Chapter 1, Problem 1.55P

Annealing, an important step in semiconductor materials processing, can be accomplished by rapidly heating the silicon wafer to a high temperature for a short period of time. The schematic shows a method involving the use of a hot plate operating at an elevated temperature $T_h.$ The wafer, initially at a temperature of $T_{w,i},$ is suddenly positioned at a gap separation distance L from the hot plate. The purpose of the analysis is to compare the heat fluxes by conduction through the gas within the gap and by radiation exchange between the hot plate and the cool wafer. The initial time rate of change in the temperature of the wafer, ${\left(d{T}_w/dt\right)}_i,$ is also of interest. Approximating the surfaces of the hot plate and the wafer as blackbodies and assuming their diameter D to be much larger than the spacing L, the radiative heat flux may be expressed as $q_{\text{rad}}^{"}=\sigma \left(T_H^4-T_w^4\right).$ The silicon wafer has a thickness of $d=0.78\text{mm,}$ a density of $\text{2700}{\text{kg/m}}^{\text{3}},$ and a specific heat of $875\text{J/kg}\cdot \text{K}\text{.}$ The thermal conductivity of the gas in the gap is $\text{0}\text{.0436}\text{W/m}\cdot \text{K}\text{.}$

(a) For $T_h=600°\text{C}$ and $T_{w,i}=20°\text{C,}$ calculate the radiative heat flux and the heat flux by conduction acrossa gap distance of $L=0.2\text{mm}\text{.}$ Also determine the value of ${\left(d{T}_w/dt\right)}_i,$ resulting from each of the heating modes.

(b) For gap distances of 0.2, 0.5, and 1.0 mm, determine the heat fluxes and temperature-time change as a function of the hot plate temperature for $300\le T_h\le 1300°\text{C}\text{.}$ Display your results graphically. Comment on the relative importance of the two heat transfer modes and the effect of the gap distance on the heating process. Under what conditions could a wafer be heated to $900°\text{C}$ in less than 10 s?