The Potential Effect of Multilayer Patterns on Temperature Uniformity During Rapid Thermal Processing
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Mat. Res. Soc. Symp. Proc. Vol. 389 ©1995 Materials Research Society
MODELING APPROACH The reactor scale transport model has been developed by Merchant et al. 8 . Given an input CAD file of the RTP system of interest, radiation exchange factors are calculated by a deterministic ray tracing approach based on the finite element mesh. This approach can account for diffuse and specular reflections. Wavelength and temperature dependent properties are taken into account using a three band model. These exchange factors are then incorporated into steady state and transient finite element fluid flow and heat transfer models. Radiative properties of the wafer and other surfaces in the enclosure are calculated using the matrix method of multilayer electromagnetic theory 9. Given a multilayer stack, this method predicts the reflectance and transmittance for a given wavelength, angle of incidence, and polarization (spectral directional properties). The main assumptions of the theory are that the layers are parallel, the surface is optically smooth, and the area in question is much larger than the wavelength of incident light (no edge effects). The optical constants of the most of the materials are obtained directly from the literature 1011, and are assumed to be independent of temperature. film thickness and microstucture. The temperature dependent optical constants of silicon are obtained from fitting a Drude/oscillator model to the data of Sato1 2. The spectral directional absorptance is obtained subtracting the reflectance and transmittance from unity, and the spectral directional emittance is obtained through Kirchhoffs law. For each 'band', which is defined by a spectral range and characteristic source temperature, these spectral directional properties are integrated with respect to wavelength and angle of incidence, and arithmetically averaged with respect to polarization to yield the total hemispherical properties. For band 1, the source temperature that of wafer, and integration is done from 0.4 to 4.0 4m. For band 2 the source temperature that of wafer, and integration is done from 4.0 to 20.0 pm. For the lamp band, the source temperature is 3000 K and integration is done from 0.4 to 4.0 4im. Since the die area is comprised of several different components (e.g. bare Si, field oxide), a percentage area is assigned to each component, and the radiative properties of each component are calculated using multilayer approach. Finally, an area weighted average is performed to get the final macroscopic radiative properties. This is a gross assumption, as the nature of the die surface could violate two principal assumptions of the multilayer theory. First, the surface is not generally optically smooth due to the topography of the devices. Secondly, the length scale over which a given stack exists can be comparable to the wavelength of the incident radiation. A more accurate estimation could be obtained either experimentally or through more rigorous electromagnetic theory. Given these uncertainties, though, this simple met
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