Kalman filter (step 86).

  The most suitable model is extracted by the multiple hypothesis test of the response surface model corresponding to each hypothesis, and the failure occurrence mechanism (match hypothesis) is identified (step 87). A combination of defect factor data is identified from the obtained response surface model (step 88). Here, for the identified hypothesis, a combination of failure factors that reduces the failure occurrence probability may be searched (step 89).

  As described above, as data / phenomenon analysis results from material / manufacturing / test / load / quality monitoring are accumulated, new information is actively added to unknown parts of model information and defect mechanism information related to defect phenomena. It is considered useful to express with a statistical model having a causal correlation structure. With such a model, it is possible to build a methodology for actively performing information extraction and early detection / identification of failure mechanisms.

  FIG. 4 is a diagram showing a schematic configuration of an apparatus for implementing the semiconductor device design support method described above. 4, components corresponding to those shown in FIG. 3 are denoted with the same reference numerals. The apparatus 100 includes a CAE preprocessing unit 101, a statistical model estimation unit 102, and a defect occurrence hypothesis identification unit 103, and further includes an input device 110, an output device 120, a ROM 130, a RAM 140, a material / shape / A load database 41, a CAE analysis model base 42, a response surface model base 43, an observation database 44, and a data distribution / fault physical model base 71 are provided in the vicinity.

  The CAE preprocessing unit 101 executes steps 31 to 39 in FIG. 3, the statistical model estimation unit 102 executes steps 51 to 63 in FIG. 3, and the defect occurrence hypothesis identification unit 103 performs steps 81 to 81 in FIG. 89 is executed. The input device 110 is a man-machine interface that provides user input for operating the device 100. The output device 120 is a man-machine interface for displaying and outputting various information related to the device 100 to the user. The ROM 130 and the RAM 140 store software for controlling the apparatus 100 and provide a storage area necessary for processing.

  In the following, as a more specific example, a specific flow of a semiconductor device design support method will be described for the identification problem of the thermal deformation mechanism (failure mechanism) of a semiconductor package.

  Due to the thinning of devices and the increase in relative variations inherent in materials and shapes, the thermal deformation of semiconductor packages is due to slight differences in the combination of material properties such as M-shaped, W-shaped, saddle-shaped, and convex-shaped. Shows complex deformation behavior. In addition to material characteristics and dimensional variations, measurement data includes large variations, and in order to clarify the thermal deformation mechanism, not only improvements in analysis and evaluation techniques for defect phenomena, but also CAE results and measured data There is a need for new methods for statistical identification using

  When performing information processing on actual observation monitoring data such as experimental data such as tests and inspections and numerical experimental data by simulation, except when performing experiments after sufficiently designing the experiment and removing noise, There are few cases where a statistical model can be constructed with a small number of parameters from the beginning. It is important to improve testing and observation methods and evaluation techniques so that all uncertain elements can be removed, but it is often difficult to reduce the number of parameters due to development period, cost-effectiveness, and technical problems. . In this example, when processing observation data for multiple failure phenomena using statistical methods, multivariate data including latent variables is included in all the parameters necessary to explain the complex phenomenon. Focus on estimating with. There is a Bayesian method as a method of dealing with this kind of problem. First, the overall framework will be described according to the components of information processing by the Bayesian method.

  If the likelihood function can be constructed from the observed data and the probability distribution of the entire data, there is a maximum likelihood method for estimating the parameters of the statistical model. However, if the number of model parameters is large and the number of observation data is insufficient, it may not be possible to estimate well. In such a case, conventionally, “Reparametrization” has been performed using a polynomial regression model or the like to reduce the number of parameters. However, in many cases, a practical meaning cannot be found in the obtained polynomial parameters.

