After being cut from a large single-crystal boule and polished, silicon wafers undergo a high-temperature annealing process. One technique for heating the wafer is to irradiate its top surface using high-intensity, tungsten-halogen lamps having a spectral distribution approximating that of a blackbody at 2800 K. To determine the lamp power and the rate at which radiation is absorbed by the wafer, the equipment designer needs to know its absorptivity as a function of temperature. Silicon is a semiconductor material that exhibits a characteristic band edge, and its spectral absorptivity may be idealized as shown schematically. At low and moderate temperatures, silicon is semitransparent at wavelengths larger than that of the band edge, but becomes nearly opaque above 600°C.

(a) What are the 1 % limits of the spectral band that includes 98% of the blackbody radiation corresponding to the spectral distribution of the lamps? Over what spectral region do you need to know the spectral absorptivity?

(b) How do you expect the total absorptivity of silicon to vary as a function of its temperature? Sketch the variation and explain its key features.

(c) Calculate the total absorptivity of the silicon wafer for the lamp irradiation and each of the five temperatures shown schematically. From the data calculate the emissivity of the wafer at 600 and 900°C. Explain your results and why the emissivity changes with temperature.

(d) If the wafer is in a vacuum and radiation exchange only occurs at one face, what is the irradiation needed to maintain a wafer temperature of 600°C?

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0.8 -Band edge 0.7 900°C 0.6 600°C 0.5 S00°C 8 0.4 0.3 400°C 30°C 0.2 300°C 0.1 0 1 2 3 4 5 6 7 8 9 10 Wavelength (um)