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This paper derives a renormalization formula defined on the parameter space where mapping behavior is preserved, together with the equivalent potential function. In contrast to the universal function given by Feigenbaum, the behavior near the critical point is governed by the potential function. There are several interesting features which are shared by Feigenbaum's universal function, such as the representation of the critical point in terms of the unstable fixed point of the potential function, but the mapping differs from the scaling. The one-dimensional mapping is considered as an example, and the critical point and the scaling, which are major constants characterizing chaos from the potential function, are calculated precisely. © 1998 Scripta Technica, Electron Comm Jpn Pt 3, 81(8): 41–51, 1998 [ABSTRACT FROM AUTHOR]

The nonuniform temperatures that develop over silicon wafers on insertion and removal from the furnaces used in the oxidation and diffusion processes of semiconductor fabrication are one factor in reduced yields. For example, because of temperature differences in the oxidation process there are thermal stresses and differing thermal hystereses over the wafer surface which degrade the uniformity of the oxide layer formed. The most pronounced temperature differences, occurring when the wafers are inserted into the furnace, develop mainly because of the effect of the boat μg which supports the wafers. Since the heat capacity of the boat is much greater than that of the wafers, during the slower rise in the boat temperature radiant heat exchange with the wafer bottoms keeps the temperature of the bottoms lower than the temperature of the wafer tops, which are almost unaffected by the boat. This transient in the wafer temperature depends on such process parameters as the boat insertion speed, the ratio between the wafer and furnace reactor tube diameters, and the wafer spacing. In order to analyze this temperature characteristic the process is modelled and the results of numerical simulation are compared with experiment. The model for the wafer temperatures inside the furnace is a system of coupled partial differential equations for the mostly radiant heat fluxes of the boat and wafers. With an insertion speed of 2 m/sec, both the experimental data and the model give a maximum temperature difference over the wafer surface of 230°C near the end of insertion, thus confirming the model. [ABSTRACT FROM AUTHOR]