Tim P. Schulze

Efficient kinetic Monte Carlo simulation

This paper concerns kinetic Monte Carlo (KMC) algorithms that have a single-event execution time independent of the system size. Two methods are presented-one that combines the use of inverted-list data structures with rejection Monte Carlo and a second that combines inverted lists with the Marsaglia-Norman-Cannon algorithm. The resulting algorithms apply to models with rates that are determined by the local environment but are otherwise arbitrary, time-dependent and spatially heterogeneous. While especially useful for crystal growth simulation, the algorithms are presented from the point of view that KMC is the numerical task of simulating a single realization of a Markov process, allowing application to a broad range of areas where heterogeneous random walks are the dominate simulation cost.

Coupling kinetic Monte-Carlo and continuum models with application to epitaxial growth

We present a hybrid method for simulating epitaxial growth that combines kinetic Monte-Carlo (KMC) simulations with the Burton-Cabrera-Frank model for crystal growth. This involves partitioning the computational domain into KMC regions and regions where we time-step a discretized diffusion equation. Computational speed and accuracy are discussed. We find that the method is significantly faster than KMC while accounting for stochastic fluctuations in a comparable way.

Suppression of Grain Boundaries in Graphene Growth on Superstructured Mn-Cu(111) Surface

As undesirable defects, grain boundaries (GBs) are widespread in epitaxial graphene using existing growth methods on metal substrates. Employing density functional theory calculations, we first identify that the misorientations of carbon islands nucleated on a Cu(111) surface lead to the formation of GBs as the islands coalesce. We then propose a two-step kinetic pathway to effectively suppress the formation of GBs. In the first step, large aromatic hydrocarbon molecules are deposited onto a sqrt[3]×sqrt[3] superstructured Cu-Mn alloyed surface to seed the initial carbon clusters of a single orientation; in the second step, the seeded islands are enlarged through normal chemical vapor deposition of methane to form a complete graphene sheet. The present approach promises to overcome a standing obstacle in large scale single-crystal graphene fabrication.

A time-dependent formulation of the mushy-zone free-boundary problem

We present time-dependent governing equations and boundary conditions for the mushy-zone free-boundary problem that are valid in an arbitrary frame of reference. The model for time-evolving mushy zones is more complicated than in the steady case because the interface velocity w can be distinct from both the velocity of the dendrites v and the fluid velocity u. We consider the limit of negligible solutal diffusivity, where there are four types of boundary condition at the mush–liquid interface, depending on both the direction of flow across the interface and the direction of the interface motion relative to the solid phase. We illustrate these boundary conditions by examining a family of one-dimensional problems in which a binary material is chilled from a fixed cold point in the laboratory frame of reference while fluid is pumped through the resulting mushy layer at a rate Q and the mushy layer itself is translated at a rate V . This allows us to exhibit three of the four types of mushy-layer interfaces. We show that the fourth type cannot occur in this scenario.

A continuum model for the growth of epitaxial films

Abstract The continuum equations presented here model the growth of epitaxial films in terms of a local edge density ∼ ∇ h and surface concentration (number density) of adatoms. This model is more amenable to computations than existing models that feature discrete edges and solve continuum equations on each terrace; yet it offers a more detailed picture than continuum models that treat the surface height as the only dependent variable. This latter feature is especially important if one wishes to account for several species which may react on the surface of the film or at step edges to build complicated unit cells. The model is motivated by and compared with numerical solutions of rate equations which are derived from kinetic Monte-Carlo simulations. After introducing the model in a 1+1 dimensional setting, we extend it to a 2+1 dimensional setting assuming spatial derivatives become surface gradients. We also discuss extension for the case with multiple species.

The rapid advance and slow retreat of a mushy zone

We discuss a model for the evolution of a mushy zone which forms during the solidification of a binary alloy cooled from below in a tank with finite height. Our focus is on behaviours of the system that do not appear when either a semi-infinite domain or negligible solute diffusion is assumed. The problem is simplified through an assumption of negligible latent heat, and we develop a numerical scheme that will permit insights that are critical for developing a more general procedure. We demonstrate that a mushy zone initially grows rapidly, then slows down and eventually retreats slowly. The mushy zone vanishes after a long time, as it is overtaken by a slowly growing solid region at the base of the tank.

Mushy Zones with Fully Developed Chimneys

Convection in mushy zones can lead to several types of free boundaries requiring distinct boundary conditions depending on whether the interface is freezing or melting and on the direction of flow relative to the interface. Here we implement these boundary conditions for the first time to arrive at solutions for mushy zones featuring liquid inclusions and fully developed chimneys.