多重失效模式微電子資料之可靠度分析

以作者查詢圖書館館藏

、以作者查詢臺灣博碩士

、以作者查詢全國書目

、勘誤回報

、線上人數：95

、訪客IP：3.145.99.226

姓名

陳睦璿(Mu-Hsuan Chen) 查詢紙本館藏

畢業系所

統計研究所

論文名稱

多重失效模式微電子資料之可靠度分析
(Reliability Analysis for Microelectronics Data with Multiple Failure Modes)

相關論文

★ 具Box-Cox轉換之逐步加速壽命實驗的指數推論模型	★ 多元反應變數長期資料之多變量線性混合模型
★ 多重型 I 設限下串聯系統之可靠度分析與最佳化設計	★ 應用累積暴露模式至單調過程之加速衰變模型
★ 串聯系統加速壽命試驗之最佳樣本數配置	★ 破壞性加速衰變試驗之適合度檢定
★ 串聯系統加速壽命試驗之最佳妥協設計	★ 加速破壞性衰變模型之貝氏適合度檢定
★ 加速破壞性衰變模型之最佳實驗配置	★ 累積暴露模式之單調加速衰變試驗
★ 具ED過程之兩因子加速衰退試驗建模研究	★ 逆高斯過程之完整貝氏衰變分析
★ 加速不變原則之偏斜-t過程	★ 花蓮地區地震資料改變點之貝氏模型選擇
★ 颱風降雨量之統計迴歸預測	★ 花蓮地區地震資料之長時期相關性及時間-空間模型之可行性

檔案

[Endnote RIS 格式]

[Bibtex 格式]

[相關文章]

[文章引用]

[完整記錄]

[館藏目錄]

至系統瀏覽論文 (2026-6-14以後開放)

摘要(中)

微電子製程尺寸逐年微縮，致使產品產生多元的失效機制，同時失效機制亦是影響其壽命的重要因素。微電子資料常具有早夭、多峰、厚尾等特徵，且因試驗時間的限制下，亦可能產生設限資料。因此分析微電子資料頗具挑戰性，特別是當試驗樣本少時更為不易。實務上，混合模型與串聯系統模型常用以配適多重失效模式的資料，但兩模型間的本質差異卻不易區別。另一方面，工程師和試驗操作人員的專業背景與經驗，卻可以對模型提供重要的資訊。本文以獨立之對數常態分布配適各失效模式之資料，分別以混合模型和串聯系統模型剖析具多重設限之產品失效機制與可靠度分析。藉由隱藏變數與設限樣本之補值，於貝氏架構中以共軛先驗分布建立共軛結構，進而簡化計算的複雜度。將兩模型分別應用於閘極氧化層資料與電遷移資料中，顯示融合先驗資訊之貝氏方法可提供實用的可靠度分析，並可驗證專家建議之模型的適切性。

摘要(英)

The year-by-year downscaling in microelectronics results in multiple failure modes which makes an essential impact on the products’ lifetime. Features such as infant mortality, multiple failure modes, and heavy tails are common in microelectronics data. Under the limitation of experimental time, it also yields censored data. Therefore, it is a challenging task for analyzing the microelectronics data, especially with a limited amount of test units. The mixture model and the series system model are often used practically. But the essential difference between these two models is hardly distinguished. On the other hand, experts with domain knowledge and experience can provide important information for the model. In this thesis, independent log-normal distributions are considered for the failure times of different failure modes in the mixture model and the series system model which are used to analyze the failure mechanisms and reliability of multiply censored products. By imputing the latent variables and the censored observations, a Bayesian modeling with conjugate priors is constructed to simplify the computation. The proposed method is applied to gate oxide data and elctromigration data. It turns out that the Bayesian approach incorporated with the prior information provides useful reliability analysis, especially in confirming the validity of the models recommended by the experts.

關鍵字(中)

★ 混合模型
★ 串聯系統模型
★ 馬可夫鏈蒙地卡羅演算法
★ 偏差訊息法則
★ 後驗機率

關鍵字(英)

