Emergent faces in crystal etching

The time development of emergent faces in crystal etching is investigated. We present and discuss a novel computational approach, based on an intuitive geometrical derivation, for predicting an etched shape given an initial polygonal (mask) shape and a diagram of etch rate as a function of orientation. A two-dimensional geometric model is derived which determines the etched shape as a function of time. The model is both intuitive and easy to implement manually or by computer. Because the model is intuitive in nature, some results can be obtained from only partial information. In addition, the model is a first step in the transition from analysis to design. Rather than predicting the etched shape for a given original shape, often what is desired is the original mask shape needed to produce a particular etched shape. This inversion process is carried out for some special cases. The concepts of equilibrium or eigen shapes (Eshapes), limit shapes, and time scaling are introduced. Model predictions are compared with experimental results. The extension from two dimensions to three is also introduced