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C. J. Spanos Design of Experiments in Semiconductor Manufacturing Costas J. Spanos Department of Electrical Engineering and Computer Sciences University of California Berkeley, CA 94720, U.S.A. tel (510) 643 6776, fax (510) 642 2739 email spanos@eecs.berkeley.edu http://bcam.eecs.berkeley.edu L2 1 C. J. Spanos Design of Experiments • Comparison of Treatments – which recipe works the best? • Simple Factorial Experiments – to explore impact of few variables • Fractional Factorial Experiments – to explore impact of many variables • Regression Analysis – to create analytical expressions that “model” process behavior • Response Surface Methods – to visualize process performance over a range of input parameter values L2 2 C. J. Spanos Design of Experiments • Objectives: – Compare Methods. – Deduce Dependence. – Create Models to Predict Effects. • Problems: – Experimental Error. – Confusion of Correlation with Causation. – Complexity of the Effects we study. L2 3 C. J. Spanos Problems Solved • Compare Recipes • Choose the recipe that gives the best results. • Organize experiments to facilitate the analysis of the data. • Use experimental results to build process models. • Use models to optimize the process. L2 4 C. J. Spanos Comparison of Treatments • Internal and External References • The Importance of Independence • Blocking and Randomization • Analysis of Variance L2 5 C. J. Spanos The BIG Question in comparison of treatments: • How does a process compare with other processes? – Is it the same? – Is it different? – How can we tell? L2 6 C. J. Spanos Using an External Reference to make a Decision • An external reference can be used to decide whether a new observation is different than a group of old observations. • Example: Create a comparison procedure for lot yield monitoring. Do it without "statistics". • Here, I use "reference data": L2 7 C. J. Spanos Example: Using an External Reference To compare the difference between the average of successive groups of ten lots, I build the histogram from the reference data: • Each new point can then be judged on the basis of the reference data. • The only assumption here is that the reference data is relevant to my test! L2 8 C. J. Spanos Using an Internal Reference... • We could generate an "internal" reference distribution from the data we are comparing. • Sampling must be random, so that the data is independently distributed. • Independence would allow us to use statistics such as the arithmetic average or the sum of squares. • Internal references are based on Randomization. L2 9 C. J. Spanos Example in Randomization • Is recipe A different than recipe B? 660 A A B Recipe Type Etch Rate 650 640 B 630 620 630 640 650 660 0 2 4 6 8 10 12 Etch Rate Sample L2 10 C. J. Spanos Example in Randomization - cont. • There are many ways to decide this... 1.External reference distribution based on old data. 2. Approximate external reference distr. (either t or normal). 3. Internal reference distribution. 4. "Distribution free" tests. • Options 2, 3 and 4 depend on the assumption that the samples are independently distributed. L2 11 C. J. Spanos Example in Randomization - cont. • If there was no difference between A and B, then let me assume that I just have one out of the 10!/5!5! (252) arrangements of labels A and B. • I use the data to calculate the differences in means for all the combinations: L2 12 C. J. Spanos The Origin of the t Distribution The student-t distribution was, in fact, defined to approximate randomized distributions! (yB - yA) - (µA - µB) t0 = s n +n1 1 A B • For the etch example, t0 = 0.44 and Pr (t > t0) = 0.34 • Randomized Distribution = 0.33 L2 13 C. J. Spanos Example in Blocking • Compare recipes A and B on five machines. • If there are inherent differences from one machine to the other, what scheme would you use? Random Blocked AA AB ABA BA BA BA BB AB B BA L2 14 C. J. Spanos Example in Blocking - cont. • With the blocked scheme, we could calculate the A-B difference for each machine. • The machine-to-machine average of these differences could be randomized. ±d1±d2±d3±d4±d5 d= 5 d - δ ~ tn-1 sd/ n In general, randomize what you don't know In general, randomize what you don't know and block what you do know. and block what you do know. L2 15 C. J. Spanos Analysis of Variance 5 Recipe Type D4 C3 B2 A 1 610 620 630 640 650 660 Etch Rate Your Question: Are the four treatments the same or not? Your Question: Are the four treatments the same or not? The Statistician's Question: Are the discrepancies between The Statistician's Question: Are the discrepancies between the groups greater than the variation within each group? the groups greater than the variation within each group? L2 16 C. J. Spanos Calculations for our Example i=1 i=2 i=3 i=4 i=5 Avg s t2 νt (yt - y)2 1: 650 648 632 645 641 643.20 202.80 4 25.00 2: 645 650 638 643 640 643.20 86.80 4 25.00 3: 623 628 630 620 618 623.80 104.80 4 207.36 4: 645 640 648 642 638 642.60 63.20 4 19.36 s2 = R s2 = T s2 T = s2 R L2 17 C. J. Spanos Variation Within Treatment Groups First, lets assume that all groups have the same spread. Lets also assume that each group is normally distributed. The following is used to estimate their common σ: nt St = Σ (ytj - yt)2 s2 = St t j=1 nt - 1 ν s +ν s2+...