Dimitri D. Vvedensky

Growth of epitaxial graphene: Theory and experiment

A detailed review of the literature for the last 5-10 years on epitaxial growth of graphene is presented. Both experimental and theoretical aspects related to growth on transition metals and on silicon carbide are thoroughly reviewed. Thermodynamic and kinetic aspects of growth on all these materials, where possible, are discussed. To make this text useful for a wider audience, a range of important experimental techniques that have been used over the last decade to grow (e.g. CVD, TPG and segregation) and characterize (STM, LEEM, etc.) graphene are reviewed, and a critical survey of the most important theoretical techniques is given. Finally, we critically discuss various unsolved problems related to growth and its mechanism which we believe require proper attention in future research.

Growth kinetics and step density in reflection high-energy electron diffraction during molecular-beam epitaxy

The kinetics of molecular‐beam epitaxy are examined by means of Monte Carlo simulations in combination with a new approach for monitoring surface growth, i.e., by calculating the evolution of the surface step density. The evolution of the step density is shown to have a remarkable correspondence to that of the measured reflection high‐energy electron diffraction (RHEED) specular spot intensities for III‐V semiconductor compounds. We study growth in a variety of systems, including flat and stepped surfaces, as a function of substrate temperature and draw several conclusions concerning the relation between RHEED measurements, kinetics, and growth quality. The range of validity of the kinematic approach to RHEED is discussed and the importance of multiple scattering in the high step density regime is highlighted.

Layer Korringa-Kohn-Rostoker electronic structure code for bulk and interface geometries

Abstract A program is presented which implements the layer Koringer-Kohn-Rostoker theory for the electronic structure of both bulk systems and those characterised by two-dimensional periodicity. The one-electron Green function is obtained by recursively assembling the layers of the system, permitting the study of interface regions embedded in otherwise perfect host materials. The program enables the calculation of self-consistent charge densities and localised states.

Multiscale modelling of nanostructures

Most materials phenomena are manifestations of processes that are operative ove ra vast range of length and time scales. A complete understanding of the behaviour of materials thereby requires theoretical and computational tools that span the atomic-scale detail of first-principles methods and the more coarsegrained description provided by continuum equations. Recent efforts have focused on combining traditional methodologies—density functional theory, molecular dynamics, Monte Carlo methods and continuum descriptions— within a unified multiscale framework. This review covers the techniques that have been developed to model various aspects of materials behaviour with the ultimate aim of systematically coupling the atomistic to the continuum descriptions. The approaches described typically have been motivated by particular applications but can often be applied in wider contexts. The selfassembly of quantum dot ensembles will be used as a case study for the issues that arise and the methods used for all nanostructures. Although quantum dots can be obtained with all the standard growth methods and for a variety of material systems, their appearance is a quite selective process, involving the competition between equilibrium and kinetic effects, and the interplay between atomistic and long-range interactions. Most theoretical models have addressed particular aspects of the ordering kinetics of quantum dot ensembles, with fa rf ewer attempts at a comprehensive synthesis of this inherently multiscale phenomenon. We conclude with an assessment of the current status of multiscale modelling strategies and highlight the main outstanding issues.

Theory of homoepitaxy on Si(001): I. Kinetics during growth

Abstract Growth kinetics during Si(001) homoepitaxy are studied with a Monte Carlo computer simulation of a solid-on-solid model. The effect of the surface reconstruction, driven by the formation of dimers, is taken into account through longer residence times of atoms attached to other atoms or clusters perpendicular to the dimer-bond axis on the same layer than for atoms attached in the parallel direction. Although local fluctuations can bypass these rules, this prescription is correct on the average and leads naturally to the stability of steps for which the dimer axis is normal to the step edge in comparison to steps where the dimer axis is perpendicular to the step edge. By calculating the angle-resolved density of surface steps on the growing substrate, qualitative comparisons can be made between the simulations and the specular intensity measured in situ by reflection high-energy electron-diffraction (RHEED). Furthermore, by calculating the fractional coverages of 1 × 2 and 2 × 1 domains during growth, comparisons can be made with the fractional-order RHEED beams. Specific aspects of Si(001) homoepitaxy that are addressed are: (i) the azimuthal dependence of RHEED intensity oscillations and the formation of elongated clusters, (ii) the dependence of the growth characteristics upon the domain structure of the initial substrate, i.e., sustained RHEED oscillations for surfaces with a single 2 × 1 domain but decaying oscillations for a substrate with coexistent 2 × 1 and 1× 2 domains, (iii) the observation of RHEED oscillations at temperatures too low (≈ 300°C) to promote the mobility of surface adatoms, as evidenced by the absence of recovery upon the termination of the incident molecular beam, and (iv) a temperature-induced monolayer-to-bilayer transition in the RHEED oscillations.