  The point of the Bayesian method is that if the simultaneous distribution of two probability distributions can be expressed, the other probability distribution can be estimated when one of the random variables is observed. In constructing the simultaneous distribution, it is important to appropriately assume the data distribution and the prior distribution. In this example, the prior distribution, which has been difficult to make a reasonable assumption in the past, is converted into a probability distribution for a plurality of latent variables for the uncertainty of each failure factor by a sequence using CAE, response surface method, and Monte Carlo method. It was set by calculating. Further, it is necessary to assume the data distribution after including knowledge about the generation mechanism of defective data. In this example, a proportional hazard model is assumed for each failure mode, which includes a reliability basis function that takes into account the failure physical model of the failure phenomenon, and allows easy understanding of the influence of each latent variable.

[Target]
FIG. 5 shows a CAE analysis model of the target semiconductor package. Thermal deformation analysis is performed using this analysis model. The target failure modes are 1) coplanarity (warping amount) of the semiconductor package and 2) thermal fatigue life of the solder joint. Further, the observation data of interest is 1) warpage distribution data of the semiconductor package (analysis data is warp distribution in the x direction and y direction), and 2) thermal fatigue life data of the temperature cycle test (FIG. 9, FIG. 9). (See FIGS. 10 and 11).

  As state quantities, the following 12 variables, which are important physical quantities in the above-described defect phenomenon, were targeted (see FIG. 8). Since only the warp distribution measurement data cannot extract the values of state quantities such as the curvature of the semiconductor package and the displacement in the horizontal direction due to insufficient data and noise, these variables are directly It can be said that it is a latent variable (state quantity) that cannot be observed.

  FIG. 6 is a diagram illustrating factors that are considered to contribute to thermal deformation of the semiconductor package. Thermal deformation factors are intricately intertwined, and there can be multiple hypotheses for expressing the same deformation mode. In order to clarify the warpage mechanism, a statistical identification method combining simulation and actual monitoring (and filtering) is indispensable. FIG. 7 is a diagram illustrating an example of a thermal deformation mode in a quarter portion of the semiconductor package. The warping mode changes greatly depending on the combination of the position where peeling occurs and the material characteristics of the member.

  FIG. 12 is a diagram showing an assumed hypothesis (thermal deformation mechanism). Moreover, FIG. 13 is a figure which shows the statistical model made into object in this example. As defect data y to be processed, 1) Warp distribution of semiconductor package, 2) Thermal fatigue life distribution of solder joints are set, and statistical model (hierarchical Bayes model, response surface model) including defect factor u Configured. As described above, 12 variables, which are important physical quantities in the behavior of thermal deformation, are used as state quantities (see FIGS. 8 and 15).

In this example, a proportional hazard model with the reliability basis function of the Weibull distribution as R ₀ is set for the thermal fatigue life data as the data distribution related to the observation data. For the warp distribution (x direction, y direction), the warp distribution can be uniquely defined when the state quantity is determined in the case of the object of this time. Therefore, as the data distribution, the delta function is used as the data distribution regarding the measurement point of the warp amount. Set. The statistical model (probabilistic model) of this data distribution includes observation variables y (thermal fatigue life data in this case), latent variables (state quantities) z and model parameters θ (Weibull distribution parameters ρ, β and proportional hazard model parameters). B) is included.

The data distribution is expressed as follows (however, since the number of state quantities is 12, k = 12)
Here, B is a parameter of the statistical model, and is estimated based on observation data. R ₀ and R (y | z) can be calculated from the following equations.
In this proportional hazard model, an unknown parameter included in R ₀ and an unknown parameter of B _i are inherent.

The reliability function of the Weibull distribution is expressed as follows.
Here, the prior distribution of the hyperparameter α in the probability distribution of the parameter θ included in the statistical model of the data distribution is the expected value and confidence interval of the distribution parameter obtained from the result of applying the probability distribution to the observed data set in advance. The prior distribution was determined based on the above, and the parameter value was determined by MAP estimation based on the information criterion.

Probability distribution of state quantities:
Next, for each hypothesis of each failure mode, a probability distribution of state quantities is obtained. The state variable random variable set represents the cause-and-effect correlation between the failure factor and the state variable using a response surface model based on the state variable data set obtained as a result of the phenomenon analysis by CAE for the target numerical experiment point. After that, it is calculated by Monte Carlo simulation using a response surface model.

