| Semiconductor manufacturers routinely test integrated circuits (IC's) while they are still part of the wafer. When the IC failure rate is high, a map of the test results typically reveals clusters of failures. Such 'failure patterns' evince circuit-to-process mismatches and processing problems, providing valuable information for design and process improvement.; We select five failure patterns that resemble simple geometrical 'objects'--- Segments, Disks, Annuli, Bands , and Rings---and determine their uniform random distributions through use of the invariant pdf. Two to four geometrical variables govern an object's statistics by locating boundaries that divide the wafer map into zones. The geometry of these zones drives subsequent calculations, giving rise to eleven Shapes. One-zone shapes comprise Pass and Fail; two-zone shapes comprise Segments , A-Disks, B-Disks, A-Annuli , and B-Annuli; three-zone shapes comprise Bands, A-Rings, B-Rings, and C-Rings.; In addition to Shape, relevant features include Area, Location, and---if there are three zones--- Orientation and Curve Direction. These four new 'feature variables' are real-valued functions of the geometrical variables, changing continuously as one pattern shape transforms into another.; This formulation enables construction of a synthetic wafer map generator, valuable for creating and studying classifiers. One study employs synthetic datasets with 160,000 maps and estimates the Bayes error of the synthetic population to be less than 1.5 percent. Other studies employ several classifiers, chief among which are nearest neighbor classifier N20 and prototype classifier kh, created with synthetic maps, and nearest neighbor classifier Xdata, created with industrial maps. These classifiers are evaluated using input sets of type RAND, containing synthetic maps, and DATA, containing industrial maps.; Classification of RAND reveals that N20 is the more accurate Shape classifier, scoring 95 percent, while kh is more accurate with Area and Location, at 96 and 94 percent, respectively. Xdata accuracies are 85, 91, and 83 percent. Classification of RAND plus bit noise demonstrates that N20 is affected less by noise than either kh or Xdata. Classification of DATA finds the kh classifier to be somewhat better than either N20 or Xdata with Shape, Area, and Location accuracies around 67, 77, and 75 percent, and with Xdata accuracy comparable to that of N20. |
