Computation of Strained Epitaxial Growth in Three Dimensions by Kinetic Monte Carlo

A numerical method for computation of heteroepitaxial growth in the presence of strain is presented. The model used is based on a solid-on-solid model with a cubic lattice. Elastic effects are incorporated using a ball and spring type model. The growing film is evolved using Kinetic Monte Carlo (KMC) and it is assumed that the film is in mechanical equilibrium. The strain field in the substrate is computed by an exact solution which is efficiently evaluated using the fast Fourier transform. The strain field in the growing film is computed directly. The resulting coupled system is solved iteratively using the conjugate gradient method. Finally we introduce various approximations in the implementation of KMC to improve the computation speed. Numerical results show that layer-by-layer growth is unstable if the misfit is large enough resulting in the formation of three dimensional islands

Scaling of Heteroepitaxial Island Sizes

Monte Carlo simulations of an atomistic solid-on-solid model are used to study the effect of lattice misfit on the distribution of two-dimensional islands sizes as a function of coverage $\Theta$ in the submonolayer aggregation regime of epitaxial growth. Misfit promotes the detachment of atoms from the perimeter of large pseudomorphic islands and thus favors their dissolution into smaller islands that relieve strain more efficiently. The number density of islands composed of $s$ atoms exhibits scaling in the form \mbox{$N_s(\Theta) \sim \Theta / \langle s \rangle^2 \, g(s/\langle s \rangle$)} where $\langle s \rangle$ is the average island size. Unlike the case of homoepitaxy, a rate equation theory based on this observation leads to qualitatively different behavior than observed in the simulations.Comment: 10 pages, LaTeX 2.09, IC-DDV-94-00

Density Functional Theory of Epitaxial Growth of Metals

This chapter starts with a summary of the atomistic processes that occur during epitaxy. We then introduce density functional theory (DFT) and describe its implementation into state-of-the-art computations of complex processes in condensed matter physics and materials science. In particular we discuss how DFT can be used to calculate parameters of microscopic processes such as adsorption and surface diffusion, and how they can be used to study the macroscopic time and length scales of realistic growth conditions. This meso- and macroscopic regime is described by the ab initio kinetic Monte Carlo approach. We discuss several specific theoretical studies that highlight the importance of the different diffusion mechanisms at step edges, the role of surfactants, and the influence of surface stress. The presented results are for specific materials (namely silver and aluminum), but they are explained in simple physical pictures suggesting that they also hold for other systems.Comment: 55 pages, 20 figures, to be published "Growth of Ultrathin Epitaxial Layers", The Chemical Physics of Soild Surfaces, Vol. 8, Eds D. A. King and D. P. Woodruff (Elsevier Science, Amsterdam, 1997

Study of Strain and Temperature Dependence of Metal Epitaxy

Metallic films are important in catalysis, magneto-optic storage media, and interconnects in microelectronics, and it is crucial to predict and control their morphologies. The evolution of a growing crystal is determined by the behavior of each individual atom, but technologically relevant structures have to be described on a time scale of the order of (at least) tenths of a second and on a length scale of nanometers. An adequate theory of growth should describe the atomistic level on very short time scales (femtoseconds), the formation of small islands (microseconds), as well as the evolution of mesoscopic and macroscopic structures (tenths of seconds). The development of efficient algorithms combined with the availability of cheaper and faster computers has turned density functional theory (DFT) into a reliable and feasible tool to study the microscopic aspects of growth phenomena (and many other complex processes in materials science, condensed matter physics, and chemistry). In this paper some DFT results for diffusion properties on metallic surfaces are presented. Particularly, we will discuss the current understanding of the influences of strain on the diffusion (energy barrier and prefactor) of a single adatom on a substrate. A DFT total energy calculation by its nature is primarily a static calculation. An accurate way to describe the spatial and temporal development of a growing crystal is given by kinetic Monte Carlo (KMC). We will describe the method and its combination with microscopic parameters obtained from ab initio calculations. It is shown that realistic ab initio kinetic Monte Carlo simulations are able to predict an evolving mesoscopic structure on the basis of microscopic details.Comment: 25 pages, 6 figures, In: ``Morphological Organisation during Epitaxial Growth and Removal'', Eds. Z. Zhang, M. Lagally. World Scientific, Singapore 1998. other related publications can be found at http://www.rz-berlin.mpg.de/th/paper.htm

Modeling the Elastic Energy of Alloys: Potential Pitfalls of Continuum Treatments

Some issues that arise when modeling elastic energy for binary alloys are discussed within the context of a Keating model and density functional calculations. The Keating model is based on atomistic modeling of elastic interactions in binary alloy using harmonic springs with species dependent equilibrium lengths. It is demonstrated that the continuum limit for the strain field are the usual equations of linear elasticity for alloys and that they correctly capture the coarse-grained displacement field. In addition, it is established that Euler-Lagrange equation of the continuum limit of the elastic energy will yield the same strain field equation. However, a direct calculation of the elastic energy of the atomistic model reveals that the continuum expression for the elastic energy is both qualitatively and quantitatively incorrect. This is because it does not take atomistic scale compositional non-uniformity into account. Importantly, we also shows that finely mixed alloys tend to have more elastic energy than segregated systems, which is the opposite of predictions by some continuum theories. It is also shown that for strained thin films the traditionally used effective misfit for alloys systematically underestimate the strain energy. In some models, this drawback is handled by including an elastic contribution to the enthalpy of mixing which is characterized in terms of the continuum concentration. The direct calculation of the atomistic model reveals that this approach suffers serious difficulties. It is demonstrated that elastic contribution to the enthalpy of mixing is non-isotropic and scale dependent. It also shown that such effects are present in density-functional theory calculations for the Si/Ge and Ag/Pt systems. This work demonstrates that it is critical to include the microscopic arrangements in any elastic model to achieve even qualitatively correct behavior

Level Set Approach to Reversible Epitaxial Growth

We generalize the level set approach to model epitaxial growth to include thermal detachment of atoms from island edges. This means that islands do not always grow and island dissociation can occur. We make no assumptions about a critical nucleus. Excellent quantitative agreement is obtained with kinetic Monte Carlo simulations for island densities and island size distributions in the submonolayer regime.Comment: 7 pages, 9 figure

Fluctuations and scaling in models for particle aggregation

We consider two sequential models of deposition and aggregation for particles. The first model (No Diffusion) simulates surface diffusion through a deterministic capture area, while the second (Sequential Diffusion) allows the atoms to diffuse up to \ell steps. Therefore the second model incorporates more fluctuations than the first, but still less than usual (Full Diffusion) models of deposition and diffusion on a crystal surface. We study the time dependence of the average densities of atoms and islands and the island size distribution. The Sequential Diffusion model displays a nontrivial steady-state regime where the island density increases and the island size distribution obeys scaling, much in the same way as the standard Full Diffusion model for epitaxial growth. Our results also allow to gain insight into the role of different types of fluctuations.Comment: 25 pages. Minor changes in the main text and in some figures. Accepted for publication in Surface Scienc