An inverse heat transfer method to provide near-isothermal surface for disc heaters used in microlithography

In microlithography, the fabrication method for semiconductors and MEMS devices, the post-exposure baking process involves the baking (heating) of a 300 mm diameter, ∼1 mm thick silicon wafer substrate with a disc heater to a set point temperature (TSET) triggering the photo-chemical reaction undergone by the photo-resist applied on the wafer. For a known loss occurring due to the convection boundary conditions at the top and side of the disc heater surface, providing a steady state heat power (QT, W) as a constant heat flux (q″, W/m2) over the heater bottom surface (A, m2) would result in a fixed temperature difference ΔT (=TMAX − TMIN) on the heater top surface. Minimizing this heater surface ΔT – an imprint of which is transferred to the heated wafer – is crucial for determining the accuracy of the semiconductor circuit pattern etched on the silicon wafer. To reduce this ΔT further (ΔT → ΔTMIN) for identical steady state heat power QT, a cost-effective method of two-zone redistribution of the heater bottom surface heat fluxes (two heat fluxes q1″ and q2″ given, respectively, to the inner and the outer-zones) is proposed. This inverse heat transfer problem in steady state is verified using numerical methods and scaling analysis from first principles. For given convection heat losses and TSET, the achievable heater surface ΔTMIN decreases as the split radius increases. Also, there exists a critical split radius (rc) below which no energy need be given to the inner-zone to achieve ΔTMIN (i.e., q1″=0). This rc value is predicted using the theoretical scaling analysis and was found to match excellently with the value obtained from numerical methods. The variations of heater surface ΔT  , q1″/q2″, and rc were found to be independent of the TSET and dependent only on the heat losses. Limiting values of achievable heater surface ΔTMIN for various split locations dividing the two-zones of heat flux are also presented.