Vol. 34, No. 2 Journal of Semiconductors February 2013

A multivariate process capability index with a spatial coefficient�

Wang Shaoxi(王少熙)�, Wang Mingxin(王明辛), Fan Xiaoya(樊晓桠),Zhang Shengbing(张盛兵), and Han Ru(韩茹)

School of Software and Microelectronics, Northwestern Polytechnical University, Xi’an 710021, China

Abstract: After analyzing the multivariate Cpm method (Chan et al. 1991), this paper presents a spatial multi-variate process capability index (PCI) method, which can solve a multivariate off-centered case and may providereferences for assuring and improving process quality level while achieving an overall evaluation of process quality.Examples for calculating multivariate PCI are given and the experimental results show that the systematic methodpresented is effective and actual.

Key words: process; process capability index; multivariate; off-centerDOI: 10.1088/1674-4926/34/2/026001 EEACC: 0170N

1. Introduction

A process capability index (PCI) is a numerical summarythat compares the behavior of a product or process charac-teristic to engineering specifications. These measures are alsooften called capability or performance indices or ratios; weuse capability index as the generic term. A capability indexrelates the voice of the customer (specification limits) to thevoice of the process. A large value of the index indicates thatthe current process is capable of producing parts that, in alllikelihood, will meet or exceed the customer’s requirements.Over the last two decades, most of the development in PCIfocuses on a univariate process, such as Kane (1986), Chanet al . (1988), Choi & Owen (1990), and Boyles (1994)Œ1�6�.However, it is not uncommon in semiconductor manufactur-ing that one often encounters processes which involve sev-eral correlated variables of interest. In such a situation, sim-ply calculating the PCI’s of individual variables and combin-ing them together will inevitably fail to value the level ofprocesses. Therefore, it is more desirable to assess the pro-cess capability using a multivariate process capability index(MPCI).

In recent years, several attempts to develop a MCPI havebeenmade by various researchers such as Taam et al. (1993)Œ7�,Chen (1994)Œ8�, Pearn, Kotz and Johnson (1992)Œ9�, Wang andDu (2000), Kotz and Lovelace (1998)Œ10� and others. However,most of the existing MPCI require that the process be normallydistributed, and the index is largely dependent on the variance-covariance structure of the underlying distribution. The othersare reluctant to apply them in practice due to the difficulty ofcomputing.

Inspired by the recent works of Chan et al. (1991)Œ11�, aspatial multivariate process capability index is presented in thepaper. The MPCI can be applied for an off-centered process.Solving the coefficient used for an off-central case is so simplebut the complex multivariate cases cannot be summarized. Themodel and its modification have been given in this paper.

2. Univariate PCI

A process capability index relates the engineering specifi-cation (usually determined by the customer) to the observed be-havior of the process. The capability of a process is defined asthe specification width and the natural tolerance spread of theprocess. Some basic capability indices that have been widelyused in the semiconductor manufacturing industry include Cpand Cpk, explicitly defined as followsŒ1�:

Cp DUSL � LSL

6�; (1)

Cpk D min�USL � �

3�;

� � LSL3�

�; (2)

where USL and LSL are the upper and the lower specifica-tion limits respectively, � is the process mean; � is the processstandard deviation. Equation (1) is effective when the mean ofthe process data is equal to the median of specification limits;however, Equation (2) is used when the mean of the processdata is separate from the median of specification limits. Someresearchers build up a relationship between Cp and Cpk, whichcan be shown as follows:

Cpk D .1 � k/Cp; (3)

k DM � �

.USL � LSL/=2: (4)

Here, M is the specification interval midpoint.

3. MPCI

As a general case, define X as a m � n sample matrix,where m is the number of semiconductor process quality char-acteristics measured on a part and n is the number of partsmeasured. That is, each column in the matrix represents the m

* Project supported by the Natural Science Foundation of Shaanxi Province (No. 2012JQ8048) and the Basic Research Foundation of North-western Polytechnical University (No. JC20110232).

