Sampling Approaches to Metrology in Semiconductor Manufacturing

Transcription

1 Sampling Approaches to Metrology in Semiconductor Manufacturing Tyrone Vincent 1 and Broc Stirton 2, Kameshwar Poolla 3 1 Colorado School of Mines, Golden CO 2 GLOBALFOUNDRIES, Austin TX 3 University of California, Berkeley CA (IMPACT) Metrology Sampling 1 / 3

2 Outline Approach 1 Approach 2 Modeling and Identification 3 Experimental Results 4 Conclusion (IMPACT) Metrology Sampling 2 / 3

3 Context Approach the progression for Advanced Process Control (APC) places increased demand on metrology lot level variation wafer level variation within-wafer variation (IMPACT) Metrology Sampling 3 / 3

4 Goal Approach decrease the number of measurements needed to determine key features on wafers (IMPACT) Metrology Sampling 4 / 3

5 Approach Approach develop a correlation model of metrology measurements for a particular process conduct an a priori analysis to determine the optimally informative sites that should be measured by minimizing an expected prediction error at the unmeasured sites. use a subset of measurement sites together with the correlation model to predict measurements at sites which are not measured (IMPACT) Metrology Sampling 5 / 3

6 Prediction Process Approach Identification Set Prediction Set m sites measured q sites measured m q sites predicted (IMPACT) Metrology Sampling 6 / 3

7 Outline Modeling and Identification 1 Approach 2 Modeling and Identification 3 Experimental Results 4 Conclusion (IMPACT) Metrology Sampling 7 / 3

8 Modeling and Identification Sources of Variation L gate (f, d, k) = L + L a (k) + L b (f, d, k) + L c (f, k) +L d (d, k) + L e (d) + L f (f, d, k) L gate random variation layout dependent variation field level systematic variation and bias wafer level systematic variation and bias (f) (e) (d) (c) (b) (a) pos (IMPACT) Metrology Sampling 8 / 3

9 Metrology Model 1 Modeling and Identification wafer sequence spatial variation y k (p) = ȳ(p) + i C i(p)x i,k + n k (p) parameters y k (p) - measurements at position p on wafer k C i (p) - variation basis function x i,k - scaling factor for variation function i n k (p) - measurement noise (IMPACT) Metrology Sampling 9 / 3

10 Modeling and Identification Identification Process 1 vectorized model y k = [ y k (p 1 ) y k (p 2 ) y k (p m ) ] T x k = [ x 1,k x 2,k x n,k ] T n k = [ n k (p 1 ) n k (p 2 ) n k (p m ) ] T y k = ȳ + Cx k + n k case 1: x k iid random sequence identification process: C and covariance of x k identified using Principle Component Analysis (IMPACT) Metrology Sampling 1 / 3

11 Metrology Model 2 Modeling and Identification wafer sequence spatial variation Measurement Wafer # time variation y k = ȳ + i C i(p)x i,k + n k x 1,k+1. = A x n,k+1 x 1,k. x n,k +w k (IMPACT) Metrology Sampling 11 / 3

12 Modeling and Identification Identification Process 2 Case 2: x k time correlated x k+1 = Ax k + w k y k = ȳ + Cx k + n k parameters A, C and covariance of w k identified using Canonical Correlation Analysis (IMPACT) Metrology Sampling 12 / 3

13 Prediction Modeling and Identification measured data y M k = ȳm + C M x k + n M k unmeasured values z U k = ȳu + C U x k - M - U common variable x k can be estimated from yk M and used to predict zk U (IMPACT) Metrology Sampling 13 / 3

14 Modeling and Identification Performance Prediction models include uncertainty in process variation and measurement error performance prediction possible S U := tr Cov [ ] ẑk U zu k (IMPACT) Metrology Sampling 14 / 3

15 Modeling and Identification Measurement Sequencing greedy algorithm for minimizing prediction error 1 Set M =, U = {1,..., m} 2 For each element i U, calculate S U {i}. 3 Select the element j for which trs U {j} is minimum. 4 Remove j from U and add it to M. 5 If M < q goto to step 2, otherwise quit. (IMPACT) Metrology Sampling 15 / 3

16 Modeling and Identification Implementation model requires data - how do we get it? re-modeling may be needed - how do we check? (IMPACT) Metrology Sampling 16 / 3

17 Modeling and Identification Implementation Process 1 predicted features measured features initial data set {}}{ { { lower rate measurements to monitor performance (IMPACT) Metrology Sampling 17 / 3

