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Hierarchical indices to detect equipment condition changes with high dimensional data for semiconductor manufacturing

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Abstract

During semiconductor manufacturing process, massive and various types of interrelated equipment data are automatically collected for fault detection and classification. Indeed, unusual wafer measurements may reflect a wafer defect or a change in equipment conditions. Early detection of equipment condition changes assists the engineer with efficient maintenance. This study aims to develop hierarchical indices for equipment monitoring. For efficiency, only the highest level index is used for real-time monitoring. Once the index decreases, the engineers can use the drilled down indices to identify potential root causes. For validation, the proposed approach was tested in a leading semiconductor foundry in Taiwan. The results have shown that the proposed approach and associated indices can detect equipment condition changes after preventive maintenance efficiently and effectively.

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Acknowledgments

This research was partially supported by National Science Council, Taiwan (NSC100-2628-E-007-017-MY3; NSC102-2622-E-007-013), Taiwan Semiconductor Manufacturing Company (100A0259JC), and National Tsing Hua University under the Toward World-Class University Project (101N2074E1). The authors deeply appreciate constructive comments and suggestions from the anonymous reviewers as well as the supports and inputs of domain experts including Dr. Yi-Chun Chen and Dr. Chun-Ju Wang.

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Correspondence to Chen-Fu Chien.

Appendix

Appendix

This section shows the derivation of an approximate power function for the proposed approach. The subscript ijk is omitted in this section. Assume that (1) holds for \(t\le t_0\) and

$$\begin{aligned} M^{*}(t):y(t)=\beta _0^*+\beta _1^*t+\varepsilon ^{*}(t),\;\varepsilon ^{*}(t){\mathop {\sim }\limits ^{iid}} N(0,\sigma _1^2 ), \end{aligned}$$
(10)

with \(\beta _0^*,\,\beta _1^*\), and \(\sigma _1\) unknown for \(t>t_0\); that is, equipment condition changes occur at \(t_{0}\). Let

$$\begin{aligned} \mu (t)&= \beta _0 +\beta _1 t \\ \psi _0^2 (t)&= \sigma _0^2 \left[ {1+\frac{1}{n(t_0 )}+\frac{\left( {t-{\bar{t}}} \right) }{\sum _{i=1}^{n(t_0 )} {\left( {t_i -{\bar{t}}} \right) ^{2}} }} \right] \end{aligned}$$

and

$$\begin{aligned} \mu ^{*}(t)&= \beta _0^*+\beta _1^*t \\ \psi _1^2 (t)&= \left[ {\sigma _1^2 +\frac{\sigma _0^2 }{n(t_0 )}+\frac{\sigma _0^2 \left( {t-\bar{{t}}} \right) }{\sum _{i=1}^{n(t_0 )} {\left( {t_i -{\bar{t}}} \right) ^{2}} }} \right] \end{aligned}$$

denote the mean and variance functions for a single measurement before and after \(t_{0}\), respectively. Note that \(n_0^*=n(t_0)+1\). The probability that the first measurement after \(t_{0}\) is outside of the PI is

$$\begin{aligned}&P\left\{ {\left| {\frac{y\left( {t_{n_0^*} } \right) -{\hat{\mu }}_{t_0 } \left( {t_{n_0^*} } \right) }{{\hat{\psi }}_{t_0} (t_{n_0^*})}}\right| >T_{n(t_0 )-2,1-\alpha _0 /2} } \right\} \nonumber \\&\quad \!\!=\!\!P\left\{ \left| {\frac{{\left( {\mu \left( {t_{n_0^*} } \right) \!\!+\!\!\varepsilon \left( {t_{n_0^*} } \right) \!-\!\hat{{\mu }}_{t_0 } \left( {t_{n_0^*} } \right) \!\!+\!\!\delta \left( {t_{n_0^*} } \right) } \right) }/{\psi _1 (t_{n_0^*} )}}{{{\hat{\psi }}_{t_0} (t_{n_0^*})}/{\psi _0(t_{n_0^*})}}} \right| ^{2}\nonumber \right. \\&\qquad \left. >\frac{\psi _0^2 (t_{n_0^*})}{\psi _1^2 (t_{n_0^*})}T_{n(t_0)-2,1-\alpha _0 /2}^2 \right\} , \end{aligned}$$
(11)

where

$$\begin{aligned} \delta \left( {t_{n_0^*} } \right) =\mu ^{*}\left( {t_{n_0^*} } \right) -\mu \left( {t_{n_0^*} } \right) . \end{aligned}$$

