Continuum level-set model for anisotropic wet etching of patterned sapphire substrates
We present a continuum modeling approach to simulate anisotropic wet etching of single-crystal sapphire employing mixtures of sulfuric acid and phosphoric acid. Wet etching of sapphire leads to the formation of crystal facets with high Miller–Bravais indices. The resulting complex three-dimensional topographies can be exploited to optimize the patterning of sapphire substrates which are employed for gallium nitride based light-emitting diodes. Due to the strong impact of the three-dimensional sapphire topography on the light extraction efficiency of the final device, precise control over the wet etching process, in particular etchant mixture, etch time, and temperature, is highly important. We present our model in the context of process technology computer-aided design, where we use the level-set method to track the evolution of the three-dimensional etch profile over time. In order to describe the intricate anisotropy of wet etched sapphire, we propose a flexible interpolation method for the etch rate distribution, which incorporates experimentally characterized crystal facets and deduces local extrema in the distribution based on local convexity/concavity considerations. The developed model has been calibrated and evaluated based on scanning electron microscopy and atomic force microscopy characterizations from the literature. Our model enables accurate sapphire etching simulations, where the emerging and disappearing crystal facets show a good agreement with experimental observations for several etchant mixtures and temperatures.