113 research outputs found
Introduction to Step Dynamics and Step Instabilities
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
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
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
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
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
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
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
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
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
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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