Modeling of a Radiative RTP-Type Furnace Through an Inverse Design: Mathematical Model and Experimental Results

The present study considers the inverse thermal design of an existing radiatively heated chamber for rapid thermal processing (RTP) of semiconductor wafers and compares the numerical results from the inverse mathematical model with experimental measurements directly taken on the physical system. The purpose of this study is to calculate with an inverse approach the power input of the chamber heating elements in order to achieve specific temperature and heat flux distributions on the design surface (wafer). Once a reasonable solution is found, the experimental confirmation takes place. The study starts with characterizing and modeling the system formed by an axisymmetric vacuum chamber designed to reproduce the major characteristics of a generic RTP process in the semiconductor industry, such as for rapid thermal annealing of wafers. A Monte Carlo model is employed for the radiative heat transmission using measured specular/diffuse reflectivities for the radiation shields, and diffuse emissivities for all other surfaces. The actual geometry of the chamber is modeled. The inverse problem is then solved by using the Conjugate Gradient Method. Using the calculated input powers of the heaters to maintain an isothermal wafer as the initial setting for the instrumented radiative chamber, experimental measurements of the steady temperature distributions on the elements of the chamber and on the wafer surface are taken. The study ends by comparing the experimental and numerical results to determine the necessary enhancement to apply to the system model with the final objective of developing an effective design tool using an inverse approach.