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

Abstract

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 ( T SET) 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 ( Q T, W) as a constant heat flux ( q″, W/m 2) over the heater bottom surface (A, m 2) would result in a fixed temperature difference Δ T (= T MAX − T MIN) on the heater top surface. Minimizing this heater surface Δ T – 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 Δ T further (Δ T → Δ T MIN) for identical steady state heat power Q T, a cost-effective method of two-zone redistribution of the heater bottom surface heat fluxes (two heat fluxes q 1 ″ and q 2 ″ 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 T SET, the achievable heater surface Δ T MIN decreases as the split radius increases. Also, there exists a critical split radius ( r c) below which no energy need be given to the inner-zone to achieve Δ T MIN (i.e., q 1 ″ = 0 ). This r c 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 Δ T, q 1 ″ / q 2 ″ , and r c were found to be independent of the T SET and dependent only on the heat losses. Limiting values of achievable heater surface Δ T MIN for various split locations dividing the two-zones of heat flux are also presented.