  Here, for the state value of continuous value, discretization is performed after specifying the number of divisions and each division range, and after the discretization of the probability distribution of the state quantity, a data set whose probability density is not 0 is extracted. did. In this case, a normal distribution is set for the failure factor u, a Monte Carlo simulation is performed in the range of the average value ± 20%, and each state quantity is discretized by dividing it into 10 parts between the maximum value and the minimum value. A data set of state quantities whose probability density is not 0 was calculated. Since it becomes a large-scale Monte Carlo simulation, it is thought that there is a need to speed up the calculation processing by distributed parallelization or the like in order to further subdivide and increase the accuracy.

Response surface model parameter values:
Using the result of thermal deformation simulation using the structural analysis software ABAQUSV Ver6.4, response surface models were created for each of the 12 state quantities by the response surface method based on the experimental design method L18. The parameter values included in the 12 response surface models are shown in FIG. Here, u represents an equivalent failure factor. The state quantity z is as shown in FIG. For reference, FIGS. 16 and 17 show the influence analysis results of each factor by analysis of variance based on the L18 experiment design method (in the case of hypothesis 2).

Estimating statistical model parameters:
Based on the target observation data set, the data distribution set above, the probability distribution of the parameter θ, and the random variable data set of the state quantity z, the parameter θ included in the statistical model of the data distribution (in the case of the current target) Weibull distribution parameters ρ, β and proportional hazard model parameters B) are estimated for each hypothesis. The EM method was used to estimate the parameter θ this time.

Hypothesis testing by statistical identification:
In this example, hypothesis verification using a Kalman filter and a multiple hypothesis test was attempted (see FIGS. 18, 19, 20, 21, and 22). First, the response surface model of each hypothesis was expressed in a state space (see FIG. 19).

  Based on the parameter θ obtained by the estimation of the statistical model corresponding to the identified hypothesis, the maximum error was ± 10% or less as a result of the fitness evaluation (error evaluation) between the data distribution and the observation data. . In order to reproduce the thermal deformation measurement results and reliability test results for the semiconductor package targeted this time, the anisotropy of the material properties of the interposer surface layer, the decrease in the apparent elastic modulus due to stress relaxation of the package resin, and the equivalent line It became clear that it was necessary to consider the increase in expansion rate.

  In the following, a supplementary description will be given of statistical methods such as the parameter estimation method used in the present embodiment.

[MAP estimation]
The distribution of the future observation value z estimated based on the data y is represented by P (z | y). The goodness of P (z | y) can be evaluated by the magnitude of its average log likelihood ∫Q (z) logP (z | y) dz. Q (z) is a generally unknown true distribution of the random variable Z. It is considered that a model having a larger average log likelihood has a better approximation to the true distribution Q (z). This criterion is called an information criterion. It is possible to compare the optimality of distributions averaged by posterior distribution from the viewpoint of information criterion. Consider a case where the generation mechanism Q (y) of the data y and z can be written by a function P (y | θ ^* ). In the following, the case where the θ distribution π (θ ^* ) is known will be considered.

Information standard ABIC:
Akaike et al. Defined the following information criterion ABIC and proposed a method for determining the hyperparameter θ by minimizing the ABIC. The second item is a correction term based on the dimension of θ, which enables model comparison / selection of data distribution and prior distribution.
That is, the information criterion ABIC (α, w) is
It is represented by

The average log likelihood of P (z | y) is
Can be expressed as
Since an estimation method that provides a good estimate only for a specific z is not necessary, the above average log likelihood is averaged with the distribution of data z and is expressed as follows.
Since θ itself is unknown, the following equation is obtained by averaging with the distribution of θ.

The last expression takes the maximum value when:
Log (s) ≦ s−1 for any s> 0 from the logarithmic function form
From now on,

If the true prior distribution of the parameters is known, the mixture distribution under the posterior distribution obtained by the Bayesian method:
Is the optimal prediction distribution when the data y is observed. π (θ | y) The value of θ that maximizes if it is sharp at a single peak:
With the parameter value
Is an approximate prediction distribution, and the parameter estimates are
Becomes the estimated MAP value.

