Introduction to Step Dynamics and Step Instabilities

Abstract

This paper provides an elementary introduction to the basic concepts used in describing epitaxial crystal growth in terms of the thermodynamics and kinetics of atomic steps. Selected applications to morphological instabilities of stepped surfaces are reviewed, and some open problems are outlined. (2000). Primary 80A22; Secondary 35R35. Mathematics Subject Classification

Nonlinear Dynamics of Surface Steps

“…The exchange of atoms between the step and the surrounding terraces is another source of nonlocality in the motion of the steps, since it necessitates the solution of a moving boundary value problem for the concentration of adatoms on the terraces [2,8]. A particular case in this class of problems is the interior model for the electromigration of vacancy islands introduced in [12], and studied in detail in [16,24].…”

“…with appropriate boundary conditions at the step edge (see [8] for a general discussion). If the exchange of atoms with the step edge is rapid, so that thermal equilibrium is maintained at the boundary at all times, a circular stationary solution drifting at constant speed against the force direction can be found [12].…”

“…The equations of motion for the steps can be obtained from the solution of a one-dimensional moving boundary value problem for the adatom concentration on the terraces. This procedure has been reviewed in detail elsewhere [8]. Here we start the discussion directly from the nonlinear equations of motion, regarded as a (physically motivated) many-dimensional dynamical system.…”

“…Comparison with (1.9) shows that f ± are linear functions with slopes γ ± , such that the stability condition reads γ + > γ − . In addition to the linear terms depending on the nearest neighbor step positions, (1.14) contains nonlinear next-nearest-neighbor contributions arising from repulsive thermodynamic step-step interactions of entropic and elastic origin [3,8], which drive the relaxation of the step train to its (equidistant) equilibrium shape. The sublimation rate for the model (1.14) is R = (γ + + γ − )l, hence for strongly conserved dynamics one has to set γ + = −γ − .…”

“…The analysis of the nonlinear dynamics of step bunches is greatly simplified if it is possible to perform a continuum limit of the problem, thus passing from the discrete dynamical system (1.14) to a partial differential equation [8,47]. Coarse graining the discrete equations of motion (1.14), one arrives first at a "Lagrangian" continuum description for the step positions x i or the terrace sizes l i = x i+1 − x i by converting the layer index i into a continuous surface height h = ih 0 (here h 0 denotes the height of an elementary step) [43,48].…”