Atomistic Simulation and Elastic Theory of Surface Steps

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ABSTRACT Atomistic computer simulations and anisotropic elastic theory are employed to determine the elastic fields of surface steps and vicinal surfaces. The displacement field of and interaction energies between steps on an {001) Ni surface are determined using atomistic simulations and EAM potentials. The step-step interaction energy found from the simulations is consistent with a surface line force dipole elastic model of a step. We derive an anisotropic form for the elastic field associated with a surface line force dipole using a two dimensional surface Green tensor for a cubic elastic half-space. Both the displacement fields and step-step interaction energy predicted by the theory are shown to be in excellent agreement with the simulations. I.

INTRODUCTION

The elastic fields of steps on surfaces play a key role in a number of important surface phenomena, including: interactions between surface steps, epitaxial growth, surface reconstruction and crystal shape. The interaction between surface defects is caused by the elastic lattice distortions which can extend far into the bulk of the crystal. Unlike for topological defects, such as dislocations, where the Burger's vector alone determines the displacement field, there is no topological relationship that uniquely specifies the elastic field of a step in terms of its magnitude and orientation. Marchenko and Parshin [1] were the first to suggest that the elastic fields of surface steps can be described in terms of a traction distribution upon a flat surface. They argued that the source of these tractions is associated with the surface stress. The discontinuity of the surface at a surface stress sets up a force moment along the surface step of magnitude proportional to the surface stress. Marchenko and Parshin further argued that this force moment must be compensated by a surface force dipole oriented perpendicular to the surface along the line of the surface step. They

estimated the magnitude of this vertical force dipole to be the product of the surface stress t and the step height b. Working within isotropic elasticity, Marchenko and Parshin predicted that the interaction energy between two identical steps (per unit length of the step) is: Eint

2(l - 1v)2+(,rb)2 + ] 12 nE

(1)

do

where tb is the magnitude of the vertical force dipole and ý is the magnitude of the force dipole in the lateral direction (providing one exists), do is the separation between two steps, and E and v are the Young's modulus and Poisson ratio of the crystal, respectively. This interaction is repulsive between like steps and decays quadratically with the reciprocal step separation. Several additional attempts have been made to study step-step interactions theoretically [2,3,4] in order to understand the energetics of vicinal surfaces in terms of a periodic array of identical steps on flat high-symmetry surfaces. Atomistic simulations of vicinal surface energetics have been performed using different descriptions of atomic interactions [3,4]. These simulations show that the interacti