  If the true prior distribution of parameters is known, the use of MAP estimation is justified from the viewpoint of the information criterion. It is a strict condition that the true prior distribution of parameters is known, but if the assumed parametric prior distribution family {π (θ | α, w)} includes a true prior distribution, the ABIC minimization method is used. It can be considered that the MAP estimation based on the prior distribution model selected in the above is justified.

Information standard EIC:
If {π (θ | α, w)} contains a true prior distribution, MAP estimation gives a good estimate in terms of information criteria. Even if {π (θ | α, w)} does not include a true prior distribution, the average log likelihood of the MAP estimation model:
There are other options, such as the log-likelihood of a non-Bayesian model:
If it was better, it would be better to use that estimate.

The information criterion AIC is obtained from the following equation. That is, AIC is
It is. The second terms S and T are random variables that are independent of y but follow the same distribution as y. This part is
Whether or not is the maximum likelihood estimate of θ based on data y, the second term uses a set of independent random numbers {S _i } and {T _i } that follow the same distribution as y.
And approximate calculation.

The EIC uses the bootstrap sample y ^* as {S _i } and {T _i }.
EIC is an information amount standard defined by the following equation. That is, EIC is
It is. in this case,
Does not have to be the maximum likelihood estimator, and may be MAP estimation. The same model parameter selection method can be selected by EIC comparison in the same way that model selection can be performed by AIC comparison.

The actual calculation of EIC is then according to:
_{^{However, y Y i = {y Y}} 1, y Y 2, ..., y Y K} is the bootstrap samples. In calculating the EIC, resampling of residuals may be more useful than resampling of raw data.

[Hyperparameter selection and identification problems]
If the values of the hyperparameters are different, naturally different models are estimated from the same observation data. Prior distribution is introduced in order to utilize prior information as much as possible, but if the prior distribution includes hyperparameters, it is necessary to give a selection method. In order to solve this problem systematically, Akaike et al. Have proposed a method that utilizes an information criterion based on the maximum likelihood method.

  The prior distribution of latent variables and the data distribution determine the simultaneous distribution of data and latent variables. If the data distribution or prior distribution depends on hyperparameters (in this example, parameter α included in the data distribution, assumed probability distribution of failure factors, and response surface model parameter w), the simultaneous distribution depends on the hyperparameters Will do. If a set of data and latent variables is formally considered as a model of data distribution, hyperparameters can be estimated by the maximum likelihood method after observing the data and setting the latent variables as missing values.

  For the latent variable (missing value), the periphery of the latent variable is integrated and the latent variable z is integrated-out as a peripheral distribution, or a temporary value is substituted. In the hyper parameter estimation, MAP estimation based on an information amount standard such as ABIC or ECI can be used. For estimation of the parameter θ of the data distribution, it is possible to use EM estimation excellent in calculation efficiency. In this example, w is treated as a definite value, and after estimating only α as a hyper parameter by the MAP estimation method, the parameter θ is estimated by the EM method. When w is also considered as a hyper parameter, it is necessary to obtain the information criterion after recalculating the prior distribution for each updated value of w. When bootstrap sampling is performed, both the case where only the observation data y is targeted and the case where the sampling is repeated for both the observation data y and the latent variable z are possible. The information criterion EIC based on the bootstrap method allows the use of MAP estimation when the prior distribution model does not include a true distribution.

[Maximum likelihood estimation]
The statistical model is one of the mathematical models that express the target phenomenon. For the purpose of statistical estimation, the unknown parameters of the statistical model assumed for the phenomenon are estimated, and the data is smoothed, identified, and predicted. is there. A characteristic of the statistical model is that a variable that defines data has irregularity, and the uncertainty is expressed as a random variable that follows a certain probability distribution.

