Anisotropic dynamics of a vicinal surface under the meandering step instability

We investigate the nonlinear evolution of the Bales-Zangwill instability, responsible for the meandering of atomic steps on a growing vicinal surface. We develop an asymptotic method to derive, in the continuous limit, an evolution equation for the two-dimensional step flow. The dynamics of the crystal surface is greatly influenced by the anisotropy inherent to its geometry, and is characterized by the coarsening of undulations along the step direction and by the elastic relaxation in the mean slope direction. We demonstrate, using similarity arguments, that the coalescence of meanders and the step flow follow simple scaling laws, and deduce the exponents of the characteristic length scales and height amplitude. The relevance of these results to experiments is discussed.Comment: 10 pages, 7 figures; submitted to Phys. Rev.

Advances in the theory of III-V Nanowire Growth Dynamics

Nanowire (NW) crystal growth via the vapour_liquid_solid mechanism is a complex dynamic process involving interactions between many atoms of various thermodynamic states. With increasing speed over the last few decades many works have reported on various aspects of the growth mechanisms, both experimentally and theoretically. We will here propose a general continuum formalism for growth kinetics based on thermodynamic parameters and transition state kinetics. We use the formalism together with key elements of recent research to present a more overall treatment of III_V NW growth, which can serve as a basis to model and understand the dynamical mechanisms in terms of the basic control parameters, temperature and pressures/beam fluxes. Self-catalysed GaAs NW growth on Si substrates by molecular beam epitaxy is used as a model system.Comment: 63 pages, 25 figures and 4 tables. Some details are explained more carefully in this version aswell as a new figure is added illustrating various facets of a WZ crysta

Level set methods for higher order evolution laws

A numerical treatment of non-linear higher-order geometric evolution equations with the level set and the finite element method is presented. The isotropic, weak anisotropic and strong anisotropic situation is discussed. Most of the equations considered in this work arise from the field of thin film growth. A short introduction to the subject is given. Four different models are discussed: mean curvature flow, surface diffusion, a kinetic model, which combines the effects of mean curvature flow and surface diffusion and includes a further kinetic component, and an adatom model, which incorporates in addition free adatoms. As an introduction to the numerical schemes, first the isotropic and weak anisotropic situation is considered. Then strong anisotropies (non-convex anisotropies) are used to simulate the phenomena of faceting and coarsening. The experimentally observed effect of corner and edge roundings is reached in the simulation through the regularization of the strong anisotropy with a higher-order curvature term. The curvature regularization leads to an increase by two in the order of the equations, which results in highly non-linear equations of up to 6th order. For the numerical solution, the equations are transformed into systems of second order equations, which are solved with a Schur complement approach. The adatom model constitutes a diffusion equation on a moving surface. An operator splitting approach is used for the numerical solution. In difference to other works, which restrict to the isotropic situation, also the anisotropic situation is discussed and solved numerically. Furthermore, a treatment of geometric evolution equations on implicitly given curved surfaces with the level set method is given. In particular, the numerical solution of surface diffusion on curved surfaces is presented. The equations are discretized in space by standard linear finite elements. For the time discretization a semi-implicit discretization scheme is employed. The derivation of the numerical schemes is presented in detail, and numerous computational results are given for the 2D and 3D situation. To keep computational costs low, the finite element grid is adaptively refined near the moving curves and surfaces resp. A redistancing algorithm based on a local Hopf-Lax formula is used. The algorithm has been extended by the authors to the 3D case. A detailed description of the algorithm in 3D is presented in this work.In der Arbeit geht es um die numerische Behandlung nicht-linearer geometrischer Evolutionsgleichungen höherer Ordnung mit Levelset- und Finite-Elemente-Verfahren. Der isotrope, schwach anisotrope und stark anisotrope Fall wird diskutiert. Die meisten in dieser Arbeit betrachteten Gleichungen entstammen dem Gebiet des Dünnschicht-Wachstums. Eine kurze Einführung in dieses Gebiet wird gegeben. Es werden vier verschiedene Modelle diskutiert: mittlerer Krümmungsfluss, Oberflächendiffusion, ein kinetisches Modell, welches die Effekte des mittleren Krümmungsflusses und der Oberflächendiffusion kombiniert und zusätzlich eine kinetische Komponente beinhaltet, und ein Adatom-Modell, welches außerdem freie Adatome berücksichtigt. Als Einführung in die numerischen Schemata, wird zuerst der isotrope und schwach anisotrope Fall betrachtet. Anschließend werden starke Anisotropien (nicht-konvexe Anisotropien) benutzt, um Facettierungs- und Vergröberungsphänomene zu simulieren. Der in Experimenten beobachtete Effekt der Ecken- und Kanten-Abrundung wird in der Simulation durch die Regularisierung der starken Anisotropie durch einen Krümmungsterm höherer Ordnung erreicht. Die Krümmungsregularisierung führt zu einer Erhöhung der Ordnung der Gleichung um zwei, was hochgradig nicht-lineare Gleichungen von bis zu sechster Ordnung ergibt. Für die numerische Lösung werden die Gleichungen auf Systeme zweiter Ordnungsgleichungen transformiert, welche mit einem Schurkomplement-Ansatz gelöst werden. Das Adatom-Modell bildet eine Diffusionsgleichung auf einer bewegten Fläche. Zur numerischen Lösung wird ein Operatorsplitting-Ansatz verwendet. Im Unterschied zu anderen Arbeiten, die sich auf den isotropen Fall beschränken, wird auch der anisotrope Fall diskutiert und numerisch gelöst. Außerdem werden geometrische Evolutionsgleichungen auf implizit gegebenen gekrümmten Flächen mit Levelset-Verfahren behandelt. Insbesondere wird die numerische Lösung von Oberflächendiffusion auf gekrümmten Flächen dargestellt. Die Gleichungen werden im Ort mit linearen Standard-Finiten-Elementen diskretisiert. Als Zeitdiskretisierung wird ein semi-implizites Diskretisierungsschema verwendet. Die Herleitung der numerischen Schemata wird detailliert dargestellt, und zahlreiche numerische Ergebnisse für den 2D und 3D Fall sind gegeben. Um den Rechenaufwand gering zu halten, wird das Finite-Elemente-Gitter adaptiv an den bewegten Kurven bzw. den bewegten Flächen verfeinert. Es wird ein Redistancing-Algorithmus basierend auf einer lokalen Hopf-Lax Formel benutzt. Der Algorithmus wurde von den Autoren auf den 3D Fall erweitert. In dieser Arbeit wird der Algorithmus für den 3D Fall detailliert beschrieben

