The construction of orientation-dependent crystal growth and etch rate functions I. Mathematical and physical aspects

For mathematical analysis and computer simulation of the shape evolution of crystals, we need a continuum description of crystal growth or etching, ra ther than the conventional atomistic description. This allows the mathemati cal integration of the interface process with other transport steps that ar e usually also described by continuum equations, like diffusion, viscous fl ow, and chemical reactions. For this reason we need a function R(n,T,C,p,.. .): the growth or etch rate as a function of the surface orientation n and of experimental variables such as temperature, composition, pressure, etc. of the parent phase. In this article we describe a logical construction met hod for such growth or etch rate functions. The virtue of our method is tha t the n variable covers the full unit sphere, i.e., all minima due to diffe rent crystal facets are expressed in the R function. The orientation depend ence of the growth or etch rate of interfaces (three dimensional) and of st eps on a facet (two dimensional) is described in a way which is logically b ased on the kink/step motion (KSM) growth model. The building blocks of the growth/etch rate function are the elementary KSM functions, plus a number of constants which are to be determined by a parameter-fitting procedure bu t do have an obvious physical meaning. For instance, for each face a roughe ning parameter enters into the function, expressing the effect of the rough ening transition for this face. This compares favorably with a Fourier seri es or spherical harmonics expansion for which the constants that appear hav e no specific relevance for the growth/etch mechanism. In this article we i ntroduce the mathematical toolbox which is required for the "nonlinear netw ork" formalism and we use this formalism for the construction of growth/etc h rate functions. In Part II we work out a practical case and compare a set of accurately measured etch rate data (silicon crystals in concentrated KO H solutions) with a network etch rate function. (C) 2000 American Institute of Physics. [S0021-8979(00)01612-1].