  Statistical model variables are broadly divided into observation variables that are observed as direct data and latent variables that are not directly observable. A latent variable may be referred to as a state quantity, pixel, missing value, or hidden variable. By introducing latent variables, a more flexible model can be constructed. There are many statistical models that include latent variables. For example, in the case of the mixed distribution estimation problem, it cannot be originally observed from which element distribution the observation data is generated. In addition, hidden Markov models used in speech recognition and state space models used in time series analysis are models having latent variables. The EM method is a general numerical solution method for obtaining a maximum likelihood estimated value of a model parameter of a statistical model having these latent variables. It was devised in 1977 by Dempter, Laird, and Rubin, and is also used for parameter estimation such as graphical modeling and covariance structure analysis.

Maximum likelihood estimation of incomplete data:
The observation data is D, and the set of latent variables corresponding to each of the observation data is Z. At this time, D is called incomplete data and (D, Z) is called complete data. Consider a case where a latent variable takes a discrete value and the probability distribution of complete data is given as p (D, Z; θ). However, θ represents an unknown parameter. The log-likelihood function for the observed data (incomplete data) can be expressed by the following formula (if the latent variable is a continuous value, the sum can be replaced with an integral).

Since it is marginalized with respect to Z, the likelihood function in the statistical model including latent variables is also called a marginal likelihood function. The maximum likelihood estimate of the unknown parameter θ given the observation data D is the parameter value that maximizes the log-likelihood function L:
Is defined as Where argmax (*) represents the variable that maximizes *. Intuitively, the maximum likelihood estimated value is a parameter value that most supports the observation data D. However, in the maximum likelihood estimation method, the support degree is evaluated by a log likelihood function. Since the above maximization belongs to a nonlinear optimization problem and cannot be solved analytically, it is necessary to use a nonlinear optimization method such as Newton's method. The EM method is an analysis method for estimating more efficiently.

[EM method]
In the EM method, instead of directly maximizing the above-mentioned log likelihood function, the conditional expected value of the log likelihood function of complete data, which is often a simpler expression, is maximized. In the EM method, a maximum likelihood estimate is obtained by successive iteration of two types of steps. Assuming that the estimated value of the parameter at the t-th iteration is θ ^(t) , first, in the expected value calculation step (E step), the complete data log likelihood function:
Expected conditional value of:
Calculate Here, the symbol E {f (x | y)} represents a conditional expected value of f (x) when y is given.
Is the posterior distribution of Z when D is observed, and is calculated by the following equation from Bayes' theorem.

The calculation of the Q function at E step is intuitively the posterior distribution of latent variables calculated based on the current parameter estimates:
It can be interpreted that the incomplete information is tentatively supplemented with its reliability. Next, θ that maximizes the Q function in the maximization step (M step) is set as an estimated value of the parameter in the (t + 1) th iteration. These EM steps are repeated until the convergence condition is satisfied.

  In the EM method, when calculating the Q function, the sum over the values of all possible latent variables is calculated. At this time, if the number of values that the latent variable can take becomes an exponential order, it is difficult to calculate the sum in a realistic time. Therefore, in this example, the latent variable z is expressed by a response surface model related to a failure factor, and a data set related to a combination of latent variables whose probability density is not 0 in a probability distribution related to the latent variable by Monte Carlo simulation based on the response surface model in advance. Extraction reduces the calculation scale.

When the observation variable y = {y ₁ , y ₂ ,..., Y _M } and the latent variable Z = {z ₁ , z ₂ ,..., Z _M }, the likelihood function for the observation data can be written as follows. Here, θ = (θ _y , θ _z ) represents a model parameter.
The Q function in step E is
Can be written.
The simultaneous distribution of (y, Z) is as follows:

The following equation can be calculated from Bayes' theorem.
A Monte Carlo method that approximates the E step by sampling can be applied to the calculation of the E step.

Unlike maximum likelihood estimation, Bayesian estimation calculates a prediction distribution instead of a prediction value. In addition, Bayesian estimation has an advantage that estimation with high generalization ability is possible with respect to maximum likelihood estimation when observation data is small. Thus, in Bayesian estimation, the posterior distribution based on observed data plays a central role. In Bayesian estimation, all unknown quantities are treated as random variables. For a statistical model (probability model) having an unknown parameter θ, a model structure M, and a latent variable set Z, the following graphical model is usually assumed. In this case, the observation data D indicates that it is independent of the model structure M (p (D | Z, M) = p (D | Z)) when Z is given.

