Reconstructing the 3D etch rate distribution of silicon in anisotropic etchants using data from vicinal {1 0 0}, {1 1 0} and {1 1 1} surfaces

We consider the reconstruction of the complete three-dimensional distribution of etch rates for crystalline silicon in a total of six markedly different etching conditions. The procedure is based on data points that, on the unit sphere, are located along the high-symmetry lines connecting the three main surface orientations Si{1 1 1}, Si{1 1 0} and Si{1 0 0}. Novel, compact formulas are presented in order to perform trilinear, triquadratic and higher order interpolations in h k l space. A wide variety of surface triangulations and tessellations are proposed in order to apply the trilinear and higher order formulas. A statistical analysis concludes that trilinear interpolation over three particular triangulations, and triquartic interpolation over a specific tessellation provide the best reconstructions for the six considered etching conditions. By combining the three triangulations and tessellation using a weighted average, the mean error is found to be less than 13% for reconstructions involving noisy experimental etch rates, while it decreases to 2–5% for less unruly distributions. The results strongly indicate that the complete orientation dependence of the etch rate can be derived from an alternative, more feasible experiment than the traditional hemispherical specimen.