Asymptotic Analysis of the Shape and Composition of Alloy Islands inEpitaxial Solid Films

Abstract

We consider the formation of solid drops(“islands”) occurring in the growth of strained solidfilms. Beginning from a detailed model for the growth of an alloy filmthat incorporates the coupling between composition, elastic stress andthe morphology of the free boundary, we develop an asymptoticdescription of the shape and compositional nonuniformity of smallalloy islands grown at small deposition rates. A key feature of theanalysis is a “thin domain” scaling in the island which enablesrecasting the free boundary problem into a set of integrodifferentialequations for the shape and composition profile. Theintegrodifferential system can be decomposed into two parts: one partgives a single integrodifferential equation for the shape analogous tothat obtained for a single-component island determined by Shanahan andSpencer (2002), and the other part gives the composition profile interms of the shape. Thus, the shape of an alloy island is identical tothat of a single-component island with the same system parameters, butwith a nonuniform composition that depends on the stress-compositioncoupling and alloy solution thermodynamics. Finally, we characterizethe structure and magnitude of the compositional nonuniformity andalso interpret our theoretical results in the context of SiGe alloyfilms.