The stability of growing or evaporating crystals

Ghez, R.; Cohen, Hirsh; Keller, Joseph B.

doi:10.1063/1.352928

Cited by 27 publications

(26 citation statements)

References 28 publications

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“…The adatom density solves the diffusion equation on each terrace with suitable boundary conditions for atom attachment-detachment at the step edges. A variety of such boundary conditions have been considered in the literature [21,9,16,27,45,4,2]. Here we focus on the simplest class of step models that is rich enough to include the effects of (i) step edge curvature; (ii) pairwise step interactions; and (iii) finite, kinetic rates of atom attachment-detachment at step edges.…”

Section: Bcf Approachmentioning

confidence: 99%

Continuum Relaxation of Interacting Steps on Crystal Surfaces in $2+1$ Dimensions

Margetis¹,

Kohn²

2006

Multiscale Model. Simul.

142

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Abstract. We study the relaxation of crystal surfaces in 2+1 dimensions via the motion of interacting atomic steps. The goal is the rigorous derivation of the continuum limit. The starting point is a discrete scheme from the Burton, Cabrera, and Frank model, which accounts for diffusion of point defects ("adatoms") on terraces and attachment-detachment of atoms at step edges. It is shown that the macroscopic adatom current involves a tensor mobility, which describes different fluxes in directions parallel and transverse to step edges, although the physics of each terrace is assumed isotropic. When the steps are everywhere parallel (straight or circular) the tensor character of the mobility is unimportant; in the general (2+1)-dimensional setting, however, it is crucial. Our methods consist of (i) the solution of the diffusion equation for adatoms via the separation of local space variables into "fast" and "slow," and (ii) the treatment of the step chemical potential for a wide class of step energies and repulsive interactions. Previous works using similar methods were mainly restricted to (1+1)-dimensional or axisymmetric geometries and entirely missed the tensor character of the mobility. The continuum limit of the step flow yields a fourth-order partial differential equation for the surface height profile.

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Section: Bcf Approachmentioning

confidence: 99%

Continuum Relaxation of Interacting Steps on Crystal Surfaces in $2+1$ Dimensions

Margetis¹,

Kohn²

2006

Multiscale Model. Simul.

142

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show abstract

“…Here, one should note the efforts of Ghez et al [1] and Keller et al [2] to clarify the stability of a step flow in the more complex situation when the surface diffusion of the adatoms is not assumed to be very fast. Their mathematical treatment is more sophisticated than the original Burton, Cabrera and Frank theory [3] in the sense that the quasi-static approximation is replaced by a Stefan-like problem for moving steps.…”

Section: Introductionmentioning

confidence: 99%

“…The results obtained in [2] show a train of fast moving steps to be unstable. The treatment in [1,2], however, neglects the step-step repulsion which is an important stabilizing factor.…”

Section: Introductionmentioning

confidence: 99%

Evaporation and growth of crystals: Propagation of step-density compression waves at vicinal surfaces

Ranguelov

Stoyanov

2007

Phys. Rev. B

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We studied the step dynamics during crystal sublimation and growth in the limit of fast surface diffusion and slow kinetics of atom attachmentdetachment at the steps. For this limit we formulate a model free of the quasi-static approximation in the calculation of the adatom concentration on the terraces at the crystal surface. Such a model provides a relatively simple way to study the linear stability of a step train in a presence of step-step repulsion and an absence of destabilizing factors (as Schwoebel effect, surface electromigration etc.). The central result is that a critical velocity of the steps in the train exists which separates the stability and instability regimes. Instability occurs when the step velocity exceeds its critical valuewhere K is the step kinetic coefficient, Ω is the area of one atomic site at the surface, and the energy of step-step repulsion iswhere l is the interstep distance. Integrating numerically the equations for the time evolution of the adatom concentrations on the terraces and the equations of step motion we obtained the step trajectories. When the step velocity exceeds its critical value the plot of these trajectories manifests clear space and time periodicity (step density compression waves propagate on the vicinal surface). This ordered motion of the steps is preceded by a relatively short transition period of disordered step dynamics.

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“…However, in some growth cases, these convection terms can be important to the growth stability [14,19]. In principle, they are necessary to obtain the conservation of mass in a region that includes a portion of the boundaries.…”

Section: Problem Descriptionmentioning

confidence: 99%

Finite element method for epitaxial growth with attachment–detachment kinetics

Bänsch

Haußer

Lakkis

et al. 2004

Journal of Computational Physics

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An adaptive finite element method is developed for a class of free or moving boundary problems modeling island dynamics in epitaxial growth. Such problems consist of an adatom (adsorbed atom) diffusion equation on terraces of different height; boundary conditions on terrace boundaries including the kinetic asymmetry in the adatom attachment and detachment; and the normal velocity law for the motion of such boundaries determined by a two-sided flux, together with the one-dimensional "surface" diffusion. The problem is solved using two independent meshes: a two-dimensional mesh for the adatom diffusion and a one-dimensional mesh for the boundary evolution. The diffusion equation is discretized by the first-order implicit scheme in time and the linear finite element method in space. A technique of extension is used to avoid the complexity in the spatial discretization near boundaries. All the elements are marked, and the marking is updated in each time step, to trace the terrace height. The evolution of the terrace boundaries includes both the mean curvature flow and the surface diffusion. Its governing equation is solved by a semi-implicit front-tracking method using parametric finite elements. Simple adaptive techniques are employed in solving the adatom diffusion as well as the boundary motion problem. Numerical tests on pure geometrical motion, mass balance, and the stability of a growing circular island demonstrate that the method is stable, efficient, and accurate enough to simulate the growing of epitaxial islands over a sufficiently long time period.

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The stability of growing or evaporating crystals

Cited by 27 publications

References 28 publications

Continuum Relaxation of Interacting Steps on Crystal Surfaces in $2+1$ Dimensions

Continuum Relaxation of Interacting Steps on Crystal Surfaces in $2+1$ Dimensions

Evaporation and growth of crystals: Propagation of step-density compression waves at vicinal surfaces

Finite element method for epitaxial growth with attachment–detachment kinetics

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