113 research outputs found

    Introduction to Step Dynamics and Step Instabilities

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    This paper provides an elementary introduction to the basic concepts used in describing epitaxial crystal growth in terms of the thermodynamics and kinetics of atomic steps. Selected applications to morphological instabilities of stepped surfaces are reviewed, and some open problems are outlined.Comment: To appear in the Proceedings of the Oberwolfach workshop on Multiscale Modeling in Epitaxial Growt

    Local electromigration model for crystal surfaces

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    We analyze the dynamics of crystal surfaces in the presence of electromigration. From a phase field model with a migration force which depends on the local geometry, we derive a step model with additional contributions in the kinetic boundary conditions. These contributions trigger various surface instabilities, such as step meandering, bunching and pairing on vicinal surfaces. Experiments are discussed

    Anisotropic diffusion in continuum relaxation of stepped crystal surfaces

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    We study the continuum limit in 2+1 dimensions of nanoscale anisotropic diffusion processes on crystal surfaces relaxing to become flat below roughening. Our main result is a continuum law for the surface flux in terms of a new continuum-scale tensor mobility. The starting point is the Burton, Cabrera and Frank (BCF) theory, which offers a discrete scheme for atomic steps whose motion drives surface evolution. Our derivation is based on the separation of local space variables into fast and slow. The model includes: (i) anisotropic diffusion of adsorbed atoms (adatoms) on terraces separating steps; (ii) diffusion of atoms along step edges; and (iii) attachment-detachment of atoms at step edges. We derive a parabolic fourth-order, fully nonlinear partial differential equation (PDE) for the continuum surface height profile. An ingredient of this PDE is the surface mobility for the adatom flux, which is a nontrivial extension of the tensor mobility for isotropic terrace diffusion derived previously by Margetis and Kohn. Approximate, separable solutions of the PDE are discussed.Comment: 14 pages, 1 figur

    Vicinal silicon surfaces: from step density wave to faceting

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    This paper investigates faceting mechanisms induced by electromigration in the regime where atomic steps are transparent. For this purpose we study several vicinal orientations by means of in-situ (optical diffraction, electronic microscopy) as well as ex-situ (AFM, microprofilometry) visualization techniques. The data show that faceting proceeds in two stages. The first stage is short and leads to the appearance of a step density wave, with a wavelength roughly independent of the surface orientation. The second stage is much slower, and leads to the formation of a hill-and-valley structure, the period of which depends on the initial surface orientation. A simple continuum model enables us to point out why the wavelength of the step density wave does not depend on the microscale details of the surface. The final wavelength is controlled by the competition between elastic step-step interaction and facet edge energy cost. Finally, the surface stress angular dependence is shown to emerge as a coarsed-grained picture from the step model.Comment: 26 pages, 9 figure

    Analysis of a combined influence of substrate wetting and surface electromigration on a thin film stability and dynamical morphologies

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    A PDE-based model combining surface electromigration and wetting is developed for the analysis of morphological stability of ultrathin solid films. Adatom mobility is assumed anisotropic, and two directions of the electric field (parallel and perpendicular to the surface) are discussed and contrasted. Linear stability analyses of small-slope evolution equations are performed, followed by computations of fully nonlinear parametric evolution equations that permit surface overhangs. The results reveal parameter domains of instability for wetting and non-wetting films and variable electric field strength, nonlinear steady-state solutions in certain cases, and interesting coarsening behavior for strongly wetting films.Comment: Submitted to the special issue "Nanoscale wetting of solids on solids" of the journal Comptes Rendus Physique (Olivier Pierre-Louis, Univ. Lyon, Editor

    Analytic Formulas for the Orientation Dependence of Step Stiffness and Line Tension: Key Ingredients for Numerical Modeling

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    We present explicit analytic, twice-differentiable expressions for the temperature-dependent anisotropic step line tension and step stiffness for the two principal surfaces of face-centered-cubic crystals, the square {001} and the hexagonal {111}. These expressions improve on simple expressions that are valid only for low temperatures and away from singular orientations. They are well suited for implementation into numerical methods such as finite-element simulation of step evolution.Comment: 10 pages; reformatted with revtex (with typos corrected) from version accepted by SIAM--Multiscale Modeling and Simulation on Nov. 21, 2006; greatly expanded introduction, several minor fixes (mostly stylistic

