Theoretical and Experimental Study of Ultra-smooth Surface Based on Gaussian Topography

Abstract

All natural surface heights conform to the law of Gaussian distribution. Thanks to this law, researchers can measure the topography of different surfaces. To function, modern precision optical instruments need extremely smooth surfaces whose RMS is usually controlled under 0.5nm. A Surface that meets this condition is called ‘Ultra smooth Surface.’ Researchers normally use X-ray reflectance test, Atomic Force Microscope (AFM) scanning and Contour scanning when they characterize smooth surfaces. However, not all the optimal conditions of the research instruments can be met in real experiments. Hence, when scientists want to measure ultra-smooth surfaces, results of these methods are often limited. Furthermore, important statistics such as the fractal index and the lateral correlation length cannot be obtained, and redesigning the current measurement devices doesn’t help much to increase accuracy. X-ray scattering not only empowers scientists with scattering models that determine the surface roughness of a given object, but also offers a PSD curve that functions on a wider spectrum. In this experiment, I put two 15mm*15mm silicon chips on the X-ray scattering device, used the glancing incidence of 0.1 degrees and 0.15 degrees respectively, and analyzed the data. Data from the Detecting Scan is analyzed by the First Order Vector Perturbation theory, and a Power Spectrum Density (PSD) is obtained. Adding on to that the Global Optimization Algorithm—Simulate Anneal Arithmetic, the PSD function is fully fitted, Thus, the surface roughness sigma, the fractal index h, and the lateral correlation length, a, is obtained. Comparing the PSD function of X-ray scattering with results obtained by the Atomic Force Microscope and Contour scanning, this project proved the validity and strength of using the new method of X-ray scattering to measure surface roughness.