Stochastic Models of Epitaxial Growth
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Stochastic Models of Epitaxial Growth
Dionisios Margetis1, Paul N. Patrone2, and T. L. Einstein2 1 Department of Mathematics, and Institute for Physical Science and Technology, and Center for Scientific Computation and Mathematical Modeling, University of Maryland, College Park, MD 20742, U.S.A. 2
Department of Physics, University of Maryland, College Park, MD 20742, U.S.A.
ABSTRACT We study theoretical aspects of step fluctuations on vicinal surfaces by adding conservative white noise to the Burton-Cabrera-Frank model in one spatial dimension. We consider material deposition from above, as well as entropic and elastic-dipole step repulsions. Two approaches are discussed: (i) the linearization of stochastic equations when fluctuations are small, which captures correlations; and (ii) a mean field approach, which leaves out correlations but captures nonlinearities. Comparisons to kinetic Monte-Carlo simulations are presented.
INTRODUCTION The fluctuations of steps on surfaces of crystalline materials such as silicon have been the subject of active experimental interest. A goal is to determine dominant pathways of atomic mass transport by observing the terrace width probability density or distribution (TWD) [1, 2]. Developing a complete theory of stochastic effects on vicinal surfaces poses a challenge. Crucial, yet largely unresolved, issues include: (1) the derivation of noise models from microscopic principles, and (2) the rigorous description of statistics for terrace widths in large systems. An approach that partly circumvents issue (1) is to add ad hoc noise to the celebrated BurtonCabrera-Frank (BCF) model [3, 4] of step flow. In this vein, a systematic kinetic description of terrace-width fluctuations in terms of a mean field was proposed recently [5] on the basis of hierarchies for terrace correlation functions in one space dimension (1D). This formalism was recently applied to a large system of interacting steps in the absence of material deposition under the assumption that the noise obeys a second-order conservative scheme [6]. Here, we extend the formulation of [6] to the case with material deposition of flux F from above. In addition, we discuss possible physical implications that stem from the analysis, and illustrate limitations of approximations for the (intrinsically nonlinear) stochastic equations of motion. We show how the step interactions and deposition flux can jointly cause narrowing of the TWD, thus enriching the recent related work by Hamouda, Pimpinelli and Einstein [2] with step energetics.
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THEORY: MODELING AND ANALYSIS The step geometry is shown in figure 1. The j-th terrace is the region xj
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