Nonlinear Dynamics of Surface Steps

Krug, Joachim

doi:10.1002/9783527629374.ch5

Cited by 6 publications

(5 citation statements)

References 65 publications

(133 reference statements)

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“…This perspective has been enriched with kinetic and energetic effects; see, e.g., Refs. [18][19][20][21][22][23]. The main idea is to view steps as boundaries moving by mass conservation.…”

Section: Background: Bcf Modelmentioning

confidence: 99%

One-dimensional model of interacting-step fluctuations on vicinal surfaces: Analytical formulas and kinetic Monte Carlo simulations

2010

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We study analytically and numerically a one-dimensional model of interacting line defects (steps) fluctuating on a vicinal crystal. Our goal is to formulate and validate analytical techniques for approximately solving systems of coupled nonlinear stochastic differential equations (SDEs) governing fluctuations in surface motion. In our analytical approach, the starting point is the Burton-Cabrera-Frank (BCF) model by which step motion is driven by diffusion of adsorbed atoms on terraces and atom attachment-detachment at steps. The step energy accounts for entropic and nearest-neighbor elastic-dipole interactions. By including Gaussian white noise to the equations of motion for terrace widths, we formulate large systems of SDEs under different choices of diffusion coefficients for the noise. We simplify this description via (i) perturbation theory and linearization of the step interactions and, alternatively, (ii) a mean-field (MF) approximation whereby widths of adjacent terraces are replaced by a self-consistent field but nonlinearities in step interactions are retained. We derive simplified formulas for the time-dependent terrace-width distribution (TWD) and its steady-state limit. Our MF analytical predictions for the TWD compare favorably with kinetic Monte Carlo simulations under the addition of a suitably conservative white noise in the BCF equations.

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“…This perspective has been enriched with kinetic and energetic effects; see, e.g., Refs. [18][19][20][21][22][23]. The main idea is to view steps as boundaries moving by mass conservation.…”

Section: Background: Bcf Modelmentioning

confidence: 99%

One-dimensional model of interacting-step fluctuations on vicinal surfaces: Analytical formulas and kinetic Monte Carlo simulations

2010

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“…It was hypothesized soon after based on a simple Burton–Cabrera–Frank (BCF) type of model that it is the directional asymmetry (bias) in the diffusion of the charged surface atoms (adatoms) that makes the motion of the steps from the vicinal surface unstable. − As a result, the regular step distribution of the vicinal surface is broken and a surface that is a sequence of groups of steps (bunches) separated by large terraces almost free of steps, hills and valleys structure, results. In the next years the phenomenon of SB was investigated actively by experimental techniques − analyzed theoretically, − and by numerical simulations − It becomes clear only quite recently that SB can be caused at a given temperature by both current directions across the steps but on a metal surfacethat of tungsten, W(110) . Besides semiconductor and metal crystal surfaces also the insulator ones exhibit the same, adatom electromigration induced, type of behavior .…”

Section: Introductionmentioning

confidence: 99%

Scaling and Dynamic Stability of Model Vicinal Surfaces

Krzyżewski

Załuska‐Kotur

Krasteva

et al. 2018

Crystal Growth & Design

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We propose an integrated modeling approach to the fundamental problem of vicinal crystal surfaces destabilized by step-down (SD) and step-up (SU) currents with focus on both the initial and the intermediate stages of the process. We reproduce and analyze quantitatively the step bunching (SB) instability, caused by the two opposite drift directions in the two situations of step motion mediating sublimation and growth. For this reason we develop further our atomistic scale model (vicCA) of vicinal crystal growth (Gr) destabilized by SD drift of the adatoms in order to account for also the vicinal crystal sublimation (Sbl) and the SU drift of the adatoms as an alternative mode of destabilization. For each of the four possible casesGr + SD, Gr + SU, Sbl + SD, Sbl + SU, we find a self-similar solutionthe time-scaling of the number of steps in the bunch N, = N T 2 /3, where T is the time, rescaled with a combination of model parameters. In order to study systematically the emergence of the instability, we use N further as a measure and probe the model's stability against SB on a dense grid of points in the parameter space. Stability diagrams are obtained, based on simulations running to fixed moderate rescaled times and with small-size systems. We confirm the value of the numerical prefactor in the time scaling of N, 2/ 3 by results obtained from systems of ordinary differential equations for the step velocity that contain, in contrast to vicCA, step−step repulsions. This last part of our study provides also the possibility to distinguish between diffusion-limited and kinetic-limited versions of the step bunching phenomenon.

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“…By varying the sample temperature the crystal can start to sublimate or grow in the step flow mode [1][2][3][4]. In the presence of an asymmetry in the kinetics at the steps bordering a terrace, perturbations in the step flow may grow such that the step profile undergoes a step bunching instability.…”

Section: Introductionmentioning

confidence: 99%

Anticoarsening and complex dynamics of step bunches on vicinal surfaces during sublimation

2010

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A sublimating vicinal crystal surface can undergo a step bunching instability when the attachment-detachment kinetics is asymmetric, in the sense of a normal Ehrlich-Schwoebel effect. Here we investigate this instability in a model that takes into account the subtle interplay between sublimation and step-step interactions, which breaks the volume-conserving character of the dynamics assumed in previous work. On the basis of a systematically derived continuum equation for the surface profile, we argue that the non-conservative terms pose a limitation on the size of emerging step bunches. This conclusion is supported by extensive simulations of the discrete step dynamics, which show breakup of large bunches into smaller ones as well as arrested coarsening and periodic oscillations between states with different numbers of bunches.

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Nonlinear Dynamics of Surface Steps

Abstract: A nonconserved situation where the step exchanges atoms with the terrace is treated below in Sect.1.2.4.

Cited by 6 publications

References 65 publications

One-dimensional model of interacting-step fluctuations on vicinal surfaces: Analytical formulas and kinetic Monte Carlo simulations

One-dimensional model of interacting-step fluctuations on vicinal surfaces: Analytical formulas and kinetic Monte Carlo simulations

Scaling and Dynamic Stability of Model Vicinal Surfaces

Anticoarsening and complex dynamics of step bunches on vicinal surfaces during sublimation

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