Advanced Process Controls of Underdetermined Systems of Feature Profiles in Semiconductor Manufacturing

Typical solutions of Advanced Process Controls in semiconductor manufacturing are focused on the eliminations of target offsets for the process results of some device features. Normally the process models are square and of full ranks, thus enabling unique determinations of the adjustments of control inputs. As complexity of semiconductor devices evolves from planar to 3D structures, elementary controls of simple target offsets are no longer sufficient as concurrent controls of both target and uniformity have become essential. For the latter it may refer to within-wafer uniformity for variations of the same device feature over the entire wafer surface; or it may refer to the in-situ profiles of different features at the same spot. In either case, they result in largely underdetermined systems as the number of control inputs is finite and fewer than the possible number of all control outputs. Conventional APC solutions in semiconductor manufacturing are incapable of handling these situations. This paper presents an approach using multivariate optimal control techniques to achieve the desired performances.