Transient and Stable Profiles During Anisotropic Wet Etching of Quartz

This paper presents a comprehensive analysis of the transient (short-term) and stable (long-term) cross-sectional profiles of elongated trenches formed by anisotropic wet etching on Z-cut quartz. Because all quartz planes parallel (and near-parallel) to the [0001] direction display very low (and dissimilar) etch rates, dramatically different profiles are observed depending on the particular mask alignment of the trench edges with respect to the substrate. Nevertheless, the profiles can be easily predicted by identifying the location of a few key orientations for each trench profile. This is done by measuring the etch rate distribution on a hemispherical specimen of quartz and locating the local maxima of the (signed) curvature of the etch rate for all planes parallel to the trench edges. We show that this generalizes the traditional method based on locating the local minima of the etch rate. Furthermore, our approach enables capturing accurately the etch front by applying the traditional Wulff-Jaccodine construction on just a few key orientations, thus greatly simplifying the use of that method and highlighting the simplicity and importance of the proposed key orientations. A maximum positive curvature recognition method is developed based on applying the moving average filter to the calculation of the first, second, and third order derivatives of the etch rate. This allows quick locating of all orientations that will appear on the short-term and long-term profiles, even for the complete etch rate distribution, thus helping the design of desired profiles with vertical (or nearly vertical) side walls. Furthermore, the proposed maximum positive curvature recognition method accurately explains the formation of complex facets on square cavities, as well as on the top, middle, and bottom regions of micro-needle arrays etched on quartz Z-cuts.