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Three different objective functions were used as manufacturability criteria for the gate oxidation optimization. The aim was to observe if one particular metric was better than the others for solving the problem of statistical recipe generation. Following are the objective functions used as the test cases:
The minimization and maximization were carried out using a small and a large target value, respectively. For each case, optimizations (smallest sum of square errors from defined targets) were carried out from multiple starting points. The verification runs of the best optima for each case are shown in Fig. 12.
It was found that for each of the three cases the optima for all the starting points resulted in very nearly the same values for the outputs. Moreover, the final values of the output parameters are almost identical irrespective of the objective function. On closer observation it is also found that the optimal process inputs, and , are identical for the first two cases, i.e., where and are used as objective functions. The optimal process inputs when yield was used as an objective function are slightly different.
These results are shown pictorially in Fig. 13.
The nearly identical answers in all three cases was due to the stringent constraint on . All three metrics improve with smaller values for the standard deviation of the gate oxide (). Yield is maximized if in addition to a decreasing variability the design is ``centered'' -- the mean value is chosen so that most of the distribution is within the constraints of acceptability. Similarly the signal-to-noise is maximized as the mean value increases without adversely affecting the standard deviation, or the standard deviation decreases without a significant drop in the mean value. For the case where the mean value is tightly constrained (towards the center of the window in this case) all three metrics are optimized when is minimized, subject to the constraints on . The constraint on the mean ensured that both and are strongly dominated by . This is the case even when the and are not independent. The difference in the results between the yield maximization and the remaining metrics can be attributed to the fact that the distribution of the gate oxide thickness results in larger yield loss in the tail regions when signal to noise or standard deviation alone are optimized.
The dependence of and on was explored by repeating the same optimizations without the constraints on the . Results (shown schematically as dashed curves in Fig. 13) indicated that both and were strong functions of both and . Moreover, the sensitivities of the yield and signal to noise ratios to the mean and the standard deviation were significantly different. Maximizing produced the same results with and without the constraint on . On the other hand, the minimization and maximizations produced identical results which were significantly different from the corresponding results with the constraints on . The optimizations were dominated by decreasing the variability of the gate oxide thickness and the resulting process conditions produced extremely low yields, i.e. the optimizer was able to push the process conditions near the edge of the acceptability region.
This example demonstrates that DOE/Opt can be used to assist in statistical process design, an integral part of design for manufacturability. Several different objective functions can be defined and the results of using different statistical criteria for optimization can be explored. We have been able to derive and calculate multi-dimensional yield by encapsulating a statistical process simulator, FABRICS. Finally, we have been able to determine the process conditions that maximize the value of the yield using the optimizer NPSOL in DOE/Opt.