Numerical ellipsometry: High accuracy modeling of thin absorbing films in the n–k plane

Abstract

A major challenge for those utilizing ellipsometry is numerical processing of the measured data. Our recent work shows how the transcendental, multivalued equations arising from the physics of reflection from layers can be solved in the n–k plane. This approach applies the mathematics of Complex Analysis to solve the equations numerically. The work presented extends the n–k method to obtain solutions within the accuracy limit of each measurement. The system treated here is that of a thin absorbing film (chromium) overlying a known substrate (silicon). Solutions for a three-layer model of the chromium film including film–substrate and film–air interfacial layers result in a mean square error (MSE) on the order of 0.01, a significant improvement over a single-layer model. Relaxing the constraint of vertical homogeneity provides a six-layer model with the same interfacial layers and four sublayers of chromium. The chromium layers have near-identical values of optical properties and an MSE of essentially zero (10−13). It is anticipated that additional methods will be needed for other classes of problems.