Calculation Model and Optimization of Epitaxial Layer Thickness based on Multi-beam Interference Using Nonlinear Least Squares Method

Abstract

Addressing the systematic errors introduced by dual-beam interference models in epitaxial layer thickness calculations and considering the robustness of actual interference phenomena, this study establishes a calculation model for the relationship between reflectance and epitaxial layer thickness under multi-beam interference conditions. This research is significant for improving the accuracy and reliability of thickness measurement in semiconductor manufacturing processes, which is a critical parameter for controlling device performance and yield. First, the necessary conditions for multi-beam interference are deduced, including identical frequencies, consistent vibration directions, constant phase differences, light source characteristics, and low absorption rates of the medium. Their impact on thickness calculation accuracy is analyzed, concluding that higher precision amplifies noise effects. The model innovatively incorporates the Airy formula, combining it with the Fresnel equations and the Zeilemann dispersion formula to derive an analytical expression for total reflectance. By applying nonlinear least squares fitting to experimental spectral data from silicon wafers, an optimal thickness of 30.0312 μm was obtained with a relative deviation of only 0.02%, indicating negligible error consistent with practical conditions. Simultaneously, fitting data from silicon carbide epitaxial layer attachments using this precise model yielded a thickness of 8.0192μm, partially eliminating systematic errors introduced by simplified models.