† Corresponding author. Email: shaoxi.wang@gmail.comReceived 10 July 2012, revised manuscript received 6 August 2012 © 2013 Chinese Institute of Electronics

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J. Semicond. 2013, 34(2) Wang Shaoxi et al.measurements recorded from a sampled part. These n observa-tions represent samples drawn from a multivariate distributionwith correlation among the m variables. Engineering specifi-cations for the processes are assumed to exist for each of the m

dimensions. Analogously to univariate process capability in-dices, multivariate capability indices also relate the allowedprocess region such as some measure of the specification re-gion, to the actual process region such as some measure of theprocess region. Therefore, a multivariate process capability in-dex is called MCp which is defined as:

MCp Dvol:.R1/

vol:.R2/; (5)

where R1 represents an engineering specification region andR2 is for a scaled 99.73% percent process region. In particular,if the process data satisfies the multivariate normal, then R2

is for an elliptical region. For Eq. (5), its computation in highdimension variables is complex and difficult. When the vari-able number in multivariate normal distribution is greater thanthree, achieving MPCI results using Eq. (5) is very difficult.Therefore, we need to transform the form and build a simplemodel.

The specification for the i th quality variable xi is usuallygiven by the triple of the lower specification limit LSLi , meanvalue �i and upper specification limit USLi . For the m pro-cess quality characteristics, a positive definite covariance ma-trix † D .�ij /, i; j D 1; : : :; m is commonly assumed. Thismultivariate PCI, here denoted by MCp presented by Chanet al. (1991), and is given as follows:

MCp D

QmiD1 .USLi � LSLi /

Volume.R/; (6)

whereQm

iD1 .USLi � LSLi / is the volume of the tolerance re-gion and R is the natural region. If we assume that X !

Nm.�; †/, then the natural region is the volume of an ellip-soid characterized by Eq. (7):

R D

nx 2 RC

W .x � �/T †�1.x � �/ 6 ro

; (7)

where r depends on the pre-specified probability that x falls inR (99.73% is the probability of 6� consideration). Equations(5) and (6) are easy to understand. Because the definition ofCp is just like comparing the length of the tolerance interval(bounded by specification) and the length of the natural toler-ance interval, as characterized by 6� under normality assump-tions. To extend this to a multivariate case, it seems naturalto consider the volume of the tolerance region as bounded byspecification and the volume of the natural region.

Considering the precondition for Eqs. (5) and (6) is thatthe process is centered, which means the mean of the processdata is equal to the median of the specification limits. Howeverthe process could be off-centered and that part of the naturalregion could actually be outside of the tolerance region. Wanget al. (2010)Œ12� extended Cpk to a multivariate case based onvolume and gave the MCpk as follows:

MCpk D .1 � Kv/

QmiD1 .USLi � LSLi /

Volume.R/; (8)

Kv Dvolume of deviation

0:5 � volume of tolerance region: (9)

Fig. 1. Two variable cases for the off-centered case process.

In there, the volume of deviation is built up by the toleranceregion and natural region midpoint. Now we use two variablecases to show how to achieveKv. The off-centered case processis shown in Fig. 1.

In Fig. 1, point a is the midpoint of tolerance region andpoint b is the midpoint of the natural region of the process. Weuse a and b coordinates to build a triangular �abc. Then Kvcan be shown as follows:

Kv DSabc

0:5StoleranceD

j�xjj�yj

Stolerance; (10)

where j�xj and j�yj are the deviations in the direction of thetwo variables respectively. According to Eq. (10), we madesome modifications to the coefficient. Moreover, If j�xj orj�yj is equal to zero, the case will be achieved as a one di-mensional case.

In the univariate case, k is the ratio of the distance betweenthe mean � and the target value T in a standard deviation unitdivided by half of (USL – LSL). We can get this ratio in themultivariate case in this way by modifying the equation below:

k D

jT � �j

�

0:5 �USL � LSL

�

: (11)

Then, the multivariate process capability index can beachieved:

MCpk0D .1 � k/MCp;

k DŒ.� � T /0†�1.� � T /�1=2

0:5 � Œ.USL � LSL/0†�1.USL � LSL/�1=2: (12)

4. Experimental analysis

Firstly, we use bivariate process data satisfying the bivari-ate normal to certificate the advantage of the multivariate pro-cess capability index using Eq. (12). Then we present an ex-ample with its application. Meanwhile, for the low dimensionnormal distribution, we can get an MPCI value using the yield.

