August_EDFA_Digital

edfas.org 9 ELECTRONIC DEV ICE FA I LURE ANALYSIS | VOLUME 24 NO . 3 The S2S t-test response is identical whether sampled randomly or sequen- tially because each individual site is identically and independently distrib- uted from ever other site, regardless of order. Under w2w and l2l , the individual wafers will vary, with tighter distribu- tions relative to the overall population while the locations of those distribu- tionswill driftbetween different wafers/ lots (e.g., Fig. 4). Furthermore, there are different relative sample sizes at each level (# sites vs # wafers vs # lots). To properly account for these effects, we need to explicitly introduce them into the analysis by specifying lots and wafers as nested random effects within a mixed-effects model. In R, [4] nested random effects are easily handled by the lme (linear mixed- effectsmodels) functionwithin the nlme (linear and nonlinear mixed effects models) package [5] via a passed argu- ment of random = list(~1|lot, ~1|wafer). Figure 10 shows the ANOVA results obtained for lme models derived from our W2W MC across selected sample sizes. Curve a in Fig. 10 depicts the results derived from random samples of 5 wafers of 13 sites each for a total of 65 sites, comparable in size to the 63 sequential siteW2Wshown in curve b of Fig. 9. Note that the lme results (curve a, Fig. 10) “corrects” the effective α back to ~0.05, but further depresses theeffective power relative to the t-test results (curve b, Fig. 9). By explicitly declaring wafers as a random effect, the lme estimates thewafer-level variance component and can make an allowance for it based on thenumber ofwafers sampled.With that added allowance, effective α is held constant near ~0.05 while effective power tracks with sample size. For smaller wafer sample sizes, a larger allowance is required to reflect the larger uncertainty in thewafer-level variance estimate which limits precision to detect δ/σ shifts. As more wafers are sampled, we gain power for detecting smaller δ/σ shifts. For our W2WMC (with 50% w2w /50% s2s ), we need to sample about 15 wafers to achieve our target power of 0.8 at δ/σ = 0.5 (curve d, Fig. 10). Figure 11 shows the ANOVA results obtained for lme models derived from our L2L MC across selected sample sizes. Curves a and b both reflect 65 site samples but with markedly different results. The samples in curve a were compiled by selecting 5 wafers from a single lot (1 lot x 5 wafers x 13 sites/wafer = 65 total sites) while those in curve b were compiled by selecting 1 wafer each from 5 lots (5 lots x 1 wafer/lot x 13 sites/wafer = 65 total sites). Both cases were identically configured for nested lot/ wafer randomeffects, but since only 1 lot was represented Fig. 9 The t-test OC curves under sequential sampling are dramatically impacted by w2w and l2l . Fig. 10 W2W: lme OC curves under sequential sampling of wafers.