JP2009076772A - Process monitoring method - Google Patents

Abstract

<P>PROBLEM TO BE SOLVED: To provide a process monitoring method by an MT method using principal component analysis, the process monitoring method being capable of unequivocally determining the number of principal components. <P>SOLUTION: In the process monitoring method, a principal component is set based upon data acquired in manufacturing processes of a product, the square value MD2 of a Mahalanobis distance is calculated based upon the principal component, and the quality of the product is predicted based upon MD2. Then the number (s) of principal components is determined to be a value (e) at which an indicator function IND(e) has a minimum value, where (g) is the number of principal components generated when principal component analysis is carried out with respect to a correlation coefficient matrix R between items of data, (e) is an integer of 1 to (g), (n) is the number of sets of data, (p) is the number of items, λ<SB>j</SB>is a j-th eigen value, ER is an error amount, and IND(e) the indicator function associated with the error amount ER. <P>COPYRIGHT: (C)2009,JPO&INPIT

Description

  The present invention relates to a process monitoring method, and more particularly to a process monitoring method using an MT method (Mahalanobis-Taguchi method).

  2. Description of the Related Art In a semiconductor device manufacturing process in which a semiconductor device is manufactured by performing various processes on a semiconductor wafer, a technique for performing process management by an MT method (Mahalanobis-Taguchi method) is known (see, for example, Patent Document 1). Process management by the MT method is obtained from data acquired in the manufacturing process, that is, QC (Quality Control) data obtained from the manufacturing apparatus in the middle of the process, and semiconductor wafers during or after processing. Based on QC data, especially QC data obtained from TEG (Test Element Group), the square value of the Mahalanobis distance (hereinafter referred to as “MD2”) is calculated, and the product (semiconductor device) is produced in the middle of the process. It is a method of predicting and issuing an alarm when necessary.

The calculation procedure of MD2 that has been conventionally performed is as follows.
It is assumed that the number of data items acquired during the process, that is, the number of parameters is p, and the number of data sets is n. The acquired data includes QC data obtained from a manufacturing apparatus and QC data obtained from a wafer. The QC data obtained from the manufacturing apparatus includes, for example, the temperature of heat treatment performed in a certain manufacturing apparatus. The QC data obtained from the wafer includes, for example, the width of the gate electrode formed on the wafer. On the other hand, n is the number of wafers, for example. At this time, if the j-th (j is an integer from 1 to p) data in the i-th (i is an integer from 1 to n) data is a _ij , the data of the entire process has each element a _ij It can be expressed as a matrix of n rows and p columns. Further, the average value of the data for the j th parameter and m _j, the standard deviation and sigma _j, the normalized data x _ij normalized data a _ij, can be calculated by the following formula 1.

Then, assuming that the correlation coefficient between the j-th parameter and the k-th parameter (k is an integer from 1 to p) is r _jk , the correlation coefficient r _jk is given by the following equation 2. The correlation coefficient of the entire process can be expressed as a p-by-p correlation coefficient matrix R with each element r _jk . At this time, the correlation coefficient r _jj that is a diagonal element, that is, j = k is 1, and the correlation coefficient r _jk that is a non-diagonal element, that is, j ≠ j, is r _jk = r _kj. . The inverse matrix of the correlation coefficient matrix R is R- ¹ .

The data of the i-th data set is a p-th order vector, that is, a p-row and one-column matrix X _i (= x _i1 , x _i2 ,..., x _ip ), and its transposed matrix, that is, one row p When a matrix of columns is ^t X _i , MD2 can be calculated by the following Equation 3. The suffix “t” indicates transposition of a matrix or a vector.

  Then, an appropriate threshold value is set for MD2, and when the MD2 of a certain data set becomes larger than this threshold value, it is determined that the quality of the product (semiconductor device) represented by this data set is poor.

However, the conventional techniques described above have the following problems.
Problem 1 is that, in general, some of the parameters acquired in the manufacturing process of a semiconductor device have multicollinearity. In the presence of multicollinearity, the inverse matrix R ⁻¹ cannot be defined. For this reason, it is necessary to perform an artificial operation of removing one of the plurality of parameters having multi-collinearity while leaving one parameter.

  Problem 2 is that not all parameters acquired in the manufacturing process of a semiconductor device generally affect the product performance, and there are extra parameters that do not affect the product performance. It is done. If MD2 is calculated including such extra parameters, the product performance prediction ability is reduced. For this reason, it is necessary to select only the parameters effective for prediction of the performance from the acquired parameters.

