Investigation of effective-medium models of microscopic surface roughness by spectroscopic ellipsometry

Using measured dielectric function data from 2.1 to 5.5 eV for chemical-vapor-deposition---grown smooth amorphous ($a$-Si) and microscopically rough fine-grained polycrystalline ($p$-Si) films, we show that the dielectric properties of microscopically rough layers of thicknesses 100-500 \AA{}A are accurately modeled in the effective-medium approximation. These microscopically rough layers show essentially no macroscopic light scattering, and thus are inaccessible to measurement by usual scattering techniques. The unambiguous identification of microscopic roughness, as opposed to, e.g., an overlying oxide, is shown to require a spectroscopic capability. Statistical-analysis techniques are introduced to determine model parameters systematically and objectively, and also to establish correlations and confidence limits that show which parameters are defined by the data and which are statistically indeterminate. A best-fit five-parameter model for the sample with the thickest surface region shows that the density profile is characteristic of hemispherical, not pyramidal, irregularities. This indicates that surface roughness arises from a three-dimensional nucleation and growth process in these samples. In a comparison of the three one-parameter effective-medium models, Bruggeman and Maxwell Garnett(2) theories are found to adequately represent the data, while the Lorentz-Lorenz model, previously used exclusively to model roughness in single-wavelength applications, predicts only qualitatively the spectral dependence and gives poor results.