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edfas.org ELECTRONIC DEV ICE FA I LURE ANALYSIS | VOLUME 24 NO . 3 4 EDFAAO (2022) 3:4-10 1537-0755/$19.00 ©ASM International ® IMPACTS OF NESTED VARIANCE COMPONENTS ON SEMICONDUCTOR ELECTRICAL TEST SAMPLING David Potts, Scott Hildreth, and Binod Kumar G. Nair GlobalFoundries, Malta, New York david.potts@gf.com INTRODUCTION Semiconductor manufacturing is a complex and expensive process. Once a failure mechanism has been identified, it must be eliminated or at least contained as best as possible. The most egregious cases can hope- fully be detected and removed from production at point of first occurrence, but circuit performance can still be impactedby the cumulative effects of variousmechanisms throughout the entire process. The inline wafer electrical test (WET) offers an early read point to assess the overall health of the process via measurements taken on special structures placed throughout the wafer. However, due to time and equipment constraints, typically only a sample of these test sites are actively monitored in production. Complicating the situation is the inherent nature of the semiconductor manufacturing system itself, with some operations being iteratively executed within sub-regions across a given wafer, others being run on the entire wafer at once and still others being applied to batches of wafers. This results in a nested variance structure under which different physical mechanisms will exhibit varying sen- sitivities to site-to-site ( s2s ), wafer-to-wafer ( w2w ), and lot-to-lot ( l2l ) levels of variation. This article uses Monte Carlo simulations to demonstrate some impacts these hierarchical variance components can exert on percep- tions of WET performance. MONTE CARLO CONFIGURATION We will use three different Monte Carlo (MC) random number generation processes to simulate semiconductor manufacturing populations for some performancemetric. Each of these populations will include 4000 lots with 25wafers per lot and13 sites perwafer, for a total of 1,300,000 sites over 100,000 wafers. The overall distribution of the simulated performance metric, Y, will be identical across all three populations (standard normal with μ = 0 and σ 2 = 1), but they will vary in the manner of which they were constructed. We will refer to these populations as S2S, W2W, and L2L. MC S2S : for this population, we simply run 1,300,000 trials of a standard normal random number generation, with individual “wafers” beingdefinedby consecutive runs of 13 “sites” and individual “lots” by consecutive runs of 25 wafers. In this population, there is nothing systemati- cally different between the lots or wafers. All of the varia- tion occurs at the site-level. In other words, it has 0% l2l , 0% w2w , and 100% s2s variance components. MC W2W : for this population, we first run 100,000 trials of a randomnumber generator, normally distributed with μ = 0 and σ 2 = 0.5 (σ = 0.71). Each of these values are replicated 13x, to simulate a wafer-level component of variation in thepopulation (e.g., thewafermeanacross the 13 sites of a particular wafer). We then run an independent sample of 1,300,000 trials of a randomnumber generator, normally distributed with μ = 0 and σ 2 = 0.5 (σ = 0.71), to represent the site-level of variation. Adding these two vectors together produces an overall standard normal population distribution (μ = 0 and σ 2 = 1) but now with 0% l2l , 50% w2w , and 50% s2s variance components. MC L2L : for this population, we first run 4000 trials of a randomnumber generator, normally distributedwithμ = 0 and σ 2 = 0.25 (σ = 0.5). Each of these values are replicated 325x, to simulate a lot-level component of variation in the population (e.g., the lot mean across the 25 wafers, each containing 13 sites within a particular lot). We then run a second independent sample of 100,000 trials of a random number generator, normally distributed with μ = 0 and σ 2 = 0.25 (σ = 0.5). Each of these values are replicated 13x, to simulate a wafer-level component of variation in our population (e.g., the wafer mean across the 13 sites of a particular wafer). Finally, we run a third sample of 1,300,000 trials of a randomnumber generator, normally distributed with μ = 0 and σ 2 = 0.5 (σ = 0.71), to represent the site-level of variation. Adding these three