Analytical models are used to optimize thermal process trajectories subject to restrictions on allowable wafer stresses and required uniformity of thermally activated processes. The maximum operating temperature is first determined by comparing calculated gravitational stresses with the temperature-dependent wafer strength. At temperatures less than this maximum, the difference between wafer strength and gravitational stress is used to determine the allowable stress and allowable temperature variation across the wafer as a function of temperature. This allowable temperature difference is then converted to an allowable ramp rate based on an analytical model of radial heat transfer in a batch furnace. Radial temperature variations predicted by this model are shown to be in good agreement with measurements made by others. Finally, these models are used to calculate optimum process trajectories and maximum batch furnace throughputs subject to prescribed thermal process uniformity. For single-wafer processors, measured temperature histories are used to construct an empirical thermal response model that can be used for stress analysis and process optimization.
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