The shape evolution of two-dimensional islands through periphery diffusion biased by an electromigration force is studied numerically using a continuum approach. We show that the introduction of crystal anisotropy in the mobility of edge atoms induces a rich variety of migration modes, which include oscillatory and irregular behavior. A phase diagram in the plane of anisotropy strength and island size is constructed. The oscillatory motion can be understood in terms of stable facets which develop on one side of the island and which the island then slides past. The facet orientations are determined analytically.PACS numbers: 66.30.Qa, The manipulation of nanostructures by macroscopic forces is likely to become a key ingredient in many nanotechnology applications. Understanding the influence of external fields on the shape evolution of nanoscale surface features is therefore of considerable importance. As a first step in this direction we analyze here the effects of an electric current on single-layer islands on a crystalline surface. The islands evolve under surface electromigration, the directed motion of adsorbed atoms due to the slight force transmitted by collisions with the conduction electrons in the sample [1].Electromigration along interfaces and grain boundaries is the most persistent and menacing reliability problem in integrated circuit technology [2,3]. Correspondingly, much work has been devoted to electromigration-induced void formation and breakdown in metallic conductor lines [3], and the capacity for quantitative numerical modeling has been demonstrated at least for simple void geometries [4,5]. A major obstacle to achieving predictive power in such studies, however, is the insufficient control over the complex internal structure of the polycrystalline samples. Hence an important motivation for investigating electromigration-induced effects on simple, well-controlled nanoscale morphologies, such as step patterns on vicinal surfaces [6] and single layer islands [7], is to bridge the gap between the microscopic mechanisms of electromigration and their consequences on technologically relevant length and time scales.Electromigration of islands has been modeled previously using Monte Carlo simulations [8] and continuum theory [9]. The continuum approach to island shape evolution, which treats the island edge as a smooth curve, has been successfully applied to a range of problems including the diffusion [10] and sintering [11] of islands, and the pinch-off of vacancy clusters [12]. Here we focus on the regime of periphery diffusion (PD), where the dominant kinetic process is the migration of atoms along the island boundary. The shape then follows a local evolution law, without coupling to the adatom concentration on the surrounding terrace.We extend the model of [9] by including crystal anisotropy in the adatom mobility. It was observed recently in the context of step flow growth [13] that crystalline anisotropy can change the behavior of step patterns in a qualitative way. In the present case, it lea...
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.
Ostwald ripening in homoepitaxy in the submonolayer regime is studied by means of numerical simulations based on a step flow model, which accounts for attachment-detachment kinetics at the island boundaries. Diffusion-limited ripening and the crossover to attachment-limited ripening is investigated. The simulations indicate that the coarsening kinetics of the average island radius is described by a t a power law, where 1/3ഛ a ഛ 1/2. Here a takes the value 1 / 3, if the ripening is purely diffusion-limited ͑infinite attachment rate at the island boundaries͒, and increases with decreasing attachment rate, approaching the value a =1/2 if the ripening becomes attachment-limited. For the diffusion-limited regime, the numerical simulations are shown to correspond with the predictions of the mean-field theory proposed by Yao et al., for both the scaling behavior of the average island size as well as the island size distribution in the late stage. Approaching the attachmentlimited regime, the numerical results meet the predictions of the classical mean-field theory of Lifshitz, Slyozov, and Wagner for attachment-limited ripening. We also analyze the influence of anisotropic edge energies and edge diffusion.
Grain size increases when crystals respond to dynamic equilibrium in a saturated solution. The pathway to coarsening is generally thought to be driven by Ostwald ripening, that is, simultaneous dissolution and reprecipitation, but models to describe Ostwald ripening neglect solid−solid interactions and crystal shapes. Grain coarsening of calcite, CaCO 3 , is relevant for biomineralization and commercial products and is an important process in diagenesis of sediments to rock during geological time. We investigated coarsening of pure, synthetic calcite powder of sub-micrometer diameter crystals and aged it in saturated solutions at 23, 100, and 200°C for up to 261 days. Scanning electron microscopy (SEM) and Brunauer−Emmett−Teller (BET) surface area analysis showed rapid coarsening at 100 and 200°C. Evidence of particle growth at 23°C was not visible by SEM, but high resolution X-ray diffraction (XRD) data demonstrated steady growth of nanometer crystallites. The results can be described by theory where grains coarsen preferentially by aggregation at early times and high temperatures and by Ostwald ripening at later stages. Crystal form and dimension are influenced by the transition from one growth mechanism to the other. This has been poorly described by mean field coarsening models and offers predictive power to grain coarsening models.
We study anisotropic surface diffusion of curves with a small corner energy regularization. The regularization allows the use of nonconvex free energy densities and turns the evolution law into a 6th order geometric equation. Using a semi-implicit time discretization, we present a variational formulation of this equation for parametric curves, leading to a discretization based on linear finite elements. The resulting linear system is shown to be uniquely solvable. Numerical examples include the convergence of closed curves to the Wulff shape and the evolution of a thermodynamically unstable surface into a hill-valley structure and its subsequent coarsening.
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