Dynamic roughening: Nucleation and stochastic equations

A crystal surface, which in equilibrium shows a thermal roughening transition, may roughen, under growth conditions, beneath the equilibrium transition temperature. After discussing a simple nucleation approach, this topic is treated from a point of view based on the Chui–Weeks stochastic differential equation. Numerical solutions are presented. Special care has been made in order to get a reliable algorithm and a correct sampling for numerical integration. Scaling properties are also discussed: they appear to be essential to choose good values of the parameters ν, Y, D. A finite size effect in equilibrium, revealed by numerical computations and related to dynamic roughening, is discussed. The ratio D/ν between the coefficients of the noise and Laplacian terms plays the role of temperature; off equilibrium, the dependence of the growth velocity on the flux and on the strength of the layering potential shows a non-trivial, non-linear behaviour, going over from a Becker–Döring to a Wilson–Frenkel growth mode