Parameters used for process management, that is, data items are selected using a two-level orthogonal table. This is because the determination accuracy deteriorates if there are extra items.
FIG. 5 is a diagram illustrating a two-level orthogonal table used for parameter selection.
In the orthogonal table 6 shown in FIG. 5, symbols A to H indicate parameters. On the other hand, each row indicates a condition for calculating MD2, that is, a combination of parameters used for the calculation. Of the numbers written in the respective columns of the columns in which the symbols A to H are described, “1” indicates that the parameter is used for the calculation of MD2, and “2” indicates that the parameter is not used. The orthogonal table 6 is used when the number P of parameters is 8 or less, and when P is 9 or more, a larger two-level orthogonal table is used. Y1 to Yn represent MD2 values calculated for each of the n data sets. Furthermore, η indicates the desired SN ratio calculated for each condition. Then, a condition that maximizes the desired SN ratio, that is, a combination of parameters is determined. Such a complicated operation is required for item selection.

  As described above, in the above-described conventional method, an artificial operation of selecting one parameter from a plurality of parameters having multicollinearity is necessary, and if this operation is not performed appropriately, the product performance will be improved. The selection of parameters that affect the performance is also inappropriate.

Therefore, a method using principal component analysis (PCA) is known as another method for calculating MD2. Principal component analysis is a method of synthesizing (compressing) a number of correlated parameters, setting a number of principal components that are not correlated with each other, and performing analysis using the principal components. The j-th principal component vector of the above-described correlation coefficient matrix R is f _j , the j-th eigenvalue is λ _j , the j-th eigenvector is w _j , and a matrix having this eigenvector w _j as an element is W, When a diagonal matrix having the eigenvalue λ _j as a diagonal element is Λ, the correlation coefficient matrix R can be modified as in the following Equation 4. The j-th principal component vector f _j is given by the following mathematical formula 5.

From the above formula 4, the following formula 6 is established for the inverse matrix R- ¹ . Further, from Equation 3 and Equation 6 below, MD2 for the i-th data set X _i is as shown in Equation 7 below.

  In this way, by using principal component analysis, MD2 can be easily calculated without considering multicollinearity between parameters by using mutually independent principal components. Thereby, the above-mentioned problem 1 can be avoided.

However, this method has a problem 3 described below. That is, the higher the principal component, that is, the greater the value of _j in the principal component vector f _j , the closer the eigenvalue λ _j becomes to 0 and the principal component vector f _j also approaches the 0 vector. Become stable. For this reason, it is necessary to appropriately terminate the calculation of principal component vectors and to optimize the number of principal component vectors. However, the optimal number of principal component vectors depends on the problem to be handled, so it must be determined uniquely. It is difficult. As for the above-described problem 2, if the number of principal component vectors cannot be set appropriately, the selection of effective parameters may be inappropriate. Therefore, this method cannot solve the problems 2 and 3.

JP-A-9-330858

  An object of the present invention is to provide a process monitoring method based on the MT method using principal component analysis, which can uniquely determine the number of principal components.

According to one aspect of the present invention, a step of setting a principal component based on data acquired in a manufacturing process of a product, a step of calculating a square value of the Mahalanobis distance based on the principal component, and the Mahalanobis distance Predicting the product performance based on a square value, and the number of principal components is the number of principal components generated when a principal component analysis is performed on a correlation coefficient matrix between items of the data. The number is g, the integer from 1 to g is e, the number of data sets is n, the number of items is p, the jth eigenvalue is λ _j , the error amount is ER, and the index function is IND When (e) is set, a process monitoring method is provided in which the value of e is such that the index function IND (e) given by the following formula takes a minimum value.

According to another aspect of the present invention, a step of setting a principal component based on data acquired in a manufacturing process of a product, a step of calculating a square value of a Mahalanobis distance based on the principal component, and the Mahalanobis Predicting the product performance based on a square value of the distance, and the number of principal components is a main component generated when a principal component analysis is performed on a correlation coefficient matrix between items of the data. When the number of components is g, the integer from 1 to g is e, the number of data sets is n, the number of items is p, the e-th eigenvalue is λ _e, and L is a variable, There is provided a process monitoring method characterized in that a value (e-1) obtained by subtracting 1 from a value of e such that a value of F calculated by a mathematical formula exceeds a reference value is provided.

According to still another aspect of the present invention, a process of calculating an inverse matrix of a correlation coefficient matrix between items of the data based on data acquired in a product manufacturing process, and a Mahalanobis based on the inverse matrix A step of calculating a square value of the distance, and a step of predicting the product performance based on the square value of the Mahalanobis distance, and the data item was subjected to principal component analysis on the correlation coefficient matrix Sometimes the number of principal components generated is g, the integer from 1 to g is e, the number of data sets is n, the number of items is p, the jth eigenvalue is λ _j , and the error amount Is ER and the index function is IND (e), the value of e such that the index function IND (e) given by the following formula takes the minimum value is the number of principal components of the data, Select based on principal components Process monitoring wherein the the those are provided.

