Monitoring Wafer Geometric Quality using Additive Gaussian Process

Transcription

1 Monitoring Wafer Geometric Quality using Additive Gaussian Process Linmiao Zhang 1 Kaibo Wang 2 Nan Chen 1 1 Department of Industrial and Systems Engineering, National University of Singapore 2 Department of Industrial Engineering, Tsinghua University May 23, 2013

2 Outline 1 Introduction 2 Statistical Quantification using AGP Model 3 Statistical Monitoring of Geometric Quality 4 Case Studies 5 Conclusion and Future Directions

3 Motivation Integrated Circuits 3 / 42

4 Motivation Semiconductor Fabrication Process Ingot Slicing Lapping Polishing Cleaning Wafer Inspection Reject Disposal Accept Front End Back End Chips 4 / 42

5 Motivation Challenges Transistor size: 32nm 28nm 22nm 16nm 14nm Wafer size: 130mm 150mm 200mm 300mm 450mm 5 / 42

6 Motivation Wafer Preparation Process IC Companies Higher Integration Wafer Fabs Larger Diameter Require Cause Good Wafer Quality Bad 6 / 42

7 Introduction AGP Model Statistical Testing Case Studies Conclusion References Motivation Wafer s Geometric Quality Contact method: touching probes; Non-contact method: wavelength scanning interferometer; Measurements 60 Thicker 40 x Thinner x Engineers problem: how to check whether the surface is desirable? 7 / 42

8 Motivation Testing Problem / 42

9 Problem formulation Framework Surface as the Response Variable Modeling Monitoring Process Control Without covariate Regression with covariates Design optimization Change detection Design optimization Run-to-Run control Fault diagnostics 9 / 42

10 Problem formulation Difficulties Complete measurement of the wafer is slow Geometric profile is too complex to be modeled by parametric functions Measurements on different surfaces might not be aligned well Deviations (errors) are spatially correlated 10 / 42

11 Problem formulation State of the art One sample model: Gaussian process (Jin, Chang, and Shi 2012), PDE-constrained Gaussian process (Zhao, Jin, Wu, and Shi 2011) Only applicable for a single surface Primitive testing: summary indicators of the whole profile Total Thickness Variation (TTV), Bow, Warp, Site TIR (Doering and Nishi 2007); Need to fill in the gap 11 / 42

12 Review of GP Gaussian Process Y (x) = µ + Z(x) with PD covariance function k(x i, x j ) Suitable for spatially correlated data (Cressie 1993); Able to approximate complex function (Sacks et al. 1989); Able to evaluate prediction error (Santner et al. 2003) Prediction Sample True Function MSE of Prediction / 42

13 Review of GP Gaussian Process with Errors Errors present in physcial processes or stochastic simulations Y (x) = µ + Z(x) + ɛ(x) ɛ(x i ) are i.i.d. normally distributed: Σ + σ 2 I ɛ(x i ) are independently and normally distributed, but var(ɛ(x i )) = σ 2 (x i ): Σ + Λ (Ankenman et al. 2010) ɛ(x i ) are correlated, then? Cycle time estimation 4 50th Quantile Regression Curve 85th Predicted Mean Samples Standard Cycle Time Quantile Cycle Time Quantile Throughput x 011 Fig. 5. G/G/1 quantile regression curve with empirical quantile estimates. 13 / 42

14 AGP Model Data Characteristics Profile Value (x 21, y 22 ) (x 11, y 11 ) (x 21, y 22 ) (x 12, y 12 ) f (x) + ɛ 1 (x) f (x) f (x) + ɛ 2 (x) Location 14 / 42

15 AGP Model AGP Model Y i (x) = f (x) + ɛ i (x) Standard surface Deviation surface Assumption f (x) is a realization of GP(µ, s( )) ɛ i (x) is a realization of GP(0, v( )) f (x) and ɛ i (x) are independent ɛ i (x) and ɛ j (x) are independent for i j 15 / 42

16 AGP Model Distributional view A Gaussian process can be used as a prior probability distribution over functions in Bayesian inference (Rasmussen and Williams 2006). 1.5 Realization 1 Realization 2 Generated Value x Linear model: Y (x) = f (x) + ɛ AGP model: Y (x) = f (x) + ɛ(x) i.i.d F ɛ i.i.d GP(0, v( )) 16 / 42

17 AGP Model Model Estimation Estimate the model parameters β [µ, σ 2 1, θ 1, σ 2 2, θ 2] from observations Profile Value (x 21, y 22 ) (x 11, y 11 ) (x 21, y 22 ) (x 12, y 12 ) f (x) + ɛ 1 (x) f (x) f (x) + ɛ 2 (x) Location 17 / 42

