The Lattice Kinetic Monte Carlo Method as a Versatile Tool for Simulating Diverse Micro and Mesoscale Phenomena

Lattice-based, or ‘on-lattice’, kinetic Monte Carlo simulations are attractive because of their relative computational simplicity and efficiency, and have been employed to simulate an enormous range of non-equilibrium physical, chemical and biological phenomena. In a kinetic Monte Carlo simulation (also referred to as dynamic Monte Carlo or the Gillespie method), which is typically applied to a collection of discrete particles, rates must first be specified for a set of ‘events’, such as a hop of an atom from one lattice site to another or a reaction between two particles. The universe of events and the associated rates are input to the kinetic Monte Carlo simulation, and must be obtained by some other means. The simulation is then carried out by executing events in a biased stochastic sequence. In this talk, variants of the lattice kinetic Monte Carlo method are applied to several distinct situations, ranging from microstructure evolution in semiconductor crystals, to cellular aggregate formation in blood flow, to coarse-grained simulations of phase evolution in generalized liquid-vapor systems. These diverse examples are used to illustrate the simplicity and flexibility of the general lattice kinetic Monte Carlo framework as a powerful computational tool, but also the potential pitfalls related to its application in certain situations. In this regard, special emphasis is placed on the discussion of (1) reduced degree-of-freedom representations (via coarse-graining) and the concomitant loss of entropy, and (2) simulation of systems of particles subject to external advective fields such as fluid flow

Entropic Origins of Stability in Silicon Interstitial Clusters

The role of entropy in the thermodynamic properties of small interstitial clusters in crystalline silicon is investigated using an empirical potential. It is shown that both vibrational and configurational entropies are potentially important in setting the properties of small silicon interstitial clusters and, in particular, contribute to the formation of “magic” sizes that exhibit special stability, which have been inferred by experimental measurements of dopant diffusion. The results suggest that a competition between formation energy and entropy of small clusters could be linked to the selection process between various self-interstitial precipitate morphologies observed in ion-implanted crystalline silicon

Detailed Microscopic Analysis of Self-interstitial Aggregation in Silicon. I. Direct Molecular Dynamics Simulations of Aggregation

A comprehensive atomistic study of self-interstitial aggregation in crystalline silicon is presented. Here, large-scale parallel molecular dynamics simulations are used to generate time-dependent views into the selfinterstitial clustering process, which is important during post-implant damage annealing. The effects of temperature and pressure on the aggregation process are studied in detail and found to generate a variety of qualitatively different interstitial cluster morphologies and growth behavior. In particular, it is found that the self-interstitial aggregation process is strongly affected by hydrostatic pressure. {111}-oriented planar defects are found to be dominant under stress-free or compressive conditions while {113} rodlike and planar defects are preferred under tensile conditions. Moreover, the aggregation pathways for forming the different types of planar defect structures are found to be qualitatively different. In each case, the various cluster morphologies generated in the simulations are found to be in excellent agreement with structures previously predicted from electronic-structure calculations and observed experimentally by electron microscopy. Multiple empirical interatomic potential models were employed and found to generally provide similar results leading to a fairly consistent picture of self-interstitial aggregation. In a companion article, a detailed thermodynamic analysis of various cluster configurations is employed to probe the mechanistic origins of these observations

Feature Activated Molecular Dynamics: An Efficient Approach for Atomistic Simulation of Solid-State Aggregation Phenomena

A new approach is presented for performing efficient molecular dynamics simulations of solute aggregation in crystalline solids. The method dynamically divides the total simulation space into “active” regions centered about each minority species, in which regular molecular dynamics is performed. The number, size and shape of these regions is updated periodically based on the distribution of solute atoms within the overall simulation cell. The remainder of the system is essentially static except for periodic rescaling of the entire simulation cell in order to balance the pressure between the isolated molecular dynamics regions. The method is shown to be accurate and robust for the Environment-Dependant Interatomic Potential (EDIP) for silicon and an Embedded Atom Method (EAM) potential for copper. Several tests are performed beginning with the diffusion of a single vacancy all the way to large-scale simulations of vacancy clustering. In both material systems, the predicted evolutions agree closely with the results of standard molecular dynamics simulations. Computationally, the method is demonstrated to scale almost linearly with the concentration of solute atoms, but is essentially independent of the total system size. This scaling behavior allows for the full dynamical simulation of aggregation under conditions that are more experimentally realizable than would be possible with standard molecular dynamics

