Parameterization of the optical function of hydrogenated amorphous carbon by means of B-splines

Abstract

ellipsometry (SE) is a non-invasive optical diagnostic that measures the change in polarization of light reflected on a thin film. To extract both the optical function and thickness of the film from SE data, a (multi-layered) model is required that describes the interaction of the incident light with the film. For amorphous materials this interaction is commonly modeled by the Tauc-Lorentz oscillator and is used to parameterize the optical function [1]. However, a fully mathematical Kramers-Kronig consistent description of the optical function by means of B-splines is also possible [2]. The B-spline parameterization requires no pre-existing knowledge about the interaction of light with the film. The layer structure for this model consists of a substrate, the bulk layer, of which the optical function is represented by Bsplines, and a roughness layer. The roughness is modeled by an effective medium approximation of 50% bulk material and 50% voids. This layer structure is verified by cross-sectional scanning electron microscopy (SEM) measurements. The roughness is in good agreement with values determined by atomic force microscopy (AFM). When the B-spline model is applied to SE data obtained during growth, it has been found that the optical function for every measured thickness is the same, thereby ascertaining the homogeneity of the bulk layer of the a-C:H. During etching of a-C:H with a hydrogen plasma, the optical function of the film - due to the homogeneity of the bulk material - can be fixed throughout the entire etch process, which enables real time in situ monitoring of the thickness evolution. Further parameterization of the dielectric function, as found by the B-spline model, by a physics-based model is possible. For a-C:H films, the bulk layer is described by two Tauc-Lorentz oscillators, from which the sp2/sp3 ratio has been, tentatively, determined from SE data up to 6.5 eV [3]. In all, the B-spline model is an accurate and fast method to determine thickness, roughness and optical constants for numerous types of thin films, including - as has been shown - hydrogenated amorphous carbon. The determined film properties can also be used as input parameters for physicsbased models