From the above, the simultaneous distribution can be decomposed as:
Here, p (D, Z | θ, M) represents a likelihood function of complete data (D, Z) under a known model structure, and p (θ | M) and p (M) are parameters. Represents the prior distribution of the model structure. The simultaneous posterior distribution of all unknowns with the observation data set D obtained from the above equation is given by the following equation from Bayes' theorem.

The denominator of the above equation is equivalent to p (D). In addition, the posterior distribution of the parameter θ
As obtained. If the posterior distribution is obtained, the posterior prediction distribution for the new sample d can be obtained by the following equation.
The above equation can be obtained approximately using Monte Carlo simulation by discretizing the variables.

Overview for explaining a statistical model of a hierarchical structure in which multiple design variables are input variables, measurable quantities related to multiple failure modes are output variables, and multiple potential state variables are interposed between them. Figure. Explanatory drawing which illustrates a general failure physical model. The flowchart which shows the semiconductor device design support method which concerns on one Embodiment of this invention. The figure which shows schematic structure of the apparatus for enforcing the semiconductor device design support method shown in FIG. The figure which shows the CAE analysis model of the semiconductor package made into the object in the identification problem of the thermal deformation mechanism (defect mechanism) of a semiconductor package. The figure which shows the defect factor in the identification problem of the thermal deformation mechanism (defect mechanism) of a semiconductor package. The figure which shows the example of the thermal deformation mode in the 1/4 part of the semiconductor package in the identification problem of the thermal deformation mechanism (defect mechanism) of a semiconductor package. The figure which shows the state quantity in the identification problem of the thermal deformation mechanism (defect mechanism) of a semiconductor package. The figure which shows one of the observation data in the identification problem of the thermal deformation mechanism (defect mechanism) of a semiconductor package. The figure which shows another one of the observation data in the identification problem of the thermal deformation mechanism (defect mechanism) of a semiconductor package. The figure which shows another one of the observation data in the identification problem of the thermal deformation mechanism (defect mechanism) of a semiconductor package. The figure which shows the hypothesis (thermal deformation mechanism) assumed in the identification problem of the thermal deformation mechanism (defect mechanism) of a semiconductor package. The figure which shows the statistical model made into the object in the identification problem of the thermal deformation mechanism (defect mechanism) of a semiconductor package. The figure which shows the example of the response surface model calculated | required in the identification problem of the thermal deformation mechanism (defect mechanism) of a semiconductor package. The figure which shows the state quantity in the statistical model shown in FIG. The figure which shows the influence analysis result of each factor calculated | required in the identification problem of the thermal deformation mechanism (defect mechanism) of a semiconductor package. FIG. 17 is a continuation diagram of FIG. 16, showing the influence analysis result of each factor obtained in the identification problem of the thermal deformation mechanism (defective mechanism) of the semiconductor package. The figure which shows the flow of hypothesis verification using the Kalman filter and the multiple hypothesis test in the identification problem of the thermal deformation mechanism (defect mechanism) of a semiconductor package. The figure which shows the state space expression of the response surface model of each hypothesis. The figure which shows the search range of an equivalent defect factor. The figure which shows the hypothesis verification by the Kalman filter and the multiple hypothesis test in the identification problem of the thermal deformation mechanism (defect mechanism) of a semiconductor package. The figure which shows the identification result in the identification problem of the thermal deformation mechanism (defect mechanism) of a semiconductor package.