The left side of the last inequality in (11) is noncentral F distributed with degrees of freedom 1 and \(n(t_0)-2\) and non-centrality of \({\delta (t_{n_0^*})}/{\psi _1}(t_{n_0^*})\). The probability that the next \(n_{1}\) indicator readings fail the chi-square test is

$$\begin{aligned}&P\left\{ {\sum \limits _{i=n_0^*+1}^{n_0^*+n_1 } {\left( {\frac{y\left( {t_i } \right) -{\hat{\mu }}_{t_0 } \left( {t_i } \right) }{{\hat{\psi }}_{t_0} \left( {t_i } \right) }} \right) } ^{2}>\chi _{n_1 ,1-\alpha _1}^2}\right\} \nonumber \\&\quad =\!P\left\{ \sum \limits _{i=n_0^*+1}^{n_0^*+n_1 } {\left( {\frac{{\left( \mu \left( {t_i } \right) \!+\!\varepsilon ^{*}\left( {t_i } \right) \!\!-\!\!{\hat{\mu }}_{t_0 } \left( {t_i } \right) \!\!+\!\!\delta \left( {t_{n_0^*} } \right) \right) }/{\psi _1 \left( {t_i } \right) }}{{{\hat{\psi }}_{t_0}\left( {t_i } \right) }/{\psi _0 \left( {t_i } \right) }}} \right) }^{2}\right. \nonumber \\&\qquad \left. \left( {\frac{\psi _1^2 (t_i )}{\psi _0^2 (t_i )}} \right) >\chi _{n_1 ,1-\alpha _1 }^2 \right\} . \end{aligned}$$
(12)

The left side of the last inequality in (12) is the sum of the independent noncentral F distribution with degree of freedom 1 and \(n(t_0)-2\) and non-centrality of \({\delta (t_{n_0^*})}/{\psi _1}(t_{n_0^*})\) multiplied by inflation factor \({\psi _1(t_i)}/{\psi _0}(t_i)\). Indicator performance is considered to change at \(t_{0}\) if \(y\left( {t_{n_0^*}}\right) \) is outside of the PI and the following \(y\left( {t_{n_0^*+1}}\right) \) to \(y\left( {t_{n_0^*+n_1}}\right) \) fail the chi-square test. Therefore, the power function of the approach is the product of (11) and (12). As training sample size increases, (12) is the asymptotic equal to:

$$\begin{aligned}&P\left\{ \sum \limits _{i=n_0^*+1}^{n_0^*+n_1 } {\left( {\frac{\mu \left( {t_i } \right) +\varepsilon ^{*}\left( {t_i } \right) -{\hat{\mu }}_{t_0 } \left( {t_i } \right) +\delta \left( {t_{n_0^*} } \right) }{\psi _1 \left( {t_i } \right) }} \right) } ^{2}\right. \nonumber \\&\left. \quad \left( {\frac{\psi _1^2 (t_i )}{\psi _0^2 (t_i )}} \right) >\chi _{n_1 ,1-\alpha _1 }^2 \right\} . \end{aligned}$$
(13)

Assume \(\sigma _0=\sigma _1\), (13) is distributed as noncentral chi-squared distribution with degree of freedom \(n_{1}\) and noncentrality is \(\sum _{i=n_0^*}^{n_0^*+n_1}{{\delta \left( {t_i}\right) }/{\psi \left( {t_i}\right) }}\). Figure 9 shows that a larger mean drift and sample size results in higher power for the training and testing samples. If \(\delta \left( {t_i}\right) =0\), each term in the summation in (13) is distributed as inflated chi-squared distributions with degree of freedom 1 and the inflation factor \({\psi _1(t_i)}/{\psi _0}(t_i)\). Figure 10 shows the relationship between power and minimal inflation factor \({\psi _1(t_{n_0^*})}/{\psi _0}(t_{n_0^*})\). When significance levels \(\alpha _0\) and \(\alpha _1\) are set to 0.01, the overall significance level is 0.0001. Models (1) and (0) are equivalent when \(\delta ({t_i})=0\) and \({\psi _1(t_i)}/{\psi _0}(t_i)=1\). A horizontal dashed line shows the overall type one error rate. Figures 9 and 10 show that a larger sample size for the training and testing samples increases power. However, increasing the sample size increases the time required for data collection, indicating that a tradeoff between power and time exists.

Fig. 9
figure 9

Power function of \(\delta (t)\). a Training sample size, b testing sample size

Fig. 10
figure 10

Power function of \({\psi _1(t_{n_0^*})}/{\psi _0}(t_{n_0^*})\). a Training sample size, b testing sample size

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Yu, HC., Lin, KY. & Chien, CF. Hierarchical indices to detect equipment condition changes with high dimensional data for semiconductor manufacturing. J Intell Manuf 25, 933–943 (2014). https://doi.org/10.1007/s10845-013-0785-3

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