P. Petrik
Ellipsometric models for vertically inhomogeneous composite structures
Physica Status Solidi A 205 (2008) 732–738
The optical properties of thin films usually depend on the preparation conditions. For example, the microstructure of ion implantationcaused damage, the grain size of deposited polycrystalline materials, or the porosity of chemically etched silicon largely depend on the parameters of ion implantation, deposition, and etching, respectively. Generally, as a first approach, these materials can be considered as a mixture of components described by reference dielectric function data from the literature using the effective medium approximation (EMA). Interface roughness and vertical inhomogeneity can be described by the volume fraction of components changing as a function of depth. The EMA models are robust, and provide useful information like damage depth profiles of ion implanted materials, crystallinity and surface roughness of deposited materials, or porosity of chemically etched porous silicon. The requirement for EMA is that the size of the components must be smaller than the wavelength of the measuring light, but large enough to retain their bulk dielectric properties. If the component sizes are large, diffraction and scattering has to be taken into account. For example, surface roughness is routinely modeled by EMA, but in case of rough surfaces with large correlation length EMA may become invalid as approaching small wavelengths in the UV part of the spectrum. If component sizes decrease, it may not be possible to describe the components using bulk reference data, because of size effects or strain. For example, fine grained polycrystalline silicon or highly porous silicon can not be modeled by the usual mixture of singlecrystalline silicon (cSi) and amorphous silicon (aSi) for polycrystalline silicon, or mixing cSi and voids for porous silicon, but a component representing an intermediate structure between cSi and aSi has to be used to describe the vanishing longrange order. A less robust but more detailed analysis is provided by parametrization of the dielectric function of the layer or the components in an EMA composition using model dielectric functions, critical point (CP) models, oscillator models, or empirical dispersion equations. In these cases, crosscorrelation, parametercoupling, selection of sensitive parameters, local minima, and regression in a multiparameter space are crucial issues to be handled properly. Consideration of the optical penetration depth (OPD) is also important, when measuring vertically inhomogeneous semiconductor materials. Obviously, CP analysis in semiconductors is most sensitive around the CP photon energies, but the OPD is the smallest just at this region of the spectrum. In silicon, the OPDs at the E_{1} and E_{2} CP energies are about 10 nm and 5 nm, respectively. Consequently, CP analysis of structures below this depth is not possible using ex situ measurement of silicon. This work shows approaches and examples to model multicomponent materials created using different methods including ion implantation, deposition, and electrochemical etching.
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