Explanation of symbols

  DESCRIPTION OF SYMBOLS 11 ... Input variable, 12 ... State variable, 13 ... Output variable, 14 ... CAE data, 15 ... Observation data, 16 ... Failure physical model, 41 ... Material / shape / load database, 42 ... CAE analysis model base, 43 ... Response Curved surface model base, 44 ... Observation database, 71 ... Data distribution / fault physical model base, 100 ... Semiconductor device design support device, 101 ... Pre-processing unit by CAE, 102 ... Statistical model estimation unit, 103 ... Identification of defect occurrence hypothesis 110, input device, 120, output device, 130, ROM, 140, RAM.

Claims (5)

A plurality of design variables (defect factors) related to occurrence of defects are set as input variables, and measurable quantities related to a plurality of failure modes are set as output variables, and a plurality of values are set between the plurality of input variables and the plurality of output variables. In a statistical model of a hierarchical structure in which a plurality of potential state variables intervene, assuming a plurality of hypotheses regarding the failure occurrence mechanism of each failure mode, the plurality of output from the plurality of state variables for each of the hypotheses A first step of assuming a function type of data distribution to variables in consideration of a predetermined failure physical model corresponding to each failure mode;
A response surface based on the CAE analysis result data given in advance or expanding the numerical experimental points, the random variable data set related to the plurality of state variables and the discretized joint probability distribution and the random variable data set having a probability density not 0. A second step of calculating using a model and Monte Carlo simulation;
From the observation data set obtained in advance or being monitored for the plurality of input variables and the plurality of output variables, and the joint probability distribution of the plurality of state variables calculated in the second step, the hypothesized And a third step of estimating a parameter according to the function form of the data distribution.   A statistical model having a high degree of fitness is selected from the statistical model of the hierarchical structure obtained by the third step, and using the selected statistical model, a failure factor (input variable) related to the temporary occurrence of the failure occurrence mechanism is selected. The semiconductor device design support method according to claim 1, further comprising a fourth step of identifying. The first step sets a hierarchical Bayesian model considering a proportional hazard model as the data distribution,
The semiconductor device design support method according to claim 1, wherein the third step is performed using a MAP estimation method, a maximum likelihood method, an EM method, and a bootstrap method. A plurality of design variables (defect factors) related to occurrence of defects are set as input variables, and measurable quantities related to a plurality of failure modes are set as output variables, and a plurality of values are set between the plurality of input variables and the plurality of output variables. In a statistical model of a hierarchical structure in which a plurality of potential state variables intervene, assuming a plurality of hypotheses regarding the failure occurrence mechanism of each failure mode, the plurality of output from the plurality of state variables for each of the hypotheses A first means for assuming a function type of data distribution to variables in consideration of a predetermined failure physical model corresponding to each failure mode;
A response surface based on the CAE analysis result data given in advance or expanding the numerical experimental points, the random variable data set related to the plurality of state variables and the discretized joint probability distribution and the random variable data set having a probability density not 0. A second means for calculating using a model and Monte Carlo simulation;
From the observation data set obtained in advance or being monitored for the plurality of input variables and the plurality of output variables, and the joint probability distribution of the plurality of state variables calculated in the second step, the hypothesized And a third means for estimating a parameter according to the function form of the data distribution. A plurality of design variables (defect factors) related to occurrence of defects are set as input variables, and measurable quantities related to a plurality of failure modes are set as output variables, and a plurality of values are set between the plurality of input variables and the plurality of output variables. In a statistical model of a hierarchical structure in which a plurality of potential state variables intervene, assuming a plurality of hypotheses relating to failure occurrence mechanisms in each failure mode, the plurality of outputs from the plurality of state variables for each of the hypotheses A first step of assuming a function type of data distribution to variables in consideration of a predetermined failure physical model corresponding to each failure mode;
A response surface based on the CAE analysis result data given in advance or expanding the numerical experimental points, the random variable data set related to the plurality of state variables and the discretized joint probability distribution and the random variable data set having a probability density not 0. A second step of calculating using a model and Monte Carlo simulation;
From the observation data set obtained in advance or being monitored for the plurality of input variables and the plurality of output variables, and the joint probability distribution of the plurality of state variables calculated in the second step, the hypothesized A semiconductor device design support program for causing a computer to execute a third step of estimating a parameter according to a function form of data distribution.