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

Simultaneous Etching and Oxidation of Vicinal Si(100) Surfaces: Atomistic Lattice-Gas Modeling of Morphological Evolution

Exposure of a vicinal Si(100) surface to oxygen at about 10−8Torr for temperatures between about 500 and 700°C produces etching-mediated step recession in competition with oxide island formation. Furthermore, the oxide islands can locally pin the receding steps and thus produce complex surface morphologies. An atomistic lattice-gas model is developed to describe these processes which accounts for the complex interplay between the oxygen surface chemistry and the silicon surface and step dynamics. The oxygen related-processes include dissociative adsorption, diffusion, oxide formation, and etching via SiO desorption. The silicon surface processes include: conversion of single vacancies formed by etching to divacancies and Si adatoms, anisotropic diffusion and aggregation (primarily at step edges) of these divacancies and Si adatoms, and Si ad-dimer attachment-detachment dynamics at steps which reflects anisotropic energetics. Kinetic Monte Carlo simulation of this model allows characterization of the evolving step morphologies. Steps retain some qualitative features of their equilibrium structure, i.e., alternating rough SB steps and smooth SA steps, although etching tends to produce step pairing, and pinning produces protruding “finger” morphologies. These morphological features are seen in scanning tnneling microscopy studies. We also comment on other aspects of evolution such as a mixed pit nucleation and step flow mode, and compare behavior with step flow type growth during Si molecular beam epitaxy

Instabilities in crystal growth by atomic or molecular beams

The planar front of a growing a crystal is often destroyed by instabilities. In the case of growth from a condensed phase, the most frequent ones are diffusion instabilities, which will be but briefly discussed in simple terms in chapter II. The present review is mainly devoted to instabilities which arise in ballistic growth, especially Molecular Beam Epitaxy (MBE). The reasons of the instabilities can be geometric (shadowing effect), but they are mostly kinetic or thermodynamic. The kinetic instabilities which will be studied in detail in chapters IV and V result from the fact that adatoms diffusing on a surface do not easily cross steps (Ehrlich-Schwoebel or ES effect). When the growth front is a high symmetry surface, the ES effect produces mounds which often coarsen in time according to power laws. When the growth front is a stepped surface, the ES effect initially produces a meandering of the steps, which eventually may also give rise to mounds. Kinetic instabilities can usually be avoided by raising the temperature, but this favours thermodynamic instabilities. Concerning these ones, the attention will be focussed on the instabilities resulting from slightly different lattice constants of the substrate and the adsorbate. They can take the following forms. i) Formation of misfit dislocations (chapter VIII). ii) Formation of isolated epitaxial clusters which, at least in their earliest form, are `coherent' with the substrate, i.e. dislocation-free (chapter X). iii) Wavy deformation of the surface, which is presumably the incipient stage of (ii) (chapter IX). The theories and the experiments are critically reviewed and their comparison is qualitatively satisfactory although some important questions have not yet received a complete answer.Comment: 90 pages in revtex, 45 figures mainly in gif format. Review paper to be published in Physics Reports. Postscript versions for all the figures can be found at http://www.theo-phys.uni-essen.de/tp/u/politi

Theory of Scanning Tunneling Microscopy

This lecture has been given at the 45th Spring School: Computing Solids: Models, Ab-initio Methods and Supercomputing organized at the Forschungszentrum J\"ulich. The goal of this manuscript is to review the basics behind the theory accompanying Scanning Tunneling Microscopy.Comment: 38 pages, 45th IFF Spring School: Computing Solids: Models, Ab-initio Methods and Supercomputing organized at the research center of Juelic