Effective medium approximation of ellipsometric response from random surface roughness simulated by finite-element method

We used numerical simulations based on the finite-element method (FEM) to calculate both the amplitude and phase information of the scattered electric field from random rough surfaces, which can be directly compared to ellipsometric measurements and effective medium approximation (EMA) calculations. FEM can serve as an exploration tool for the relationship between the thickness of the surface roughness evaluated by Bruggeman EMA and the morphological parameters of the surface, such as the root mean square height, the lateral auto-correlation length, and the typical average slope. These investigations are of high interest in case of poly-crystalline and amorphous materials. The paper focuses on the simulations of rough Si surfaces. The ellipsometric calculations from FEM and EMA simulations match for wavelengths of illumination much shorter than the typical feature size of the surface. Furthermore, for these cases, the correlation between the EMA thickness and the root mean square height of the roughness for a given auto-correlation length is quadratic, rather than linear, which is in good agreement with experimental measurements and analytical calculations presented in recent reports.