18 AGP Model Structure of Σ 0 { s(xij, x i k) + v(x ij, x i k), i = i cov(y ij, y i k) = s(x ij, x i k), i i i, i = 1, 2,, N 0 X IC X 1 X 2 X N0 X 1 n 1 n 1 0 X IC M 0 M 0 + X 2 n 2 n 2 X N0 0 n N0 n N0 s(x ij, x i k θ 1 ) v(x ij, x i k θ 2 ) 18 / 42

19 AGP Model MLE Given the data from all surface profiles X IC, Y IC, we can estimate β as { ˆβ = arg max 1 β 2 log[det(σ2 1S + σ2v)] 2 1 } 2 (Y IC µ1 M0 ) T (σ1s 2 + σ2v) 2 1 (Y IC µ1 M0 ). Maximizing profile likelihood: given θ 1, θ 2, the correlation matrix S, V are fixed. Then µ, σ1 2, σ2 2 can be obtained easily. µ = 1T M 0 (S + ρv) 1 Y IC 1 T M 0 (S + ρv) 1 1 M0, ρ = σ 2 2/σ 2 1 σ 2 1 = (Y IC µ1 M0 ) T (S + ρv) 1 (Y IC µ1 M0 ) M 0 19 / 42

20 AGP Model Prediction For new unmeasured site (X l, Y l ): ( ) [( ) ( ) ] Yl µ1nl Σl Σ l,0 N, Σ T l,0 Σ 0 Y IC µ1 M0 Y l Y IC N( µ l, Σ l ), where µ l = µ1 nl + Σ l,0 Σ 1 0 (Y IC µ1 M0 ) Σ l = Σ l Σ l,0 Σ 1 0 ΣT l,0 Σ l,0 may have a different form depending on whether Y l are taken from existing profiles or new ones. 20 / 42

21 AGP Model Prediction Demonstration 4 3 Predicted Mean Samples Standard Predicted Variance Predicted mean Predicted variance 21 / 42

22 T 2 Test Statistical Testing Profile Value Location Whether the new profile deviates from f (x) within acceptable region Statistical testing based on the samples (where to sample?) 22 / 42

23 T 2 Test T 2 Test If the new surface conforms with the model, Y l N( µ l, Σ l ) Reducing surface comparison to multivariate normal data comparison H 0 : Y l N( µ l, Σ l ) H 1 : Y l N( µ l, Σ l ). Testing statistic: T 2 l = (Y l µ l ) T Σ 1 l (Y l µ l ), Under H 0, T 2 l χ 2 n l. Reject H 0 when T 2 l > H T. 23 / 42

24 Generalized likelihood ratio test GLR Test Only focus on a certain class of alternative models Another deviation source is considered as the alternative models Y l (x) = f (x) + ɛ l (x) + ξ(x) ξ(x) is a realization of another GP(δ, w( )). Suitable to model the global change effects Testing hypothesis H 0 :Y l (x) = f (x) + ɛ l (x) H 1 :Y l (x) = f (x) + ɛ l (x) + ξ(x) 24 / 42

25 Generalized likelihood ratio test GLR Test With finite number of observations Testing hypothesis: H 0 :Y l N( µ l, Σ l ) H 1 :Y l N( µ l + δ1 nl, Σ l + Σ w ) for some nonzero δ, γ 2, θ l GLR statistic: [ ] sup δ,γ 2,θl det( Σ l + Σ w ) 1/2 exp (Y l µ l δ1 nl ) T ( Σ l + Σ w ) 1 (Y l µ l δ1 nl )/2 R l = 2 ln [ ] det( Σ l ) 1/2 exp (Y l µ l ) T 1 Σ l (Y l µ l )/2 R l equal mixture χ χ2 2 asymptotically under H 0. Reject H 0 when: R l > H R. 25 / 42

26 Generalized likelihood ratio test Summary N 0 IC Units n i on Unit i AGP Model ( µ l, Σ l ) X l New Unit Y l T 2 Test GLR Test Accept Continue Reject Disposal 26 / 42

27 Approximation and Estimation Performance Approximation Performance Standard profile (Shpak 1995): f (x) = sin(x) + sin(10x/3) + log(x) 0.84x + 3 Spatially correlated error: ɛ(x) GP(0, 0.05 v( 5)) Predicted mean f(x) Measurements AGP OGP Predicted variance AGP OGP x OGP Model: Y i (x) = µ + ɛ i (x) x 27 / 42

28 Approximation and Estimation Performance Bias and RMSE of MLE Accuracy of the MLE with different sample size: (N 0, n 0 ) µ = 1 σ 2 = 0.2 θ 1 = 3 τ 2 = 0.05 θ 2 = 10 (10,10) Bias RMSE (10,20) Bias RMSE (20,10) Bias RMSE (20,20) Bias RMSE / 42