+ νks2 2 s2 = 1 1 2 2 k = SR = SR R ν1 + ν2 +...+ νk N - k νR • This is an estimate of the unknown, within group s - square. • It is called the within treatment mean square L2 18 C. J. Spanos Variation Between Treatment Groups • Let us now form Ho by assuming that all the groups have the same mean. • Assuming that there are no real differences between groups, a second estimate of sT2 would be: k Σ nt(yt - y)2 s2 = t=1 = ST T k-1 νT This is the between treatment mean square If all the treatments are the same, then the within and If all the treatments are the same, then the within and between treatment mean squares are estimating the same between treatment mean squares are estimating the same number! number! L2 19 C. J. Spanos What if the Treatments are different? If the treatments are different then: k s2 estimates σ2 + T Σ nt τ2/ (k - 1) t t=1 where τt ≡ µt - µ • In other words, the between treatment mean square is inflated by a factor proportional to the difference between the treatments! L2 20 C. J. Spanos Final Test for Treatment Significance Therefore, the hypothesis of equivalence is rejected if: s2 T is significantly greater than 1.0 s2 R s2 T ~ F This can be formalized since: 2 k-1, N-k sR L2 21 C. J. Spanos More Sums of Squares A measure of the overall variation: k nt SD = Σ Σ (ytj - y)2 s2 = SD = SD t=1 j=1 D N - 1 νD Obviously (actually, this is not so obvious, but it can be proven): SD = ST + SR and ν D = ν T + ν R L2 22 C. J. Spanos ANOVA Table Source Sum DFs Mean sq of Var of sq 2 between ST vT (k-1) sT 2 within SR vR (N-k) sR 2 total SD vD (N-1) sD L2 23 C. J. Spanos ANOVA Table (full) Source Sum DFs Mean sq of Var of sq 2 average SA vA ( 1 ) sA between ST vT (k-1) s2 T within SR vR (N-k) s2 R total S v (N) L2 24 C. J. Spanos Anova for our example... Data File: CompEtch Sum of Deg. of Mean Source Squares Freedom Squares F-Ratio Prob>F Between Recipe 1.3836e+3 3 4.6120e+2 1.6126e+1 4.29e-5 Error 4.5760e+2 16 2.8600e+1 Total 1.8412e+3 19 L2 25 C. J. Spanos Decomposition of Observations Y=A+T+R In Vector Form: yti y yt - y yti - yt . = . + . + . . . . . . . . . N 1 k-1 N-k The term degrees of freedom refers to the dimensionality of the space each vector is free to move into. L2 26 C. J. Spanos Geometric Interpretation of ANOVA Y=A+D Easy to prove that A ⊥ D. D = R +T Easy to prove that R ⊥ T and A ⊥ R. Y D R Y A T L2 27 C. J. Spanos Model and Diagnostics yti = µt + eti eti ~ N (0, σ 2) So, the "sufficient statistics" are: s2 , y1, y2,..., yk R as estimators of: σ 2, µ1, µ2,..., µk For our example: yt A: 643.20 B: 643.20 s2 28.6 R C: 623.80 D: 642.60 According to this model, the According to this model, the residuals are IIND. How do you residuals are IIND. How do you verify that? verify that? L2 28 C. J. Spanos Anova Example: Poly Deposition t 200 190 h 180 170 i 160 c 150 140 k 130 120 110 100 90 80 70 60 50 E B C D A F Recipe Are these recipes significantly different? Analysis of Variance Source DF Sum of Squares Mean Square F Ratio Model 5 26969.525 5393.91 20.9593 Error 227 58418.758 257.35 Prob > F C Total 232 85388.283 0.0000 L2 29 C. J. Spanos Residual thick. Residual Plots: 100 0.4 75 0.3 50 0.2 25 0.1 -40 -30 -20 -10 0 10 20 30 40 50 60 70 80 90 Residual thick. 25 0.75 20 15 0.5 10 0.25 5 -40 -30 -20 -10 0 10 20 30 40 Residual thick. 40 0.6 30 0.4 20 0.2 10 -40 -20 -10 0 10 20 30 40 50 60 Residual thick. 30 0.6 20 0.4 10 0.2 -40 -30 -20 -10 0 10 20 30 40 50 L2 30 C. J. Spanos Residual Plots (cont): R 90 e 80 s 70 i 60 d 50 u a 40 l 30 20 t 10 h i 0 c -10 k -20 . -30 -40 E B C D E F V1 V2 Deposition Recipe Wafer Vendor L2 31 C. J. Spanos Anova Summary • Plot Originals • Construct ANOVA table • Are the treatment effects significant? • Plot residuals versus: – treatment – group mean – time sequence – other? • ANOVA is the basic tool behind most empirical modeling techniques. (chapter 6 in BHH) L2 32

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posted: | 5/30/2010 |

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