Dynamics and fluctuations during MBE on vicinal surfaces. II. Nonlinear analysis

This paper is the natural next step beyond the linear regime presented in the preceding paper. By concentrating on the situation close to the step morphological instability threshold, we derive nonlinear evolution equations for interacting steps on a vicinal train. This treatment is coherent in that it retains only relevant nonlinearities close enough to the threshold. Our analysis provides the expression of the coefficients in terms of thermodynamic and transport coefficients. Numerical analysis of these equations reveals spatially and temporally disordered patterns. We give a criterion specifying the region where step roughness is due to both stochastic effects (associated with various sources of noise) and deterministic ones (stemming from deterministic spatiotemporal chaos). Outside this region, the roughness is dominated by either stochastic or deterministic effects. Starting from the discrete version (this is taken to mean that each step is described as an entity) of step dynamics (that is to say, each step is separately described by an evolution equation), we derive a coarse-grained evolution equation for the surface. This results in an anisotropic Kuramoto-Sivashinsky equation including propagative effects. Numerical analysis reveals situations where the original surface undergoes a secondary instability leading ultimately to a rough pattern. The surface looks as if two-dimensional nucleation were allowed. Implication and outlooks are discussed.