Understanding and Improving Materials Processing through Interpreting and Manipulating Predictive Models
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Dynamic optimization is related to the inverse problem, but rather than specifying the desired trajectories as specific constraints they are specified as least-squares inequality constraints. For example, instead of requiring exact temperature-time histories at certain points on a wafer (the inverse problem), the problem would be posed by requiring that the wafer follow a specified temperature trajectory within plus or minus 10 degrees and with no more than 5 degrees temperature variation within the wafer. Mathematically, dynamic optimization is derived from optimal control theory. Both Inverse Problems and Dynamic Optimization provide the equipment designer with powerful capabilities. One important application is to evaluate and understand trajectory feasibility and control authority. For example, consider a nominal CVD reactor design with a maximum heater power of 50 kW and a process that calls for a temperature ramp to process temperature in 20 seconds with a specified maximum temperature gradient within the wafers of 5 degrees. The inverse approaches will determine the power-time histories for the heaters. If the required power exceeds the maximum available for the given design, then the desired trajectory is infeasible. The constraints would either have to be relaxed or a alternative design developed that provided sufficient "control authority." While the inverse problems are not used to design the controller per se, they do provide important information on controllability and process feasibility. While nearly all physically based modeling is run "open loop," nearly all processing equipment is run under "closed loop" process control. There is a great need to develop the software tools that permit driving the physically based models under closed-loop control, and thus enable the concurrent engineering of equipment design and control algorithms. We seek to run the physically based simulations under the control of the very same process-control software that would be loaded into the actual hardware controllers. Certain dependent variables in the model become the "sensors" (e.g., the temperature of an element on a wafer) and boundary conditions in the model become the "actuators" (e.g., the power or temperature of heater elements), which are controlled by the process-control software. With well defined communication interfaces, the controller software and the physical model could be running simultaneously on different computer platforms -- the physical model replaces the actual hardware. There is little doubt that equipment-development times would be shortened considerably when the controller design is developed and evaluated on a validated physical simulation during the equipment design and development effort. Rapid-Thermal Processing Examples CVC Products, Inc. is developing a new rapid-thermal-processing (RTP) reactor for application to several processing steps in silicon-semiconductor fabrication, including contact silicide formation, dielectric deposition, and source-drain anneal. This new product development
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