18 Modeling and Identification Implementation Process 1 initial data set {}}{ { predicted features measured features { model expires (IMPACT) Metrology Sampling 17 / 3

19 Modeling and Identification Implementation Process 1 initial data set {}}{ { predicted features measured features { data set to { renew model }} { model expires (IMPACT) Metrology Sampling 17 / 3

20 Modeling and Identification Implementation Process 1 initial data set {}}{ { predicted features measured features { data set to { renew model }} { model expires model expires (IMPACT) Metrology Sampling 17 / 3

21 Modeling and Identification Implementation Process 1 initial data set {}}{ { predicted features measured features { data set to { renew model }} { model expires data set to { renew model }} { model expires (IMPACT) Metrology Sampling 17 / 3

22 Modeling and Identification Implementation Process 2 initial data set {}}{ { predicted features measured features { data set to renew model {}}{ model expires (IMPACT) Metrology Sampling 18 / 3

23 Outline Experimental Results 1 Approach 2 Modeling and Identification 3 Experimental Results 4 Conclusion (IMPACT) Metrology Sampling 19 / 3

24 Experimental Results Will this work? performance is process specific! best case: process variation is low order process variation is stationary validation requires real process data (IMPACT) Metrology Sampling 2 / 3

25 Experimental Results Real Process Data use for Verification poly-gate critical dimension (CD) process data from GLOBALFOUNDRIES Fab wafers with common litho-etch equipment sequence Same feature measured on 18 die. (IMPACT) Metrology Sampling 21 / 3

26 Average wafer Experimental Results Y position [3] [13] [2] Average wafer [4] [12] [1] [1] X position [5] [9] [8] [11] [6] [7] (IMPACT) Metrology Sampling 22 / 3

27 Experimental Results PCA variance contribution Variance Contribution PCA state dimension (2σ) State index Average prediction error (unmeasured sites only) n=2 n=1 n=3 n=7 n= Number of measurements per wafer A plot of average prediction error vs. # of sites measured, for different model dimensions, n. (IMPACT) Metrology Sampling 23 / 3

28 Experimental Results First four basis elements Y position PCA C matrix, column 1 [9] [4] [3] [8] [13] [5] [11] [2] [12] [1] [7] Y position PCA C matrix, column 2 [9] [4] [3] [8] [13] [5] [11] [2] [12] [1] [7] X position [1] [6] X position [1] [6] PCA C matrix, column 3 [9] [1] PCA C matrix, column 4 [9] [1].4.3 Y position [3] [13] [2] [4] [12] [1] [5] [8] [11] [6] [7] Y position [3] [13] [2] [4] [12] [1] [5] [8] [11] [6] [7] X position X position (IMPACT) Metrology Sampling 24 / 3

29 Experimental Results Data Splits for Validation Full Data Set for Litho i/etch j Split: wafer N s Identification Set: N wafers Prediction Set: M wafers m sites measured q sites measured m q sites predicted (IMPACT) Metrology Sampling 25 / 3

30 Experimental Results Prediction Error vs. Model Window Average prediction error (unmeasured sites only) N=5.1 N=899 N= Number of measurements per wafer Prediction error with M = 899 Required size of model set 2 wafers (IMPACT) Metrology Sampling 26 / 3

31 Experimental Results Prediction Error vs. Prediction Window Average prediction error (unmeasured sites only) M=1598 M= Number of measurements per wafer Prediction Error with N = 2. Performance degrades gracefully with prediction window (IMPACT) Metrology Sampling 27 / 3

32 Time correlation Experimental Results Average prediction error (unmeasured sites only).2.1 CCA M=5 PCA M= CCA M=4 PCA M= Number of measurements per wafer No improvement with more complex model, except for small number of measurements per wafer (IMPACT) Metrology Sampling 28 / 3

33 Outline Conclusions 1 Approach 2 Modeling and Identification 3 Experimental Results 4 Conclusion (IMPACT) Metrology Sampling 29 / 3

34 Conclusions Conclusions Intelligent strategies can reduce the expense of metrology without compromising effectiveness Reduced sampling realized through models which predict data at unmeasured sites The real challenge: SPC and fault detection Why do we use metrology? To tell us if something goes wrong [also closed-loop process control] What are optimal sampling strategies that don t compromise SPC or fault detection? Metrics: time to detect a process drift or failure, false alarm rates, etc Supported in part by NSF grant ECS , Berkeley IMPACT program and DOE Programs DE-ZDO and DE-FG36-8GO881. (IMPACT) Metrology Sampling 3 / 3