Stability of a circular epitaxial island

The morphological stability of a single, epitaxially growing, circular adatom island with a radially symmetric adatom distribution is studied using a Burton–Cabrera–Frank type island dynamics model. Various kinds of boundary conditions for the adatom density that include the thermodynamic equilibrium value, line tension, and attachment–detachment kinetics, and different velocity formulas with or without the one-dimensional “surface” diffusion are examined. Rigorous analysis shows that the circular island is always stable if its normalized area A is larger than a critical value. If A is less than such a critical value, and if neither the line tension nor surface diffusion is present, then there exists a critical wavenumber kc = kc(A) such that the island is only stable for wavenumbers less than kc. When the line tension or surface diffusion is present, small islands are always stable. In particular, the Bales–Zangwill instability for straight steps due to the kinetic asymmetry does not exist for small circular islands.