Bayesian inference analysis of ellipsometry data

Abstract

Variable angle spectroscopic ellipsometry is a nondestructive technique for accurately determining the thicknesses and refractive indices of thin films. Experimentally, the ellipsometry parameters ψ and Δ are measured, and the sample structure is then determined by one of a variety of approaches, depending on the number of unknown variables. The ellipsometry parameters have been inverted analytically for only a small number of sample types. More general cases require either a model-based numerical technique or a series of approximations combined with a sound knowledge of the test sample structure. In this paper, the combinatorial optimization technique of simulated annealing is used to perform least-squares fits of ellipsometry data (both simulated and experimental) from both a single layer and a bilayer on a semi-infinite substrate using what is effectively a model-free system, in which the thickness and refractive indices of each layer are unknown. The ambiguity inherent in the best-fit solutions is then assessed using Bayesian inference. This is the only way to consistently treat experimental uncertainties along with prior knowledge. The Markov chain Monte Carlo algorithm is used. Mean values of unknown parameters and standard deviations are determined for each and every solution. Rutherford backscattering spectrometry is used to assess the accuracy of the solutions determined by these techniques. With our computer analysis of ellipsometry data, we find all possible models that adequately describe that data. We show that a bilayer consisting of a thin film of poly(styrene) on a thin film of silicon dioxide on a silicon substrate results in data that are ambiguous; there is more than one acceptable description of the sample that will result in the same experimental data