A semiconductor industry roadmap for microlithography processing requires that a 300-mm-diameter silicon wafer be maintained at a steady-state temperature of 140°C to within a uniformity of 0.1 Dc. The design of a hot-plate tool to hopefully meet this requirement is shown schematically. An equalizing block (EB), on which the wafer would be placed, is fabricated from an aluminum alloy of thermal conductivity k = 75 W/m ∙ K and is heated by two ring-shaped electrical heaters. The two-zone heating arrangement allows for independent control of a main heater (MH) and a trim heater (TH), which is used to improve the uniformity of the surface temperature for the EB. Your assignment is to size the heaters, MH and TH, by specifying their applied heat fluxes, q"_mh and q"_th; (W/m²), and their radial extents, ∆r_mh and ∆r_th. The constraints on radial positioning of the heaters are imposed by manufacturing considerations and are shown in the schematic.

Use the finite-element method of FEHT to perform a conduction analysis on an axisymmetric EB of 340-mm diameter. The upper and lateral surfaces are exposed to the ambient fluid at T_∞ = 25°C with a convection coefficient of 10 W/m² ∙ K. The lower surface of the EB is adiabatic, except for the ring sectors with the uniform applied heat fluxes, q"_mh and q"_th.

(a) For an upper surface of 140°C, perform an overall energy balance on the EB to obtain an initial estimate for the applied heater fluxes. Assume that q"_mh = q"_th and that each heater extends fully over the radial limits indicated schematically. Use this estimate as a boundary condition in your FEHT model; determine the temperature distribution; and using the View/Temperature Contours command, examine the isotherms and the temperature distribution across the upper surface of the EB. Did you achieve the desired uniformity?

(b) Rerun your FEHT model with different values of the heater fluxes, until you obtain the best uniformity possible within the imposed constraints.

(c) In what manner would a non-uniform distribution of the convection coefficient across the upper surface of the EB affect the temperature uniformity? For the downward flowing gas stream used in the microlithography process, a representative distribution of the convection coefficient on the upper surface of the EB is h(r) = h_o [1+a(r/r_o) ⁿ], where ho = 5.4 W/m² ∙ K and a = n = 1.5. For this distribution and retention of a value of h = 10W/m² ∙ K at the lateral surface of the EB, can you adjust the trim heater flux to obtain improved uniformity of the surface temperature?

(d) What changes to the design would you propose for improving the surface temperature uniformity?