Decay of one-dimensional surface modulations

Israeli, Navot; Kandel, Daniel

doi:10.1103/physrevb.62.13707

Cited by 48 publications

(62 citation statements)

References 26 publications

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“…The assumption that every surface feature is composed of many steps clearly breaks down on macroscopic facets where there are no steps at all. Thus, existing continuum models fail near singular regions.A possible resolution of this problem is to solve the continuum model in non-singular regions and use matching conditions at the singular facet edges [11][12][13][14]. This is a good solution in extremely simple cases (see, e.g.…”

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confidence: 99%

Novel Continuum Modeling of Crystal Surface Evolution

Israeli

Kandel

2002

Phys. Rev. Lett.

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We propose a novel approach to continuum modeling of the dynamics of crystal surfaces. Our model follows the evolution of an ensemble of step configurations, which are consistent with the macroscopic surface profile. Contrary to the usual approach where the continuum limit is achieved when typical surface features consist of many steps, our continuum limit is approached when the number of step configurations of the ensemble is very large. The model can handle singular surface structures such as corners and facets. It has a clear computational advantage over discrete models.PACS numbers: 68.55.-a, 68.35.JaThe behavior of classical physical systems is typically described in terms of equations of motion for discrete microscopic objects (e.g. atoms). In many cases, the behavior of such systems is smooth when observed on macroscopic length and time scales. It is useful to describe these systems in terms of continuum, coarsegrained models, which treat the dynamics of the macroscopic, smoothly varying, degrees of freedom rather than the microscopic ones. Such models are more amenable to analytical treatments and have enormous computational advantages over their discrete counterparts.Many physical systems exhibit a macroscopically smooth behavior everywhere, except in small regions of space where their behavior is singular. Examples are dislocations in a crystal, cracks in crystalline material, facet edges on crystal surfaces, etc. These singular regions are of interest because they frequently drive the dynamics of the whole system. It is an interesting and important challenge to develop continuum descriptions of such systems, since standard phenomenological continuum models completely fail in the singular regions.In this Letter we address the above problem in the context of the dynamics of crystal surfaces. Below the roughening temperature the evolution of these surfaces proceeds by the nucleation, flow and annihilation of atomic steps. These steps originate either from a miscut of the initial surface with respect to a high symmetry plane of the crystal or as the boundaries of islands and vacancies which nucleate on the surface during morphological evolution (we ignore screw dislocations as sources of steps). The steps are separated by terraces, which are parallel to a high symmetry crystal orientation. Each step consists of atomic kinks separated by straight portions along closed-packed orientations. However, on length scales larger than the typical distance between kinks the step is a smooth line. Thus the evolution of the surface corresponds to the motion of discrete smooth lines.One can model step motion by solving the diffusion problem of adatoms on the terraces with boundary conditions at step edges. These conditions account for attachment and detachment of atoms to and from step edges. The local flux of adatoms at step edges and the resulting step velocities can then be calculated. This approach was introduced long ago by Burton, Cabrera and Frank [1], and was further developed by other authors [2]. Given the ar...

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confidence: 99%

Novel Continuum Modeling of Crystal Surface Evolution

Israeli

Kandel

2002

Phys. Rev. Lett.

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“…This is not true in Ref. 17, for example: there, the equations of motion for steps 1, 2, N − 1, and N are different due to step-antistep annihilations occurring at the peaks and valleys of 1D periodic corrugations.…”

Section: A Straight Stepsmentioning

confidence: 90%

“…We take q͑y͒ = ͚ m=0 c m ͑y +1͒ m/2 to be the form of q near the facet edge. 17 ͑The coefficients c m here should not be confused with the terrace adatom densities, c i , used in the main text.͒ In our numerical procedure, we used = 12 and this value gave sufficient accuracy in order to compare with step simulation data. Noting Eq.…”

Section: ͑B1͒mentioning

confidence: 99%

“…Let h 1 be the ͑constant͒ height of the facet to the left of w 1 ; hence, h͑x , t͒ = h 1 for x Յ w 1 . Then the first boundary condition follows from enforcing continuity of slope 17,18 across the facet edge. Note that a discontinuous slope, shown in Fig.…”

Section: A One-dimensional Casementioning

confidence: 99%

“…By balancing the flux of adatoms in and out of each step edge, one can write down equations of motion to describe the morphological evolution of the vicinal surface. Two commonly studied step geometries are the straight-step model 16,17 and the axisymmetric model 8,9,18 ͑see Fig. 1͒.…”

Section: Introductionmentioning

confidence: 99%

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Facet evolution on supported nanostructures: Effect of finite height

2008

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The surface of a nanostructure relaxing on a substrate consists of a finite number of interacting steps and often involves the expansion of facets. Prior theoretical studies of facet evolution have focused on models with an infinite number of steps, which neglect edge effects caused by the presence of the substrate. By considering diffusion of adsorbed atoms ͑adatoms͒ on terraces and attachment-detachment of atoms at steps, we show that these edge or finite height effects play an important role in the structure's macroscopic evolution. We assume diffusion-limited kinetics for adatoms and a homoepitaxial substrate. Specifically, using data from step simulations and a continuum theory, we demonstrate a switch in the time behavior of geometric quantities associated with facets: the facet edge position in a straight-step system and the facet radius of an axisymmetric structure. Our analysis and numerical simulations focus on two corresponding model systems where steps repel each other through entropic and elastic dipolar interactions. The first model is a vicinal surface consisting of a finite number of straight steps; for an initially uniform step train, the slope of the surface evolves symmetrically about the centerline, i.e., the middle step when the number of steps is odd. The second model is an axisymmetric structure consisting of a finite number of circular steps; in this case, we include curvature effects which cause steps to collapse under the effect of line tension. In the first case, we show that the position of the facet edge, measured from the centerline, switches from O͑t 1/4 ͒ behavior to O͑t 1/5 ͒ ͑where t is time͒. In the second case, the facet radius switches from O͑t 1/4 ͒ to O͑t͒. For the axisymmetric case, we also predict analytically through a continuum shock wave theory how the individual collapse times are modified by the effects of finite height under the assumption that step interactions are weak compared to the step line tension.

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Configurational Continuum Modelling of Crystalline Surface Evolution

Israeli¹,

Kandel²

Multiscale Modeling in Epitaxial Growth

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We propose a novel approach to continuum modelling of dynamics of crystal surfaces. Our model follows the evolution of an ensemble of step configurations, which are consistent with the macroscopic surface profile. Contrary to the usual approach where the continuum limit is achieved when typical surface features consist of many steps, our continuum limit is approached when the number of step configurations of the ensemble is very large. The model is capable of handling singular surface structures such as corners and facets and has a clear computational advantage over discrete models.

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Decay of one-dimensional surface modulations

Cited by 48 publications

References 26 publications

Novel Continuum Modeling of Crystal Surface Evolution

Novel Continuum Modeling of Crystal Surface Evolution

Facet evolution on supported nanostructures: Effect of finite height

Configurational Continuum Modelling of Crystalline Surface Evolution

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