★ mixture model
★ series system model
★ MCMC
★ DIC
★ posterior probability

論文目次

摘要 i
Abstract ii
誌謝 iii
目錄 iv
圖目錄 vi
表目錄 vii
第一章緒論 1
1.1 研究背景 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 研究動機 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 文獻探討 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.4 研究方法 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.5 本文架構 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
第二章多失效模式模型之最大概似推論 7
2.1 混合模型 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.1.1 已知成分個數時參數之最大概似估計 . . . . . . . . . . . . . . . . . 8
2.1.2 拔靴法與區間估計 . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.1.3 模型選擇 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.1.4 可靠度推論 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.1.5 失效模式之判定 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.1.6 適合度檢定 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.2 串聯系統模型 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
iv
2.2.1 非隱蔽串聯系統模型參數之最大概似估計 . . . . . . . . . . . . . . 14
2.2.2 部分隱蔽串聯系統模型參數之最大概似估計 . . . . . . . . . . . . . 15
2.2.3 拔靴法與區間估計 . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.2.4 可靠度推論 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.2.5 失效模式之判定 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
第三章多失效模式之貝氏可靠度分析 19
3.1 混合模型之貝氏架構 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.1.1 樣本失效模式已知且無設限樣本時之貝氏方法 . . . . . . . . . . . . 19
3.1.2 樣本失效模式未知且無設限樣本時之貝氏方法 . . . . . . . . . . . . 21
3.1.3 樣本失效模式未知且含設限樣本之貝氏方法 . . . . . . . . . . . . . 23
3.1.4 貝氏模型選擇 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.1.5 貝氏可靠度推論 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.1.6 貝氏適合度檢定 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.2 串聯系統模型之貝氏架構 . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.2.1 部分隱蔽串聯系統之貝氏方法 . . . . . . . . . . . . . . . . . . . . . 29
3.2.2 貝氏可靠度推論 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
第四章實例分析 33
4.1 分析步驟 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
4.2 閘極氧化層資料 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
4.3 電遷移資料 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
第五章結論與展望 53
參考文獻 54
附錄 59
A.1 定理 3.1 之證明 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
A.2 定理 3.2 之證明 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
A.3 定理 3.4 之證明 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
A.4 系理 3.4.1 之證明 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
A.5 定理 3.5 之證明 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
v
圖目錄
1.1 閘極氧化層資料直方圖。 . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
4.1 閘極氧化層資料於 π3 先驗分布時配適模型參數之收斂圖。 . . . . . . . . 35
4.2 閘極氧化層資料於先驗分布 π1 時配適模型之圖。 . . . . . . . . . . . . . . 38
4.3 閘極氧化層資料於先驗分布 π2 時配適模型之圖。 . . . . . . . . . . . . . . 39
4.4 閘極氧化層資料於先驗分布 π3 時配適模型之圖。 . . . . . . . . . . . . . . 40
4.5 閘極氧化層資料於不同先驗分布時配適模型之機率密度函數圖。 . . . . . 41
4.6 電遷移資料直方圖。 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
4.7 電遷移資料於 π6 先驗分布時配適模型參數之收斂圖。 . . . . . . . . . . . 46
4.8 電遷移資料於先驗分布 π4 下配適模型之累積分布函數圖與機率圖。 . . . 49
4.9 電遷移資料於先驗分布 π5 下配適模型之累積分布函數圖與機率圖。 . . . 50
4.10 電遷移資料於先驗分布 π6 下配適模型之累積分布函數圖與機率圖。 . . . 50
vi
表目錄
4.1 閘極氧化層資料於不同失效模式個數之配適模型 DIC 值與 AIC 值。 . . . 34
4.2 閘極氧化層資料於兩種失效模式對數常態混合模型下之參數估計結果。 . 36
4.3 閘極氧化層資料於配適模型之第一種失效模式壽命之 q-分位數估計。 . . . 36
4.4 閘極氧化層資料於配適模型之第二種失效模式壽命之 q-分位數估計。 . . . 37
4.5 閘極氧化層資料於配適模型之產品壽命之 q-分位數估計。 . . . . . . . . . 37
4.6 閘極氧化層資料於先驗分布 π1 下之失效模式分類。 . . . . . . . . . . . . 42
4.7 閘極氧化層資料於先驗分布 π2 下之失效模式分類。 . . . . . . . . . . . . 43
4.8 閘極氧化層資料於先驗分布 π3 下之失效模式分類。 . . . . . . . . . . . . 44
4.9 電遷移資料於不同失效模式個數之配適模型 DIC 值與 AIC 值。 . . . . . 46
4.10 電遷移資料於兩種失效模式對數常態串聯系統模型下之參數估計結果。 . 46
4.11 電遷移資料於配適模型之平均失效時間估計。 . . . . . . . . . . . . . . . . 47
4.12 電遷移資料於不同先驗分布時配適模型之失效模式分類。 . . . . . . . . . 51
4.13 電遷移資料於配適模型之第一個元件壽命之 q-分位數。 . . . . . . . . . . 52
4.14 電遷移資料於配適模型之第二個元件壽命之 q-分位數。 . . . . . . . . . . 52
4.15 電遷移資料於配適模型之產品壽命 q-分位數估計。 . . . . . . . . . . . . . 52

