Unified approach to the temporal evolution of surface profiles in solid etch and deposition processes

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Abstract
The differential equation describing the temporal evolution of solid surfaces subject to etching or depositon is derived. Its general properties are discussed and solutions to particular problems are obtained. The differential equation explicitly shows the evolution of the shape of the surface to be the product of a geometric factor which is common to all etch/deposition processes and a rate factor which depends on the physics of the process considered. The rate factor must in general be derived from first principles. Here we limit consideration to rate factors that describe isotropic and unidirectional processes, as well as linear combinations of these two processes. The properties of the evolution of surface profiles are studied in detail for each case. A unidirectional etch with a rate factor dependent on the local surface curvature is shown to produce good agreement with the observed shapes of deep isolation trenches in silicon. Finally, the surface evolution equation is used to expore the ability of certain processes to smooth and polish initially rough surfaces.