No abstract

Starting with a many-atom master equation of a kinetic, restricted solid-on-solid (KRSOS) model with external material deposition, we investigate nonlinear aspects of the passage to a mesoscale description for a crystal surface in 1+1 dimensions. This latter description focuses on the motion of an atomic line defect (i.e. a step), which is defined via appropriate statistical average over KRSOS microstates. Near thermodynamic equilibrium and for low enough supersaturation, we show that this mesoscale picture is reasonably faithful to the Burton-Cabrera-Frank (BCF) stepflow model. More specifically, we invoke a maximum principle in conjunction with asymptotic error estimates to derive the elements of the BCF model: (i) a diffusion equation for the density of adsorbed adatoms; (ii) a step velocity law; and (iii) a linear relation for the mass flux of adatoms at the step. In this vein, we also provide a criterion by which the adatom flux remains linear in supersaturation, suggesting a range of non-equilibrium conditions under which the BCF model remains valid. Lastly, we make use of kinetic Monte Carlo simulations to numerically study effects that drive the system to higher supersaturations -e.g. deposition of material onto the surface from above. Using these results, we describe empirical corrections to the BCF model that amount to a nonlinear relation for the adatom flux at the step edge.

Below the roughening transition, crystal surfaces have macroscopic plateaus, facets, whose evolution is driven by the microscale dynamics of steps. A long-standing puzzle was how to reconcile discrete effects in facet motion with fully continuum approaches. We propose a resolution of this issue via connecting, through a jump condition, the continuum-scale surface chemical potential away from the facet, characterized by variations of the continuum surface free energy, with a chemical potential originating from the decay of atomic steps on top of the facet. The proposed condition accounts for step flow inside a discrete boundary layer near the facet. To validate this approach, we implement in a radial geometry a hybrid discrete-continuum scheme in which the continuum theory is coupled with only a few, minimally three, steps in diffusion-limited kinetics with conical initial data.

We investigate the scaling behavior for roughening and coarsening of mounds during unstable epitaxial growth. By using kinetic Monte Carlo (KMC) simulations of two lattice-gas models of crystal surfaces, we find scaling exponents that characterize roughening and coarsening at long times. Our simulation data show that these exponents have a complicated dependence on key model parameters that describe a step edge barrier and downward transport mechanisms. This behavior has not been fully described in previous works. In particular, we find that these scaling exponents vary continuously with parameters controlling the surface current. The kinetic processes of the KMC models that we employ include surface diffusion, edge diffusion, step-edge barriers, and also account for transient kinetics during deposition via downward funneling and transient mobility. Our extensive simulations make evident the salient interplay between step-edge barrier strength and transient kinetic processes.