According to still another aspect of the present invention, a process of calculating an inverse matrix of a correlation coefficient matrix between items of the data based on data acquired in a product manufacturing process, and a Mahalanobis based on the inverse matrix A step of calculating a square value of the distance, and a step of predicting a product performance based on the square value of the Mahalanobis distance, wherein the items of data are principally related to a correlation coefficient matrix between the items of the data. G is the number of principal components generated when component analysis is performed, e is an integer from 1 to g, n is the number of data sets, p is the number of items, and the e-th eigenvalue is λ. _{When e} is L and L is a variable, a value (e-1) obtained by subtracting 1 from the value of e such that the value of F calculated by the following mathematical formula exceeds the reference value is defined as the number of the principal components, and the number Selected based on the principal components of Process monitoring wherein the is provided.

  According to the present invention, it is possible to realize a process monitoring method by an MT method using principal component analysis, which can uniquely determine the number of principal components.

Hereinafter, embodiments of the present invention will be described with reference to the drawings.
First, a first embodiment of the present invention will be described.
FIG. 1 is a block diagram illustrating a semiconductor factory in this embodiment.
In FIG. 1, the signal flow is indicated by a solid line, and the wafer flow is indicated by a broken line.

  As shown in FIG. 1, in the semiconductor factory 1 according to the present embodiment, a plurality of manufacturing apparatuses 2 that process a semiconductor wafer W are provided. As the semiconductor wafer W sequentially passes through the plurality of manufacturing apparatuses 2, a plurality of processes are sequentially performed on the semiconductor wafer W to manufacture a semiconductor device. Further, an inspection apparatus 3 is also provided between the manufacturing apparatuses 2 or on the downstream side of the manufacturing apparatus 2. The inspection device 3 inspects the semiconductor wafer in the middle of the process or after the process is completed. A plurality of manufacturing apparatuses 2 and inspection apparatuses 3 form a manufacturing line.

  The semiconductor factory 1 is provided with a process monitoring device 4. The process monitoring device 4 acquires manufacturing data such as processing conditions from the manufacturing device 2, acquires product data such as dimensions of the microstructure formed on the semiconductor wafer W from the inspection device 3, and based on these data The product performance is predicted according to the procedure described below. When it is predicted that the product is poor, an alarm is transmitted.

Next, the process monitoring method according to the present embodiment will be described.
FIG. 2 is a flowchart illustrating the process monitoring method according to this embodiment.
FIG. 3 is a graph illustrating the distribution of non-defective products and defective products with MD2 on the horizontal axis and frequency on the vertical axis.

  In the semiconductor factory 1 shown in FIG. 1, semiconductor wafers W are sequentially introduced into a production line, processed continuously by a plurality of manufacturing apparatuses 2, and continuously inspected by an inspection apparatus 3. The process monitoring device 4 continuously acquires manufacturing data from the manufacturing device 2 and continuously acquires product data from the inspection device 3 to monitor the process.

Hereinafter, the process monitoring method by the process monitoring apparatus 4 will be described in detail.
First, as shown in step S _< b _> 1 of FIG. 2, the process monitoring device 4 acquires data a _ij from the manufacturing device 2 and the inspection device 3. The data a _ij is manufacturing data or product data acquired in the manufacturing process as described above, for example, processing conditions and dimensions of the fine structure, and the like, for example, the heating temperature of the heat treatment and the width of the gate electrode. is there. Next, as shown in step S2, the data a _ij is normalized according to the above mathematical formula 1, and standardized data x _ij is created.

Next, as shown in step S3 in FIG. 2, by rotating the coordinate in the x space of normalized data x _ij, in order from the largest eigenvalue lambda _1, ie in descending order, to compute the eigenvalues lambda _j. Also, the eigenvector w _j is calculated, and the j-th principal component vector f _j is also calculated. At this time, the eigenvalue λ _j is significant up to a certain order (s-th) eigenvalue, but the subsequent eigenvalues are considered to be due to errors, and the s-th eigenvalue from the _first eigenvalue λ1. Up to λ _s is calculated, and the (s + 1) th and subsequent eigenvalues λ _j are not calculated. Similarly, for the eigenvector w _j and the principal component vector f _j , only the first to sth are calculated, and the (s + 1) th and subsequent are not calculated. A method for determining the value of s will be described later.

