A statistical analysis of single and multiple response surface modeling

This work examines the use of single response surface (SRS) and multiple re sponse surface (MRS) techniques for modeling spatial nonuniformity in semic onductor applications. Previous works halve suggested that the MRS estimati on techniques better measure the nonuniformity due to the underlying spatia l function of the process, whereas SRS estimation methods measure the tc,ta l process nonuniformity (systematic spatial nonuniformity plus random site nonuniformity). This work further highlights this fact in an analytical set ting. It is demonstrated that the MRS estimation technique is biased and th at this bias can lead to the choice of a nonoptimal process. Experimental d ata from a chemical-mechanical polishing (CMP) process confirms these obser vations and demonstrates that careful use of the MRS estimator is required in achieving meaningful results for estimating spatial nonuniformity, Modif ied versions of each method, which measure spatial nonuniformity alone, as well as versions which measure total nonuniformity, are proposed for the ca se when one is comparing discrete process settings. Analytical expressions for the expected value and variance of both the SRS and MRS estimators are determined. These are used to compare the efficiency (estimator variance) o f these modified estimators. When comparing spatial nonuniformity, it is fo und that the unbiased MRS estimator is more efficient than the SRS estimato r modified to measure spatial nonuniformity. However, it is shown that the MRS estimator, when modified to measure total nonuniformity, is not necessa rily more efficient than the SRS method. Finally, the: continuous response surface modeling case is considered, It is demonstrated how confidence inte rvals on the underlying continuous site models lead to a nonuniform bias in the response surface generated by the MRS method. This suggests that care must be taken when using the MRS technique in creating continuous response surfaces of spatial nonuniformity as a function of the process settings.