#### Year of Publication

2011

#### Degree Name

Doctor of Philosophy (PhD)

#### Document Type

Dissertation

#### College

Arts and Sciences

#### Department

Mathematics

#### First Advisor

Dr. Michel E. Jabbour

#### Abstract

Vicinal surfaces consist of terraces separated by atomic steps. In the step-flow regime, deposited atoms (adatoms) diffuse on terraces, eventually reaching steps where they attach to the crystal, thereby causing the steps to move. There are two main objectives of this work. First, we analyze rigorously the differences in qualitative behavior between vicinal surfaces consisting of infinitely many steps and nanowires whose top surface consists of a small number of steps bounded by a reflecting wall. Second, we derive the continuum model that describes the macroscopic behavior of vicinal surfaces from detailed microscopic models of step dynamics.

We use the standard theory of Burton-Cabrera-Frank (BCF) to show that in the presence of an Ehrlich-Schwoebel barrier, i.e., a preferential attachment of adatoms from the lower terraces, *N*-periodic step motions are stable with respect to step collisions. Nonetheless, for *N* > 2 step collisions may occur. Moreover, we consider a single perturbed terrace, in which we distinguish three cases: no attachment from the upper terraces (perfect ES barrier), no attachment from the lower terraces (perfect inverse ES barrier), and symmetric attachment. For a perfect ES barrier, steps never collide regardless of the initial perturbation. In contrast, for a perfect inverse ES barrier, collisions occur for any nonzero perturbation. Finally, for symmetric attachment, step collisions occur for sufficiently large outward perturbations.

To model nanowire growth, we consider rectilinear steps and concentric steps bounded by reflecting walls. In contrast to a vicinal surface with infinitely many steps, we prove analytically that the Ehrlich-Schwoebel barrier is destabilizing with respect to step collisions. We further consider nanowire growth with desorption, and prove that the initial conditions that lead to step collisions are characterized by a unique step motion trajectory.

We take as our starting point a thermodynamically consistent (TC) generalization of the BCF model to derive PDE that govern the evolution of the vicinal surface at the macroscale. Whereas the BCF model yields a fourth-order parabolic equation for the surface height, the TC model yields a system of coupled equations for the surface height and the surface chemical potential.

#### Recommended Citation

Kirby, Nicholas O., "Modeling, Analysis, and Simulation of Discrete-Continuum Models of Step-Flow Epitaxy: Bunching Instabilities and Continuum Limits" (2011). *University of Kentucky Doctoral Dissertations*. 187.

http://uknowledge.uky.edu/gradschool_diss/187