In the semiconductor industry, the production of microcircuits involves many steps. The wafer fabrication process typically builds these microcir- cuits on silicon wafers, and there are many micro- circuits per wafer. Each production lot consists of between 16 and 48 wafers. Some processing steps treat each wafer separately, so that the batch size for that step is one wafer. It is usually necessary to estimate several components of variation: within- wafer, between-wafer, between-lot, and the total variation.

(a) Suppose that one wafer is randomly selected from each lot and that a single measurement on a critical dimension of interest is taken. Which components of variation could be estimated with these data? What type of control charts would you recommend?

(b) Suppose that each wafer is tested at five fixed locations (say, the center and four points at the circumference). The average and range of these within-wafer measurements are xww and Rww, respectively. What components of variability are estimated using control charts based on these data?

(c) Suppose that one measurement point on each wafer is selected and that this measurement is recorded for five consecutive wafers. The average and range of these between-wafer measure- ments are xBW and RBW, respectively. What com- ponents of variability are estimated using control charts based on these data? Would it be neces- sary to run separate x and R charts for all five locations on the wafer?

(d) Consider the question in part (c). How would your answer change if the test sites on each wafer were randomly selected and varied from wafer to wafer?

(e) What type of control charts and rational subgroup
scheme would you recommend to control
the batch-to-batch variability?