Application of inverse problem algorithm for temperature uniformity in rapid thermal processing

Abstract

This article presents a finite-difference-method formulation to the application of inverse problem algorithms for uniform temperature tracking of several different linear ramp-up rates in rapid thermal processing. A one-dimensional thermal model and temperature-dependent thermal properties of silicon wafers are used. The required incident-heat-flux profiles for temperature uniformity across 300-mm-diameter 0.775-mm-thick silicon wafer were intuitively evaluated. Our numerical results indicate that temperature non-uniformity occurring during the ramp increase with the ramp-up rate. Although a linear ramp-up rate of 300 °C/s was used and random errors did reach 3.864 °C, the temperature over the wafer was maintained within 0.665 °C of the wafer center if the incident-heat-flux profiles were dynamically controlled according to the inverse results. These temperature non-uniformities could be acceptable in the advanced rapid thermal processing system.