Recovery kinetics during interrupted epitaxial growth

Abstract The recovery kinetics upon cessation of growth by molecular-beam epitaxy (MBE) are investigated with Monte Carlo simulations of a solid-on-solid model. The simulations are monitored by calculating both the step density and the kinematic approximation to the reflection high-energy electron-diffraction (RHEED) specular intensity. A comprehensive analysis of recovery includes (1) the microscopic origins of the recovery profile of the specular intensity in RHEED, (2) the isolation of particular recovery processes and the problems associated with interpreting recovery data in terms of elementary events, (3) the dependence of recovery characteristics upon the point in layer completion at which growth is interrupted, (4) the determination of kinetic parameters from recovery data in an experimentally-realizable manner, and so obtaining a parametrization of simple models of MBE, and (5) the influence of diffraction conditions upon the qualitative and quantitative interpretation of RHEED recovery data. Our study illustrates the considerable potential in analyzing recovery data and highlights the fact that recovery is a more discriminating test of models of MBE and of RHEED than simply growth. This derives from the ability to make quantitative comparisons between measured and calculated characteristic times without necessarily resorting to dynamical diffraction calculations.

ISLAND DYNAMICS AND THE LEVEL SET METHOD FOR EPITAXIAL GROWTH

Abstract We adapt the level set method to simulate the growth of thin films described by the motion of island boundaries. This island dynamics model involves a continuum in the lateral directions, but retains atomic scale discreteness in the growth direction. Several choices for the island boundary velocity are discussed, and computations of the island dynamics model using the level set method are presented.

Novel growth mechanism of epitaxial graphene on metals.

Graphene, a hexagonal sheet of sp(2)-bonded carbon atoms, has extraordinary properties which hold immense promise for nanoelectronic applications. Unfortunately, the popular preparation methods of micromechanical cleavage and chemical exfoliation of graphite do not easily scale up for application purposes. Epitaxial graphene provides an attractive alternative, though there are many challenges, not least of which is the absence of any understanding of the complex atomistic assembly kinetics of graphene layers. Here, we present a simple rate theory of epitaxial graphene growth on close-packed metal surfaces. On the basis of recent low-energy electron-diffraction microscopy experiments, our theory supposes that graphene islands grow predominantly by the attachment of five-atom clusters. With optimized kinetic parameters, our theory produces a quantitative account of the measured time-dependent carbon adatom density. The temperature dependence of this density at the onset of nucleation leads us to predict that the smallest stable precursor to graphene growth is an immobile island composed of six five-atom clusters. This conclusion is supported by a recent study based on temperature-programmed growth of epitaxial graphene, which provides direct evidence of nanoclusters whose coarsening leads to the formation of graphene layers. Our findings should motivate additional high-resolution imaging experiments and more detailed simulations which will yield important input to developing strategies for the large-scale production of epitaxial graphene.

Submonolayer epitaxy without a critical nucleus

Abstract The nucleation and growth of two-dimensional islands is studied with Monte Carlo simulations of a pair-bond solid-on-solid model of epitaxial growth. The conventional description of this problem in terms of a well-defined critical island size fails because no islands are absolutely stable against single atom detachment by thermal bond breaking. When two-bond scission is negligible, we find that the ratio of the dimer dissociation rate to the rate of adatom capture by dimers uniquely indexes both the island size distribution scaling function and the dependence of the island density on the flux and the substrate temperature. Effective pair-bond model parameters are found that yield excellent quantitative agreement with scaling functions measured for Fe Fe (001) .

Differential renormalization-group generators for static and dynamic critical phenomena

Abstract The derivation of differential renormalization-group (DRG) equations for applications to static and dynamic critical phenomena is reviewed. The DRG approach provides a self-contained closed-form representation of the Wilson renormalization group (RG) and should be viewed as complementary to the Callan-Symanzik equations used in field-theoretic approaches to the RG. The various forms of DRG equations are derived to illustrate the general mathematical structure of each approach and to point out the advantages and disadvantages for performing practical calculations. Otherwise, the review focuses upon the one-particle-irreducible DRG equations derived by Nicoll and Chang and by Chang, Nicoll, and Young; no attempt is made to provide a general treatise of critical phenomena. A few specific examples are included to illustrate the utility of the DRG approach: the large- n limit of the classical n -vector model (the spherical model), multi- or higher-order critical phenomena, and crit ical dynamics far from equilibrium. The large- n limit of the n -vector model is used to introduce the application of DRG equations to a well-known example, with exact solution obtained for the nonlinear trajectories, generating functions for nonlinear scaling fields, and the equation of state. Trajectory integrals and nonlinear scaling fields within the framework of ϵ-expansions are then discussed for tricritical crossover, and briefly for certain aspects of multi- or higher-order critical points, including the derivation of the Helmholtz free energy and the equation of state. The discussion then turns to critical dynamics with a development of the path integral formulation for general dynamic processes. This is followed by an application to a model far-from-equilibrium system that undergoes a phase transformation analogous to a second-order critical point, the Schlogl model for a chemical instability.