    Measuring the surface stress polar dependence

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    While measurements of the polar dependence of the surface free energy are easily available, measurements of the whole polar dependence of the surface stress of a crystal do not exist. In this paper is presented a new procedure that allows, for the first time, the experimental determination of the surface stress polar dependence of a crystal. For this purpose (1) electromigration is used to control the kinetic faceting of surface orientations that belong to the equilibrium shape of the crystal and (2) for each destabilised surface, the period of faceting as well as the crystallographic angles of the appearing facets are measured by AFM. The so-obtained data lead to a set of equations whose mathematical solution, compatible with physical constraints, gives access to the surface stress polar dependence of the whole crystal and thus to a better understanding of surface stress properties.Comment: 8 pages, 6 Figure

    Electromigration-driven Evolution of the Surface Morphology and Composition for a Bi-Component Solid Film

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    A two PDEs-based model is developed for studies of a morphological and compositional evolution of a thermodynamically stable alloy surface in a strong electric field, assuming different and anisotropic diffusional mobilities of the two atomic components. The linear stability analysis of a planar surface and the computations of morphology coarsening are performed. It is shown that the conditions for instability and the characteristic wavelength and growth rate differ from their counterparts in a single-component film. Computational parametric analyses reveal the sensitivity of the scaling exponents to the electric field strength and to the magnitude of the anisotropies difference

    Dynamics of Steps on Vicinal Surfaces

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    This theoretic work deals with the dynamics of steps on vicinal surfaces, where the bunching instability on the vicinal Si(111) is the physically relevant phenomenology. Thereby two models are studied in detail. The first one is the standard Burton- Cabrera-Frank model from 1951 with the quasi-static approximation for the adatom concentrations on the terraces, considered for non-permeable steps and in the limiting case of fast attachment/detachment kinetics and slow diffusion. In our derivation of the discrete equations we take into account higher order non-linear terms, neglected in the previous studies. We found, that those terms are present in the case of sublimation, but not in the case of growth. Analytical and numerical methods are employed in order to study the impact of these terms on the step dynamics. For both asymmetry effects, the Ehrlich-Schwoebel effect and the effect of electromigration, there is a change in the dispersion relation obtained from the linear stability analysis, whereas there is no such change in the case of growth. Due to the non-linear terms, the dynamics changes from conservative to non-conservative with respect to the crystal volume. The continuum limits of the discrete equations for both asymmetry cases yield a hint of slope selection in the so called mechanical analog of the partial differential equation. As a consequence, the scaling relations of the bunching geometry in the case of sublimation differ strongly from those in the case of growth. The numerical simulations of the discrete equations confirm these analytic results. In the non-linear regime there is anti-coarsening or arrested coarsening of the step bunches and thus there are stationary solutions with bounded maximal slope. A sensitive dependence on the initial conditions is observed. The second model we analyze was recently introduced by Ranguelov and Stoyanov. It accounts for the case of strong transparency, fast diffusion and slow attachment/detachment kinetics. This model goes beyond the approximation of quasi-static concentration profiles of adatoms. Calculations in order to reproduce Ranguelov and Stoyanov’s results for the gradient of the adatom concentration, depending on the electromigration force as well as for the linear stability analysis were carried out. Quantitative deviations were found and the corrections are presented. Finally, the equations are simulated and the dependence of the maximal slope on the different input parameters in the bunching regime is illustrated

    Single-phase-field model of stepped surfaces

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    8 pages, 5 figures.-- PACS nrs.: 81.10.Aj, 68.35.Ct, 81.15.Hi, 81.15.Aa.We formulate a phase-field description of step dynamics on vicinal surfaces that makes use of a single dynamical field, at variance with previous analogous works in which two coupled fields are employed, namely, a phase-field proper plus the physical adatom concentration. Within an asymptotic sharp interface limit, our formulation is shown to retrieve the standard Burton-Cabrera-Frank model in the general case of asymmetric attachment coefficients (Ehrlich-Schwoebel effect). We confirm our analytical results by means of numerical simulations of our phase-field model. Our present formulation seems particularly well adapted to generalization when additional physical fields are required.The present work has been partially supported by MEC (Spain) Grants No. FIS2006-12253-C06-01, No. FIS2006-12253-C06-05, and No. FIS2006-12253-C06-06, UC3M/CAM (Spain) Grant No. UC3M-FI-05-007, and CAM (Spain) Grant No. S-0505/ESP-0158.Publicad
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