MCpk00D

1

3˚�1

�field C 1

2

�D

1

3˚�1

��total C 1

2

�: (13)

�total is the total yield of the process, when the two variablesare independent, �total D �1�2, �1 and �2 are the yield of thetwo variables respectively. If the value of MCpk is closer to

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J. Semicond. 2013, 34(2) Wang Shaoxi et al.

Table 1. Experimental results for the two bivariates.

� D .200; 160/0� D 0 � D 0:3 � D 0:5

T D (180,150)0 T D (230,180)0 T D (180,150)0 T D (230,180)0 T D (180,150)0 T D (230,180)0

�1 D 6

�2 D 4

MCpk 1.214 1.096 1.2728 1.149 1.402 1.266MCpk0 0.7246 0.3124 0.759 0.3222 0.836 0.3473MCpk00 0.5733 0.27 0.58 0.2867 0.5883 0.3

Table 2. Two variable process data for Brinell hardness and tensile strength.H 143 200 160 181 148 178 162 215 161 141 175 187 187S 34.3 57.0 47.5 53.4 47.8 51.5 45.9 59.1 48.4 47.3 57.3 58.5 58.2H 186 172 182 177 204 178 196 160 183 179 194 181S 57.0 49.4 57.2 50.6 55.1 50.9 57.9 45.5 53.9 51.2 57.5 55.6

MCpk00 than that of MCpk0, we believe Eq. (10) is better thanEq. (12). The same happens to MCpk0.

The results of MCpks, MCpk0s and MCpk00s are listed inTable 1. In the table, � is the process mean; T is the targetvalue; �1 and �2 are the standard deviation of the two vari-ables respectively; � is the correlation coefficient; and we setthe upper and lower bound with � ˙ 3� .

In Table 1, the value of MCpk was achieved using Eqs. (8)and (10); the value of MCpk0 was gotten using Eqs. (8) and(12); and the value of MCpk00 was gotten using Eq. (13).According to the results of Table 1, we can find the value ofMCpk0 is closer to MCpk00 than that of MCpk. Therefore, us-ing Eqs. (8) and (12) we can obtain a reasonable multivariateprocess capability index.

Secondly, we use the Sultan (1986) bivariate process datashowed in Table 2 to examine the definition of a multivariatePCI over a rectangular zone. In this process, the Brinell hard-ness H and the tensile strength S of the output of a process areof interest.

From the data, we have n D 25, �H D 18:38, �S D 5.8,the correlation coefficient � D 0:8341, �H D 177.2 and �S D

52.32. We chose two cases to study the example shown as fol-lows.

(1) When the mean of the process data is equal to the me-dian of specification limits, we set the upper and lower bound-ries with �i ˙ 3�i i D H, S and (177, 53) for the median ofthe specification limits. The process in the case is capableŒ8�.

(2) When the mean of the process data is not equal tothe median of specification limits, we set the upper and lowerboundries with �i ˙ 3�i i D H, S and (150.45, 45.05) for themedian of the specification limits. The process in the case isnot capableŒ8�.

Then we get the multivariate PCI value for the two cases.

MCp D 1:52; MCpk D 1:23:

The MPCI value is large and does not give the process ca-pability information because the tolerance region is rectangularand nature region is elliptical, which leads to a larger value forMPCI. We can use the modified tolerance region instead of thetolerance regionŒ12� and get the results:

MCp0D 1:15; MCpk0

D 0:92;

where theMCp0 D 1.15means the centered process I is capable

and MCpk0 D 0.92 demonstrates the off-centered process II isnot capable.

5. Conclusions

The process capability ultimately decides the process qual-ity. Based on the analyzing process capability index, the pro-cess capability may be effectively assured. With the rapid de-velopment in process, the quality evaluation of processes con-cerns more than one quality characteristic. In this situation,simply calculating the univariate PCI of individual variablesand combing them together will inevitably fail to value thelevel of processes. Therefore, it is more desirable to assess theprocess capability using the multivariate Cpm method (Chanet al. 1991). This paper presents a modified spatial multivari-ate PCI method, which can solve a multivariate off-centeredcase and may provide references for assuring and improvingthe process quality level while achieving an overall evaluationof process quality. Examples for calculating multivariate PCIare given and the experimental results show that the modifiedmethod presented is effective and actual. Experimental analy-ses and practical examples using the modified method demon-strate the method is reasonable and effective.

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