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

摘要

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