Mound formation and coarsening from a nonlinear instability in surface growth

TL;DR

This paper investigates nonlinear instabilities leading to mound formation and coarsening in one-dimensional surface growth models, revealing a phase transition and differences between continuum and discrete models.

Contribution

It introduces a nonlinear instability mechanism for mound formation without the Ehrlich-Schwoebel barrier and explores phase transitions and coarsening dynamics in surface growth models.

Findings

Mound formation occurs due to nonlinear instability, not linear step-edge barriers.

A first-order phase transition separates rough and mounded phases.

Coarsening exponents differ between continuum and discrete models.

Abstract

We study a class of one-dimensional, nonequilibrium, conserved growth equations for both nonconserved and conserved noise statistics using numerical integration. An atomistic version of these growth equations is also studied using stochastic simulation. The models with nonconserved noise statistics are found to exhibit mound formation and power-law coarsening with slope selection for a range of values of the model parameters. Unlike previously proposed models of mound formation, the Ehrlich-Schwoebel step-edge barrier, usually modeled as a linear instability in growth equations, is absent in our models. Mound formation in our models occurs due to a nonlinear instability in which the height (depth) of spontaneously generated pillars (grooves) increases rapidly if the initial height (depth) is sufficiently large. When this instability is controlled by the introduction of an infinite number of higher-order gradient nonlinearities, the system exhibits a first-order dynamical phase transition from a rough self-affine phase to a mounded one as the value of the parameter that measures the effectiveness of control is decreased. We define a new ``order parameter'' that may be used to distinguish between these two phases. In the mounded phase, the system exhibits power-law coarsening of the mounds in which a selected slope is retained at all times. The coarsening exponents for the continuum equation and the discrete model are found to be different. An explanation of this difference is proposed and verified by simulations. In the growth equation with conserved noise, we find the curious result that the kinetically rough and mounded phases are both locally stable in a region of parameter space. In this region, the initial configuration of the system determines its steady-state behavior.