參考文獻

[1] Akaike, H. (1974). A new look at the statistical model identification, IEEE Transactions
on Automatic Control, 6, 716–723.
[2] Barr, D. R. and Davidson, T.(1973). A Kolmogorov-Smirnov test for censored
samples, Technometrics 15, 739–757.
[3] Berger, J. O. (1985). Statistical Decision Theory and Bayesian Analysis(Second
Edition), Springer-Verlag, New York.
[4] Celeux, G., Forbes, F., Robert, C., and Titterington, M. (2006). Deviance information
criteria for missing data models, Bayesian Analysis, 1, 651–674.
[5] Chan, V. and Meeker, W. Q. (1999). A failure-time model for infant-mortality and
wearout failure modes, IEEE Transactions on Reliability, 48, 377–387.
[6] Degraeve, R., Ogier, J. L., Bellens, R., Roussel, P. J., Groeseneken, G., and Maes,
H. E. (1998). A new model for the field dependence of intrinsic and extrinsic timedependent dielectric breakdown. IEEE Transactions on Electron Devices, 45, 472–
481.
[7] DeIorio, M., and Robert, C. P. (2002). Discussion of Spiegelhater et al. Journal of
the Royal Statistical Society: series B, 64, 629–630.
[8] Dempster, A. P., Laird, N. M., and Rubin, D. B. (1977). Maximum likelihood from
incomplete data via the EM algorithm. Journal of the Royal Statistical Society:
series B, 39, 1–38.
[9] Depas, M., Nigam, T., and Heyns, M. M. (1996). Soft breakdown of ultra-thin gate
oxide layers. IEEE Transactions on Electron Devices, 43, 1499–1504.
54
[10] Diebolt, J. and Robert, C. P. (1994). Estimation of finite mixture distributions
through bayesian sampling. Journal of the Royal Statistical Society, 56, 363–375.
[11] Efron, B. (1979). Bootstrap methods: another look at the jackknife. The Annals of
Statistics, 7, 1–26.
[12] Fan, T. H. and Hsu, T. M.(2014). Constant stress accelerated life test on a multiplecomponent series system under Weibull lifetime distributions. Communications in
Statistics - Theory and Methods, 43, 2370–2383.
[13] Fan, T. H. and Hsu, T. M.(2015). Statistical inference of a two-component series
system with correlated log-normal lifetime distribution under multiple type-I censoring.
IEEE Transactions on Reliability, 40, 376–385.
[14] Fleming, T. R., O’Fallon, J. R., O’Brien, P. C., and Harrington, D. P.(1980).
Modified Kolmogorov-Smirnov test procedures with application to arbitrarily rightcensored data. Biometrics, 36, 607–625.
[15] Gelman, A., Meng, X. L., and Stern, H. (1996). Posterior predictive assessment of
model fitness via realized discrepancies, Statistica Sinica, 6, 733–807.
[16] Hasting, W. K. (1970). Monte carlo sampling methods using markov chains and
their applications. Biometrika, 57, 97–109.
[17] Hong, Y. and Meeker. W. Q. (2011). The importance of identifying different
components of a mixture distribution in the prediction of field returns. Applied
Stochastic Models in Business and Industry, 27, 280–289.
[18] Kaplan, E. L. and Meier, P. (1958). Nonparametric estimation from incomplete
observations. Journal of the American Statistical Association, 53, 457–481.
[19] Kirkpatrick, S., Gelatt, C. D. Jr., Vecchi, M. P. (1983). Optimization by simulated
annealing. Science, 220, 671–680.
[20] Koziol, J. A. (1980). Goodness-of-fit tests for randomly censored data. Biometrika,
67, 693–696.
[21] Lee, J. C., Chen, I. C., and Hu, C. M. (1988). Modeling and characterization of
gate oxide reliability. IEEE Transactions on Electron Devices, 35, 2268–2278.
55
[22] Leytham, K. M. (1984). Maximum likelihood estimates for the parameters of
mixture distributions. Water Resources Research, 20, 896–902.
[23] Li, B., McLaughlin, P. S., Bickford, J. P., Habitz, P., Netrabile, D., and Sullivan, T.
D. (2011). Statistical evaluation of electromigration reliability at chip level. IEEE
Transactions on Device and Materials Reliability, 11, 86–91.
[24] Linde, A. V. D.(2004). DIC in variable seleciton. Statistica Neerlandica, 59, 45–56.
