Nonlinear evolution of a terrace edge during step-flow growth

Abstract

The nonlinear evolution of a terrace edge during growth in the step-flow configuration is investigated. Starting from the constitutive growth equations of Burton, Cabrera, and Frank [Philos. Trans. R. Soc. London Ser. A 243, 299 (1951)], we show that the dynamics of an isolated terrace edge, i.e., in the absence of any kind of step-step interaction, is described, close enough to the morphological instability threshold found by Bales and Zangwill [Phys. Rev. B 41, 5500 (1990)], by a nonlinear partial differential equation. This equation, although it emerges physically in a somewhat disguised manner, can be simply interpreted as representing the growth with an effective negative line tension, a stabilizing line diffusionlike process along the edge, and a quadratic nonlinearity restoring the Galilean invariance. This nonlinear equation, despite its apparent simplicity, manifests a variety of dynamics, ranging from simple steady-cellular structures to spatiotemporal chaos. When the system size is large enough, chaotic dynamics always prevails. We argue here that when the step-step interaction is important, however, one expects ‘‘regular’’ solutions, such as steady-cellular, broken-parity-traveling, and oscillatory modes, to be stable over a finite range of control parameters.