Efficient design of experiments for complex response surfaces with application to etching uniformity in a plasma reactor

Plasma etching uniformity across silicon wafers is of paramount importance in the semiconductor industry. The complexity of plasma etching, coupled with lack of instrumentation to provide real-time process information (that could be used for feedback control), necessitate that optimal conditions for uniform etching must be designed into the reactor and process recipe. This is often done empirically using standard design of experiments which, however, are very costly and time consuming.;The objective of this study was to develop a general purpose efficient design strategy that requires a minimum number of experiments, and can handle complex constraints in the presence of uncertainties. Traditionally, Response Surface Methodology (RSM) is used in these applications to design experiments to determine the optimal value of decision variables or inputs. We demonstrated that standard RSM, when applied to the problem of plasma etching uniformity, has the following drawbacks (1) inefficient search due to process nonlinearities, (2) lack of converge to the optimum, and, (3) inability to handle complex inequality constraints.;We developed a four-phase Efficient Design Strategy (EDS) based on the DACE paradigm (Design and Analysis of Computer Experiments) and Bayesian search algorithms. The four phases of EDS are: (1) exploration of the design space by maximizing information, (2) exploration of the design space for feasible points by maximizing probability of constraint satisfaction, (3) optimization of the objective and (4) constrained local search. We also designed novel algorithms to switch between the different phases.;The choice of model parameters for DACE predictors is usually determined by the Maximum Likelihood Estimation (MLE) method. Depending on the dataset, MLE could result in unrealistic predictors that show a peak-and-dip behavior. To solve this problem we developed techniques to detect the presence of peak-and-dip behavior and a new scheme based on Maximum a Posteriori (MAP) estimation.;Computer experiments using an argon plasma in an Inductively Coupled Plasma (ICP) reactor revealed that a stove-top coil with a relatively large radius resulted in better uniformity of ion flux on the wafer as compared to a stove-top coil with a relatively small radius. Solenoidal coil configurations showed an interesting behavior of non-uniformity with chamber pressure. The EDS algorithm was used to optimize a plasma reactor design/recipe problem. It was found that the uniformity of argon ion flux (under the constraint that the average ion flux was larger than a given minimum value) was optimal when the solenoidal coil was located 17.86 cm from the reactor top and the gas pressure was 9.0 mtorr.