Epitaxial Growth and Recovery: an Analytic Approach
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EPITAXIAL GROWTH AND RECOVERY: AN ANALYTIC APPROACH A. ZANGWILLt, C.N. LUSEt, D.D. VVEDENSKYt AND M.R. WILBYV tSchool of Physics, Georgia Institute of Technology, Atlanta, Georgia 30332 IThe Blackett Laboratory, Imperial College, London SW7 2BZ, UK
ABSTRACT Most detailed studies of morphological evolution during epitaxial growth and recovery make use of computer-based simulation techniques. In this paper, we discuss an alternative, analytic approach to this problem which takes explicit account of the atomistically random processes of deposition and surface diffusion. Beginning with a master equation representation of the dynamics of a solid-on-solid model of epitaxial growth, we derive a discrete, stochastic equation of motion for the surface profile. This Langevin equation is appropriate for growth studies. In particular, we are able to provide a microscopic justification for a non-linear continuum equation of motion proposed for this problem by others on the basis of heuristic arguments. During recovery, the deposition flux and its associated shot noise are absent. We analyze this process with a completely deterministic equation of motion obtained by performing a statistical average of the original stochastic equation. Results using the latter compare favorably with full Monte Carlo simulations of the original model for the case of the decay of sinusoidally modulated initial surfaces. INTRODUCTION The phrase epitazial architecture has been used [1] to describe the process of ultra-precise control of atomic kinetics at surfaces to produce structures with both aesthetic and functional appeal. A severe impediment to the achievement of this goal is the inevitable roughening of an initially flat surface during epitaxial growth. For the case of molecular-beam epitaxy (MBE), this kinetic roughening effect is clearly observed in the decay of the intensity of oscillations observed during reflection high energy electron diffraction (RHEED) measurements [2]. Under the most common experimental conditions, this occurs because statistical fluctuations both in the deposition flux and in the rates of nucleation and coalescence of two-dimensional islands leads to a loss of correlation between spatially separated regions of the surface. Ultimately, the growing surface spreads over many layers. In a few reported cases [3], a transition from layer growth to columnar growth occurs. Up to the present, essentially all theoretical analyses of surface morphological evolution during epitaxial growth have been based on microscopic computer-based methods. Rate-equation modelling has a distinguished history of application to deposition problems [4], but when applied to epitaxial growth onto singular surfaces [5], one is severly limited by the necessity to focus on the time evolution of layer-averaged quantities rather than upon the morphology of the growth front itself. In principle, molecular dynamics simulation [6] would be the method of choice. Unfortunately, realistic interatomic potential functions generally are lacking and, even when
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