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edfas.org 7 ELECTRONIC DEVICE FAILURE ANALYSIS | VOLUME 21 NO. 1 looking at each die individually. However, more informa- tion can be extracted when we start to analyze a large population of failing die together. The next section dis- cusses how RCD exploits this fact to build a defect Pareto chart, even in the presence of diagnosis noise. RCD ANALYSIS OF VOLUME DIAGNOSIS RESULTS RCD starts by assuming that the population of failing die being analyzed has been created by a fixed defect mechanism, aka root cause, distribution. A root cause can be any mechanism that leads to defects. However, a generic root cause list is used as a starting point and includes opens and shorts on all physical layers in the design. The initial assumption is that the probability of a failing die containing a defect from a root cause is the same for all die in the population. The probability distribution is unknown, and the goal of RCD is to estimate it from the diagnosis reports. To do so, a statistical model is built, which determines the probability of seeing the given set of diagnosis reports (corresponding to the failing die in the population) as a function of the probability distribution andmarginalizing over all the suspects in the report. The model also calculates the probability that the real suspect of a root cause will be found in the diagnosis report, taking design characteristics (critical areas) into account. Having created the probabilistic model, RCD estimates the unknown root cause distribution using the well-known maximum likelihood estimation (MLE) learning technique. This essentially says that the root cause distribution that maximizes the likelihood of see- ing the given set of diagnosis reports must be the correct one. Because RCD is cognizant of the diagnosis ambigu- ity in its probabilistic modeling, this ambiguity is inher- ently handled. Once the root cause distribution has been calculated, the probability that a suspect is the real defect in a failing die can be determined. Because this is the probability of a suspect being real, it can be used to pick die for FA that most likely represent each root cause in the calcula- ted distribution. CELL-AWARE DIAGNOSIS COMBINED WITH RCD RCD can build a defect Pareto chart for a given set of defective die and can handle cell-aware diagnosis results by considering the probability of a suspect locationbehav- ing in a particular way. For cell-internal defects, accurate analog simulation can determine their exact behavior because there are a limitednumber of cell types. However, for interconnect defects, each defect location can have a different neighborhood, rendering a similar accurate characterization too cost prohibitive. Because accurate interconnect defect behavior cannot be known, diagnosis algorithmsmust be “lenient”while picking these suspects to avoid loss in accuracy. This fact must be compensated for in the probability modeling by taking into consider- ation how the diagnosis algorithm is picking each suspect included in the final report. With this capability, RCD can be extended to analyze cell-internal defects as well. RESULTS To validate the cell-aware diagnosismethodology, we conducted a controlled injection experiment. A single root cause was chosen and a population of defective die were created by injecting defect locations fromthe chosen root cause in each die. As an example, a population of 200 die were created by injecting polysilicon bridges in a design. As a reference point, the failureswere diagnosedwithnon- cell-aware diagnosis and run through RCD with the only cell root causes being cell counts for each cell. Results of Fig. 5 Comparing results fromRCD vs. cell-aware diagnosis plus RCD for population with polysilicon bridges. (a) (b)