Next, as shown in step S4, with respect to the i-th data set X _i , the first to s-th eigenvalues λ _j and the principal component vector f _j are used to calculate the square of the Mahalanobis distance according to the following formula 8. The value MD2 is obtained. The following formula 8 is a formula in which p = s in the above formula 7.

  Next, as shown in step S5, the product performance is predicted based on the value of MD2. That is, as shown in FIG. 3, the non-defective product distribution 7 forming the unit space and the defective product distribution 8 deviating from the unit space are independent of each other. If the calculated MD2 exceeds the threshold value, it is determined that the product represented by the MD2 is included in the defective product distribution 8, and it is determined that the product is defective. The process monitoring device 4 issues an alarm when it determines that the product quality is poor.

Hereinafter, a method for determining the number s of eigenvalues to be calculated will be described.
FIG. 4 is a graph illustrating the index function IND (e) with e on the horizontal axis and IND (e) on the vertical axis.
Let g be the number of principal components generated when principal component analysis (PCA) is performed on a correlation coefficient matrix R between items (parameters) of data acquired by the process monitoring device 4, and an integer from 1 to g to e Where n is the number of data sets and p is the number of parameters. Then, the error amount ER is calculated by the following formula 9. J is a variable. Further, as shown in Equation 10 below, g takes the smaller value of n and p. The error amount ER corresponds to the total amount of errors caused by the (e + 1) th to gth principal components.

  Next, an index function IND (e) is obtained from the value of the ER by the following formula 11. The index function IND (e) is a function obtained empirically.

  Then, as shown in FIG. 4, a value of e is obtained such that the index function IND (e) takes a minimum value, and the value of e is set as the number s of eigenvalues. In this way, the calculation of the eigenvalue λ is terminated at the sth eigenvalue, MD2 is calculated using only the 1st to sth eigenvalues, and the (s + 1) th and subsequent eigenvalues are not used in the MD2 calculation. The index function IND (e) corresponding to can be minimized. In FIG. 4, the index function IND (e) is represented by a curve for convenience of illustration, but since the value of e is actually an integer, the value of IND (e) is also discrete. .

Next, the effect of this embodiment will be described.
As described above, according to the present embodiment, the number s of principal components can be uniquely determined in the process monitoring method based on the MT method using principal component analysis. Thereby, highly accurate process monitoring can be performed without intervening artificial operation. Further, the work of examining the correlation between factors and the work of extracting effective factors using a two-level orthogonal table are not required.

Next, a second embodiment of the present invention will be described.
The present embodiment is different from the first embodiment in the method for determining the number of main components s. G is the number of principal components generated when principal component analysis (PCA) is performed on the correlation coefficient matrix R between parameters, e is an integer from 1 to g, and n is the number of data sets. when the number of parameters is p, the e-th eigenvalue lambda _e, a conversion eigenvalues REVe defined by the following equation 12. The conversion eigenvalue REVe is an amount proportional to the standard deviation.

On the other hand, the (e + 1) th and subsequent pooled eigenvalues _{REV pooled,} calculated by the following equation 13. Note that L is a variable. Then, the value of the ratio F is calculated as shown in Equation 14 below.

  The value of F calculated by the above mathematical formula 14 corresponds to a risk factor. The reason is that the e-th eigenvalue indicates the magnitude (variance) of fluctuation of the e-th item, so that the variance calculated from the e-th eigenvalue and the variance calculated from the (e + 1) -th eigenvalue and later If the sum can be regarded as statistically equal, the items up to the (e-1) th item are valid, and the items after the eth can be determined as errors. Further, from an empirical rule, it is possible to calculate the number of data or the number of items as the degree of freedom by using the eigenvalue as the magnitude of variation. If F is about 1, the above-described determination is possible. However, what value can be said by F is determined by F test. For example, the determination is made from the F table with 1 and (ge) degrees of freedom.

  Therefore, the reference value of the risk rate is set appropriately, in one example, 5%, the value of e is increased by 1, and the value of e immediately before the value of F exceeds this reference value is set to MD2 Let s be the number of principal components used in the calculation. That is, when the value of F exceeds the reference value when e = e ′, it is determined that the influence of the e′-th principal component on MD2 is not significant, and the eigenvalues up to (e′−1) -th eigenvalues. And MD2 using the principal component vector. That is, s = e′−1.

  Other configurations in the present embodiment are the same as those in the first embodiment. Also in this embodiment, the same effect as the first embodiment described above can be obtained.

