Numerical ellipsometry: Advanced analysis of thin absorbing films in the n–k plane

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. Previously we have applied Complex Analysis in the n–k plane and this has led to a growing array of numerical research methods which achieve computational accuracy at a chosen limit (~ 10− 14) rather than achieving only a mean square value of a large number of measurements, more than 14 orders of magnitude less accurate in this case. The work presented here is the mathematics for film stacks of an arbitrary number of layers required to apply numerical methods for use with absorbing films deposited on transparent and on absorbing substrates. The approach finds intersections of projections of three “twisted curves” resulting from a single light incidence angle and light wavelength measurement for three film thicknesses at a time. The approach employed here determines the thickness and optical properties of chromium films (between 10 and 25 nm nominal thickness) deposited onto silicon substrates. A simple single film model matched the measurements to a first order only. Improved agreement is obtained by the addition of interface layers above and below the main chromium film.