Theory of off-normal incidence ion sputtering of surfaces of type A_xB_{1-x} and a conformal map method for stochastic continuum models

Bradley et. al. recently provided an explanation of nanodot and defect-free ordered ripple production from binary compounds, for normal and oblique incidence ion sputtering respectively, by the inclusion of the effect of the preferential sputtering of one type of atom relative to the other type on the surfaces of binary compounds. In this paper, we propose an extended anisotropic model of such surfaces of type AxB1-x that addresses anisotropy in the variations of the specie-composition as well. We show that this model gives the anisotropic Cuerno-Barabasi model as x approaches unity, and analyze the general properties of the model. Further, the complexity of the solutions of non-linear higher-order differential equations in general has led to the creation of great number of highly technical and computationally intensive numerical schemes. We introduce a simple conformal map method which allows for a fast and accurate simulation of the dynamical evolution of ion-sputtered surfaces based on any stochastic continuum model. An optimization algorithm for an efficient application of the scheme is also introduced. In this scheme the noise term has a physical meaning which allows one to go beyond the usual white noise approximation by actually being able to assign physical parameters to it.