Next, a third embodiment of the present invention will be described.
In the present embodiment, parameters constituting the s-th principal component are specified from the first principal component based on the value of s determined by the method of the first embodiment described above. Then, it is assumed that an item (parameter) is selected by regarding the identified parameter as an effective parameter that affects the performance of the product. Next, using these parameters, an inverse matrix R ⁻¹ of the correlation coefficient matrix R is obtained, and MD2 is calculated based on the following formula 15 instead of the above formula 8. Note that the matrix representing the i-th data set is X _i (= x _i1 , x _i2 ,..., X _ip ), the transposed matrix is ^t X _i, and the number of data sets is n.

  Other configurations in the present embodiment are the same as those in the first embodiment. Also in this embodiment, the same effect as the first embodiment described above can be obtained.

Next, a fourth embodiment of the present invention will be described.
In the present embodiment, parameters constituting the s-th principal component are specified from the first principal component based on the value of s determined by the method of the second embodiment described above. Then, in the same manner as in the third embodiment described above, an inverse matrix R ⁻¹ is obtained, and MD2 is calculated by the above formula 14. Other configurations and effects in the present embodiment are the same as those in the second and third embodiments described above.

  While the present invention has been described with reference to the embodiments, the present invention is not limited to these embodiments. For example, as long as a person skilled in the art appropriately adds, deletes, or changes a design, or adds, deletes, or changes a process for each of the above-described embodiments, the gist of the present invention is included. , Within the scope of the present invention. For example, the product to be monitored is not limited to a semiconductor device, but may be another industrial product.

It is a block diagram which illustrates a semiconductor factory in a 1st embodiment of the present invention. It is a flowchart figure which illustrates the process monitoring method which concerns on 1st Embodiment. FIG. 5 is a graph illustrating the distribution of non-defective products and defective products with MD2 on the horizontal axis and frequency on the vertical axis. FIG. 6 is a graph illustrating an index function IND (e) with e on the horizontal axis and IND (e) on the vertical axis. It is a figure which illustrates the orthogonal table of the 2 level system used for selection of a parameter.

Explanation of symbols

1 Semiconductor Factory, 2 Manufacturing Equipment, 3 Inspection Equipment, 4 Process Monitoring Equipment, 6 Orthogonal Table, 7 Good Product Distribution, 8 Defective Product Distribution

Claims (5)

A step of setting the main component based on data acquired in the manufacturing process of the product;
Calculating a square value of Mahalanobis distance based on the principal component;
Predicting the performance of the product based on the square of the Mahalanobis distance;
With
The number of principal components is g, where e is an integer from 1 to g, and g is the number of principal components generated when principal component analysis is performed on the correlation coefficient matrix between the data items. When the number is n, the number of the items is p, the j-th eigenvalue is λ _j , the error amount is ER, and the index function is IND (e), the index function IND (e ) Is a value of e that takes a minimum value.

A step of setting the main component based on data acquired in the manufacturing process of the product;
Calculating a square value of Mahalanobis distance based on the principal component;
Predicting the performance of the product based on the square of the Mahalanobis distance;
With
The number of principal components is g, where e is an integer from 1 to g, and g is the number of principal components generated when principal component analysis is performed on the correlation coefficient matrix between the data items. the number is n, the number of the item and p, the e-th eigenvalue and lambda _e, when the L a variable, 1 from the value of e as the value of F calculated by the following formula exceeds a reference value A process monitoring method characterized in that the value (e-1) is obtained by subtracting.

Calculating an inverse matrix of a correlation coefficient matrix between items of the data based on data acquired in a product manufacturing process;
Calculating a square value of Mahalanobis distance based on the inverse matrix;
Predicting the performance of the product based on the square of the Mahalanobis distance;
With
The data items include g as the number of principal components generated when principal component analysis is performed on the correlation coefficient matrix, e as an integer from 1 to g, and n as the number of sets of the data. When the number of items is p, the jth eigenvalue is λ _j , the error amount is ER, and the index function is IND (e), the index function IND (e) given by the following equation takes the minimum value. The value of e is the number of principal components of the data, and the process monitoring method is selected based on the number of principal components.

Calculating an inverse matrix of a correlation coefficient matrix between items of the data based on data acquired in a product manufacturing process;
Calculating a square value of Mahalanobis distance based on the inverse matrix;
Predicting the product performance based on the square of the Mahalanobis distance;
With
In the data item, g is the number of principal components generated when principal component analysis is performed on a correlation coefficient matrix between the data items, e is an integer from 1 to g, and the number of sets of the data the is n, the number of the item and p, the e-th eigenvalue and lambda _e, when the L and variable, 1 from the value of e as the value of F calculated by the following formula exceeds a reference value The subtracted value (e-1) is the number of the principal components, and the process monitoring method is selected based on the number of principal components.

  The process monitoring method according to claim 1, wherein the product is a semiconductor device.

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