Determination of optimal polynomial regression function to decompose on-die systematic and random variations

A procedure that decomposes measured parametric device variation into systematic and random components is studied by considering the decomposition process as selecting the most suitable model for describing on-die spatial variation trend. In order to maximize model predictability, the log-likelihood estimate called corrected Akaike information criterion is adopted. Depending on on-die contours of underlying systematic variation, necessary and sufficient complexity of the systematic regression model is objectively and adaptively determined. The proposed procedure is applied to 90-nm threshold voltage data and found the low order polynomials describe systematic variation very well. Designing cost-effective variation monitoring circuits as well as appropriate model determination of on-die variation are hence facilitated.