From crystal steps to continuum laws: Behavior near large facets in one dimension

Margetis, Dionisios; Nakamura, Kanna

doi:10.1016/j.physd.2011.03.007

Cited by 15 publications

(14 citation statements)

References 37 publications

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“…To quantify this term, we assume that the total step repulsion energy can be written as a sum of pairwise contributions, V(W) > 0, for each nearest-neighbor pair of steps separated by a terrace of width W. Then, for a specific step with adjacent upper (lower) terrace of width W U (W L ), it is straightforward to show that μ rep = V′(W L ) -V′(W U ), where the prime represents the derivative with respect to terrace width [2]. It is believed that V(W) has an inverse square form for entropic repulsion with strength, g, so that [2,25] V(W) = g/W 2 and μ rep = 2g[(W L ) -3 -(W U ) -3 ].…”

Section: Appendix B: Effect Of Step-step Repulsionmentioning

confidence: 99%

Refined BCF-type boundary conditions for mesoscale surface step dynamics

2015

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Deposition on a vicinal surface with alternating rough and smooth steps is described by a solid-on-solid model with anisotropic interactions. Kinetic Monte Carlo (KMC) simulations of the model reveal step-pairing in the absence of any additional step attachment barriers. We explore the description of this behavior within an analytic BCFtype step dynamics treatment. Without attachment barriers, conventional kinetic coefficients for the rough and smooth steps are identical, as are the predicted step velocities for a vicinal surface with equal terrace widths. However, we determine refined kinetic coefficients from a two-dimensional discrete deposition-diffusion equation formalism which accounts for step structure. These coefficients are generally higher for rough steps than for smooth steps, reflecting a higher propensity for capture of diffusing terrace adatoms due to a higher kink density. Such refined coefficients also depend on the local environment of the step and can even become negative (corresponding to net detachment despite an excess adatom density) for a smooth step in close proximity to a rough step. Our key observation is that incorporation of these refined kinetic coefficients into a BCF-type step dynamics treatment recovers quantitatively the mesoscale steppairing behavior observed in the KMC simulations.

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Section: Appendix B: Effect Of Step-step Repulsionmentioning

confidence: 99%

Refined BCF-type boundary conditions for mesoscale surface step dynamics

2015

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“…The connection of step flow to continuum theories has been studied analytically for semi-infinite 1D facets at fixed heights in surface diffusion [1,2,36]; however, only the attachment-detachment limited (ADL) regime has been addressed rigorously [1,2]. In this setting, the surface height is a convenient independent variable by which there is no need to use a free boundary for the facet; furthermore, step collapses do not occur and thus the total number of steps is preserved.…”

Section: On Past Workmentioning

confidence: 99%

Discrete and Continuum Relaxation Dynamics of Faceted Crystal Surface in Evaporation Models

Nakamura¹,

Margetis²

2013

Multiscale Model. Simul.

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Abstract. We study linkages of two scales in the relaxation of an axisymmetric crystal with a facet in evaporation-condensation kinetics. The macroscale evolution is driven by the motion of concentric circular, repulsively interacting line defects (steps) which exchange atoms with the vapor. At the microscale, the step velocity is proportional to the variation of the total step free energy, leading to large systems of differential equations for the step radii. We focus on two step flow models. In one model (called M1) the discrete mobility is simply proportional to the upper-terrace width; in another model (M2) the mobility is altered by an extra geometric factor. By invoking self-similarity at long time, we numerically demonstrate that (i) in M1, discrete slopes follow closely a continuum thermodynamics approach with "natural boundary conditions" at the facet edge; (ii) in contrast, predictions of M2 deviate from results of the above continuum approach; and (iii) this discrepancy can be eliminated via a continuum boundary condition with a geometry-induced jump for top-step collapses. At the macroscale, both step models give rise to free-boundary problems for a second-order, parabolic partial differential equation, which we study via the subgradient formalism. We discuss the interpretation of the facet height as shock and prove convergence of the solution of each discrete scheme to the (weak) entropy solution of a conservation law if steps do not interact.

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“…Therefore the methods we used for the ADL case seem not to be applicable in the DL setting. (The recent paper by Nakamura and Margetis [17] examines the DL self-similar solution using an entirely different technique, but does not address its stability.) Our paper [1] considers only a monotone step train connecting two semi-infinite terraces as in Figure 1.…”

Section: Some Open Problemsmentioning

confidence: 99%

Surface Relaxation Below the Roughening Temperature: Some Recent Progress and Open Questions

Kohn

2012

Nonlinear Partial Differential Equations

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We discuss two recent projects concerning the evolution of a crystal surface below the roughening temperature. One addresses the evolution of a monotone one-dimensional step train (joint work with Hala Al Hajj Shehadeh and Jonathan Weare). The other addresses the finite-time flattening predicted by a fourth-order PDE model (joint work with Yoshikazu Giga). For each project we begin with a discussion of the mathematical model; then we summarize the recent results, the main ideas behind them, and some related open problems.

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From crystal steps to continuum laws: Behavior near large facets in one dimension

Cited by 15 publications

References 37 publications

Refined BCF-type boundary conditions for mesoscale surface step dynamics

Refined BCF-type boundary conditions for mesoscale surface step dynamics

Discrete and Continuum Relaxation Dynamics of Faceted Crystal Surface in Evaporation Models

Surface Relaxation Below the Roughening Temperature: Some Recent Progress and Open Questions

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