29 Monitoring Performance Three Change Scenarios Y (x) = f (x) + ɛ(x) Mean (µ) Variance (σ 2 2 ) Correlation (θ 2 ) 4 3 Standard Shifted 4 3 Standard Shifted 4 3 Standard Shifted / 42

30 Monitoring Performance Performance of Different Tests Three tests to compare: Max-Min Test GLR Test T 2 Test 1.0 Mean MaxMin GLR T2 Variance Correlation Beta error Shift magnitude / 42

31 Monitoring Performance Effect of Testing Sample Size (n l ) 31 / 42

32 Monitoring Performance Effect of In Control Sample Size (N 0, n 0 ) (10,10) (20,10) (10,20) (20,20) Mean Variance Correlation T2 T2 T Beta error Mean Variance Correlation GLR GLR GLR Shift magnitude / 42

33 Real Application Monitoring Wafer Thickness Profile Data are collected from real production plant; 8 in control wafers to construct AGP model, 30 wafers to be tested; 120 measurements from each in control wafer to construct AGP model; 480 measurements from each testing wafer to conduct tests. 33 / 42

34 Real Application Demos of Thickness Profile In control wafer #2 In control wafer #7 Approximated standard profile 34 / 42

35 Real Application p-values of the Tests T 2 GLR Significant Level 0.7 p Value Wafer Surfaces 35 / 42

36 Real Application Rejected Wafers (p-values) #12 (T 2 : GLR: ) #23 (T 2 : GLR: ) #24 (T 2 : GLR: ) #26 (T 2 : GLR: ) #28 (T 2 : GLR:0) #30 (T 2 : GLR: ) 36 / 42

37 Open issues Optimal Design for AGP Nonparametric model, Fisher information matrix is not enough Ordinary space filling design for deterministic experiments does not consider geometric feature does not consider the error process 37 / 42

38 Open issues Optimality Criteria Prediction accuracy: minimize (integrated) RMSE Determine N 0, n 0 and x ij Approximation accuracy of f (x) and error process estimation σ 2 2, θ 2 Sequential allocation strategy (Ankenman et al. 2010) Detection power: minimize β error T 2 test: when only µ changes, the Mahalanobis distance δ Σ 1 l δ determines the power, where Σ l = Σ l Σ l,0 Σ 1 0 ΣT l,0 Constant mean shift: max Xl 1 Σ 1 D-optimal: max Xl det Σ 1 l l 1 = min Xl det Σ l 38 / 42

39 Open issues GP with Covariates Surface profile depends on other factors: speed, force, materials, etc. GP model GP with independent errors Ankenman et al. (2010) GP with dependent errors Multivariate output/response Co-kriging Zhou et al. (2011); Qian et al. (2008) Different distance metrics Surface response 39 / 42

40 Open issues Conclusion AGP model is suitable to approximate surface profile and quantify dependent deviations; A simple and flexible framework for process monitoring Need to further consider design issues and extend the model to the case with covariate 40 / 42

41 Reference I Ankenman, B., Nelson, B., and Staum, J. (2010), Stochastic kriging for simulation metamodeling, Operations Research, 58, Cressie, N. (1993), Statistics for Spatial Data, revised edition, vol. 928, Wiley, New York. Doering, R. and Nishi, Y. (2007), Handbook of semiconductor manufacturing technology, CRC Press, Boca Raton, FL. Jin, R., Chang, C., and Shi, J. (2012), Sequential measurement strategy for wafer geometric profile estimation, IIE Transactions, 44, Qian, P. Z. G., Wu, H., and Wu, C. F. J. (2008), Gaussian Process Models for Computer Experiments with Qualitative and Quantitative Factors, Technometrics, 50, Rasmussen, C. E. and Williams, C. K. I. (2006), Gaussian Processes for Machine Learning, MIT Press, Boston. Sacks, J., Welch, W., Mitchell, T., and Wynn, H. (1989), Design and analysis of computer experiments, Statistical science, 4, Santner, T., Williams, B., and Notz, W. (2003), The design and analysis of computer experiments, Springer, New York. Shpak, A. (1995), Global optimization in one-dimensional case using analytically defined derivatives of objective function, Computer Science Journal of Moldova, 3, Zhao, H., Jin, R., Wu, S., and Shi, J. (2011), Pde-constrained gaussian process model on material removal rate of wire saw slicing process, Journal of Manufacturing Science and Engineering, 133, Zhou, Q., Qian, P. Z. G., and Zhou, S. (2011), A Simple Approach to Emulation for Computer Models with Qualitative and Quantitative Factors, Technometrics, 53,

42 Thanks and questions