Reliability in Modeling of Spectroscopic Ellipsometry

Abstract

Spectroscopic ellipsometry (SE) [1, 2] is a powerful non-destructive technique to determine the structure of silicon-based materials such as amorphous silicon (a-Si) and hydrogenated amorphous silicon (a-Si :H). In analyzing the SE spectra, the Bruggeman effective medium approximation (EMA) [3, 4] is usually used to model a thin film as a mixture of amorphous silicon and voids. In these models, however, precise knowledge of the dielectric functions of the various components is required in order to obtain reliable estimation of the structural parameters. This requirement is particular critical to the amorphous component because the structure of amorphous silicon is known to depend considerably on the preparation procedures [5]. So far, the dielectric functions used for modeling a-Si1 x :Hx have been based on certain dielectric functions without consideration of the preparation method. It may not be clear, particularly for thin films of amorphous silicon prepared by novel method, whether it is relaxed or unrelaxed dielectric functions which are appropriate for the modeling. In the present work the scope of the problem is assessed by constructing dielectric functions for relaxed and unrelaxed a-Si1 x :Hx thin films using measured a-Si dielectric functions, and fitting, respectively, using simulated dielectric functions for bulk unrelaxed and relaxed a-Si1 x :Hx on the surface roughness and void concentration parameters. The tetrahedron model [3, 4] is applied to construct the complex dielectric functions for relaxed (e 1⁄4 e1r þ ie2r) and unrelaxed (e 1⁄4 e1ur þ ie2ur) a-Si1 x :Hx, by using the dielectric functions for relaxed and unrelaxed a-Si [5], respectively. For details we refer to the work of Mui and Smith [3], [4]. The complex dielectric functions constructed for relaxed and unrelaxed a-Si1 x :Hx components with various hydrogen concentrations are shown in Fig. 1. It is seen that the relaxed and unrelaxed spectra exhibit different peak position and amplitude. Moreover, as the hydrogen content increases, the complex dielectric functions shift to higher photon energy with lower peak values. The complex dielectric functions for the relaxed (unrelaxed) a-Si1–x :Hx film are constructed using a three-layer structure: the first layer is a surface roughness layer, modeled using the EMA, consisting of 50% voids, 50% layer 2. Layer 2 is modeled using the EMA, consisting of a mixture of relaxed (unrelaxed) a-Si1 x :Hx and voids. The substrate is smooth silicon dioxide (SiO2). Fractional void concentrations in layer 2 are varied from 0.01 to 0.26 in the simulations. The thickness of layer 1 is set to be 0.001 mm, while the film thickness (layer 2) is fixed at 1 mm. The incidence angle is set to be 75 . The constructed complex dielectric functions for relaxed a-Si1 x :Hx films are then fitted using unrelaxed a-Si :H spectra, applying a nonlinear regression based on the Levenberg-Marquardt method [6], and vice-versa. In order to avoid interference caused by the interface, the fits are done above 1.75 eV. We denote the void concentration and the thickness of the surface roughness by Cvoid and TR respectively. The fitting error of the void concentration and the thickness of surface roughness are denoted by DCvoid and DTR, respectively, i.e. Cvoid 1⁄4 Cconstructed void Cfit void , similarly for DTR. In order to reveal the quality of the modeling, we define