Chapter 1, Problem 1.56P

In the thermal processing of semiconductor materials, annealing is accomplished by heating a silicon wafer according to a temperature-time recipe and then maintaining a fixed elevated temperature for a prescribed period of time. For the process tool arrangement shown as follows, the wafer is in an evacuated chamber whose walls are maintained at $27°\text{C}$ and within which heating lamps maintain a radiant flux $q_s^{"}$ at its upper surface. The wafer is 0.78 mm thick, has a thermal conductivity of $30\text{W/m}\cdot \text{K,}$ and an emissivity that equals its absorptivity to the radiant flux $\left(\epsilon ={\alpha }_l=0.65\right).$ For $q_s^{"}=3.0\times {10}^5{\text{W/m}}^{\text{2}},$ the temperature on its lower surface is measured by a radiation thermometer and found to have a value of $T_{w,l}=997°\text{C}\text{.}$

To avoid warping the wafer and inducing slip planes in the crystal structure, the temperature difference across the thickness of the wafer must be less than $2°\text{C}\text{.}$ Is this condition being met?