Comparative study of x-ray scattering by first-order perturbation theory and generalized Harvey-Shack theory

Abstract

Application of the x-ray scattering (XRS) technique for studying super-smooth surfaces such as Si wafers is discussed. The XRS method is demonstrated to enable quantitative evaluation of power spectrum density (PSD) functions and effective roughness of super-smooth surfaces. Within the calculation of PSD functions, comparative study between first-order vector perturbation theory (FOPT) and generalized Harvey-shack theory (H-S) is performed. First-order perturbation theory which is widely accepted and has been extensively validated even for large scattered and incident angles for smooth surfaces considers the scattering amplitude as a power series in the roughness height; its scattering diagram is related to the statistical parameters of surface roughness (PSD functions) in a very simple way (linear). Therefore, PSD functions can be uniquely and directly extracted from the measured data. However, generalized Harvey-shack theory considers that scattering behavior is characterized by a surface transfer function which relates the scattering behavior to the surface topography. With the grazing incident angle less than critical value about 0.22 degree, three Si wafers with rms roughness of 0.29nm, 0.46nm and 0.67nm are inspected by XRS (λ=0.154nm) method. The calculated values are all in a good agreement with the results obtained from Atomic force microscope (AFM). However, the difference resulting from the limits of applicability of the theories used in XRS data processing appears and be analyzed. Both of the theories are not only used for optical surface characterization, but also can allow accurate predictions of image degradation due to scattering effects in grazing X-ray telescopes. The experimental schemes are also analyzed.