Numerical ellipsometry: Analysis of thin metal layers using n-k-d twisted curve methods with multiple incidence angles

Abstract

A major challenge for those utilizing ellipsometry is numerical processing of the measured data. The transcendental, multivalued equations arising from the physics of simple reflection are problematic for the least-squares numerical methods in common use. These early numerical methods require fairly accurate initial estimates, bounding to avoid local minima, and only find solutions at the bottom of a relatively flat numerical topography. Previously the authors have applied complex analysis in the n-k plane to improve visualization of the mathematics, and this has led to a growing array of new numerical methods avoiding these difficulties. The work presented here extends these new numerical methods to three dimensions (n-k-d space) for use with absorbing films deposited on transparent and on absorbing substrates. The method finds intersections of three “twisted curves” resulting from three (or more) light incidence angle measurements per wavelength. The method is employed here to determine the thickness and optical properties of chromium films (between 10 and 25 nm nominal thickness) deposited onto two different substrates, silicon and silica oxide, employing measurements made at three different angles of incidence for light wavelengths between 280 and 2500 nm.