[25] Little, R. J. A. and Rubin, D. B. (1987). Statistical Analysis with Missing Data
(Third Edition), John Wiley and Sons, New York.
[26] Marin, J. M., Mengersen, K. L., and Robert, C. P. (2005). Bayesian modelling and
inference on mixtures of distributions. Handbook of Statistics, 25, 459–507.
[27] Meeker, W. Q. (1987). Limited failure population life tests: application to integrated
circuit reliability. Technometrics, 29, 51–65.
[28] Meeker, W. Q. and Escobar, L. A. (1998). Statistical Methods for Reliability Data,
John Wiley and Sons, New York.
[29] Meng, X. L. (1994). Posterior predictive p-values, The Annals of Statistics, 22,
1142–1160.
[30] Metropolis, N., Rosenbluth, A. W., Rosenbluth, M. N., Teller, A. H., and Teller,
E. (1953). Equation of state calculations by fast computing machines. Journal of
Chemical Physics, 21, 1087–1092.
[31] Olosunde, A. and Olofintuade, S. (2022). Some inferential problems from log
Student’s T-distribution and its multivariate extension. Revista Colombiana de
Estadística, 45, 209–229.
[32] Pathak, M., Pak, J., Pan, D. Z., and Lim, S. K. (2011). Electromigration modeling
and full-chip reliability anaylsis for BEOL interconnect in TSV-based 3D ICs. IEEE/
ACM International Conference on Computer-Aided Design, 555–562.
[33] Raghavan N. and Tan C. M. (2007). Statistical analysis of mutli-censored electromigration
data using the EM algorithm. 14th International Symposium on the Physical and
Failure Analysis of Integrated Circuits, 257–262.
56
[34] Richardson, S. and Green, P. J. (1997). On Bayesian analysis of mixtures with an
unknown number of components (with discussion). Journal of the Royal Statistical
Society: series B, 59, 731–792.
[35] Rubin, D. B. (1984). Bayesianly justifiable and relevant frequency calculations for
the applied statistician, The Annals of Statistics, 12, 1151–1172.
[36] Smirnov, N. V. (1939). On the estimation of the discrepancy between empirical
curves of distribution for two independent samples. Moscow University Mathematics
Bulletin, 2, 3–14.
[37] Spiegelhalter, D. J., Best, N. G., Carlin, B. P., and Linde, A. V. L. (2002). Bayesian
measures of model complexity and fit. Journal of the Royal Statistical Society: series
B, 64, 583–639.
[38] Stathis, J. H. (1999). Percolation models for gate oxide breakdown. Journal of
Applied Physics, 86, 5757–5766.
[39] Sune, J. and Wu, E. Y. (2003). Statistics of successive breakdown events in gate
oxides, IEEE Electron Device Letters, 24, 272–274.
[40] Tan, C. M. and Raghavan, N. (2007). An approach to statistical analysis of gate
oxide breakdown mechanisms. Microelectronics Reliability, 47, 1336–1342.
[41] Tan, C. M. and Raghavan, N. (2008). Simulated annealing for mixture distribution
analysis and its applications to reliability testing. In Tech, 227–256.
[42] Tanner, M. A. and Wong, W. H. (1987). The calculation of posterior distributions
by data augmentation. Journal of the American Statistical Association, 82, 528–540.
[43] Wu, E. Y., Abadeer, W. W., Han, L. K., Lo, S. H., and Hueckel, G. R. (1999).
Challenge for accurate reliability projections in the ultra-thin oxide regime. IEEE
Annual International Reliablilty Physics Symposium.
[44] Yokogawa, S. (2015). Statistical characteristics of lifetime distribution based on
defect clustering for time-dependent dielectric breakdown in middle- and back-endof-line. Japanese Journal of Applied Physics, 54, 1–10.
57
[45] Yokogawa, S. and Tate, K. (2018). Reliability evaluation of defect accounted
time-dependent dielectric breakdown with competing-mixture distribution. IEEE
International Reliability Physics Symposium, P-GD.2-1 – P-GD.2-4.
[46] Yokogawa, S. and Kunii, K. (2019). A simple prediction method for chip-level
electromigration lifetime using generalized gamma distribution. IEEE International
Reliability Physics Symposium, 1–6.

指導教授

樊采虹(Tsai-Hung Fan)

審核日期

2023-6-15

推文