Characterization of thin films with linear inhomogeneity by spectroscopic ellipsometry

The measurement of the refractive index n and the physical thickness t of a homogeneous thin film by ellipsometry is a well-established technique. Spectroscopic ellipsometry provides n and t measurements over a broad spectral range, thus allowing determination of the dispersion in n. However, spectroscopic ellipsometry opens up the possibility of characterizing inhomogeneous thin films. This paper deals with the theoretical analysis of the sensitivity of ellipsometry to a linear refractive- index gradient within a single-layer thin film. The film is treated analytically by assuming that the reflectance depends only on the refractive indices at the surfaces of the film, thus ignoring the details of the index profile. The reflectance is also calculated by modeling the linear index gradient by a series of thin homogeneous layers and performing standard thin-film calculations. A major result is that to a first approximation, the ellipsometry of a film at a wavelength which is an odd multiple of four times the optical thickness (i.e., at the odd quarterwave points) is unaffected by the inhomogeneity of the film but depends only on the average refractive index of the film. The ellipsometric results for wavelengths which are multiples of twice the optical thickness (the halfwave points) are strongly affected by the inhomogeneity of the films. Results are presented relative to the sensitivity of ellipsometric measurements for the determination of the average index and the degree of inhomogeneity of a film.