Parameter correlation and precision in multiple-angle ellipsometry

Abstract

A general procedure for the characterization of the optical parameters of a given surface using fixed wavelength multiple-angle-of-incidence ellipsometry is developed. Quantitative evaluation of estimated parameter precision is studied considering the sum of errors squared of a least squares fit, the parameter cross-correlation at this minimum, and the confidence limits of the parameters from variance-covariance relations. Most significantly, it is shown numerically that precision is improved when the solutions exhibit a lower correlation to the remaining parameters when a correct system model is used. Using the silicon dioxide-silicon substrate system we were able to determine the complex index of silicon and the extinction coefficient of the oxide with high precision. Once this was done it was possible to estimate with reasonable confidence the refractive index and particularly the thickness of the oxide despite an observed large cross-correlation coefficient between them. A wide range of starting values for the film refractive index n(f) and thickness t(f) always converged to a small sum of errors squared, and nearly identical values for n(f) and t(f) after accurate estimates were obtained for the other parameters of the system.

Method for determination of the parameters of transparent ultrathin films deposited on transparent substrates under conditions of low optical contrast

“…Then it might become impossible to distinguish experimentally which of the parameters, the refractive index or the thickness, causes the changes observed for certain ellipsometric characteristics. The latter circumstances can restrict significantly the resources of direct optimization methods [13] for solving a standard inverse ellipsometric problem, due to increasing confidence limits occurring for the correlated parameters [10,14,15], and even result in failure to obtain a unique solution. Hence, the ellipsometric studies of essentially thin films face serious problems.…”

An Interface Enhanced Spectroscopic Ellipsometry Technique: Application to Si-SiO2

“…Spectroscopic (as opposed to single wavelength) ellipsometry measurements at a fixed angle of incidence helps to decrease the correlation, but does not overcome this problem (34,35), especially when the number of unknown parameters is more than two. In order to obtain more information about the properties of the interface, it is useful to use both SE and MAI measurements to eliminate the correlation between the thickness of the interface and its optical parameters.…”

A software program for aiding the analysis of ellipsometric measurements, simple methods

“…+ fc2e/x]2 = -^^l^= « where is an arbitrary constant and real because of translation invariance within the plane. One may write so that ^ixkaOL X = e"*"** (12) where in the ambient region, = \/i7sin0 = constant > 0, (13) with £0 being the dielectric function of the ambient region, and 0 being the angle of incidence of the light, i.e., 0<<^< y. Here, (^= 0) involves a direction normal to the surface.…”

“…From the differential representation of Majcwell's equations across a surface of discontinuity in piecewise continuous media, the tangential fields, Ey and H^^are continuous across boundaries between adjacent layers. Consequently, eq (12) becomes valid within each spatial region, i.e., ambient, layers, and substrate. Then, letting ri = \J z^Lo?…”