An Internally Consistent Approach for Modeling Solid-State Aggregation: II. Mean-Field Representation of Atomistic Processes

A detailed continuum (mean-field) model is presented that captures quantitatively the evolution of a vacancy cluster size distribution in crystalline silicon simulated directly by large-scale parallel molecular dynamics. The continuum model is parameterized entirely using the results of atomistic simulations based on the same empirical potential used to perform the atomistic aggregation simulation, leading to an internally consistent comparison across the two scales. It is found that an excellent representation of all measured components of the cluster size distribution can be obtained with consistent parameters only if the assumed physical mechanisms are captured correctly. In particular, the inclusion of vacancy cluster diffusion and a model to capture the dynamic nature of cluster morphology at high temperature are necessary to reproduce the results of the large-scale atomistic simulation. Dynamic clusters with large capture volumes at high temperature, which are the result of rapid cluster shape fluctuations, are shown to be larger than would be expected from static analyses, leading to substantial enhancement of the nucleation rate. Based on these results, it is shown that a parametrically consistent atomistic-continuum comparison can be used as a sensitive framework for formulating accurate continuum models of complex phenomena such as defect aggregation in solids

Internally Consistent Approach for Modeling Solid-State Aggregation: I. Atomistic Calculations of Vacancy Clustering in Silicon

A computational framework is presented for describing the nucleation and growth of vacancy clusters in crystalline silicon. The overall approach is based on a parametrically consistent comparison between two representations of the process in order to provide a systematic method for probing the details of atomic mechanisms responsible for aggregation. In this paper, the atomistic component of the overall framework is presented. First, a detailed set of targeted atomistic simulations are described that characterize fully the thermodynamic and transport properties of vacancy clusters over a wide range of sizes. It is shown that cluster diffusion is surprisingly favorable because of the availability of multiple, almost degenerate, configurations. A single large-scale parallel molecular dynamics simulation is then used to compute directly the evolution of the vacancy cluster size distribution in a supersaturated system initially containing 1000 uniformly distributed vacancies in a host lattice of 216,000 Si atoms at 1600 K. The results of this simulation are interpreted in the context of mean-field scaling theory based on the observed power-law evolution of the size distribution moments. It is shown that the molecular dynamics results for aggregation of vacancy clusters, particularly the evolution of the average cluster size, can be very well represented by a highly simplified mean-field model. A direct comparison to a detailed continuum model is made in a subsequent article

Vacancy Self-trapping During Rapid Thermal Annealing of Silicon Wafers

The density and spatial distribution of oxide precipitates within a crystalline silicon wafer is of paramount importance for microelectronic device yield. In this letter, the authors show how the formation of previously unconsidered, very small vacancy aggregates can explain macroscopic spatial variations in the oxide precipitate density, which are observed following certain rapid thermal annealing conditions. The formation of these nanometer-sized voids is predicted on the basis of their recent model for vacancy aggregation that accounts for high temperature entropic effects

An Enthalpy Landscape View of Homogeneous Melting in Crystals

A detailed analysis of homogeneous melting in crystalline materials modeled by empirical interatomic potentials is presented using the theory of inherent structures.We show that the homogeneous melting of a perfect, infinite crystalline material can be inferred directly from the growth exponent of the inherent structure density-of-states distribution expressed as a function of formation enthalpy. Interestingly, this growth is already established by the presence of very few homogeneously nucleated point defects in the form of Frenkel pairs. This finding supports the notion that homogeneous melting is appropriately defined in terms of a one-phase theory and does not require detailed consideration of the liquid phase. We then apply this framework to the study of applied hydrostatic compression on homogeneous melting and show that the inherent structure analysis used here is able to capture the correct pressure-dependence for two crystalline materials, namely silicon and aluminum. The coupling between the melting temperature and applied pressure arises through the distribution of formation volumes for the various inherent structures

Coarse-Grained Lattice Monte Carlo Simulations with Continuous Interaction Potentials

A coarse-grained lattice Metropolis Monte Carlo (CG-MMC) method is presented for simulating fluid systems described by standard molecular force fields. First, a thermodynamically consistent coarse-grained interaction potential is obtained numerically and automatically from a continuous force field such as Lennard-Jones. The coarse-grained potential then is used to driveCG-MMC simulations of vapor-liquid equilibrium in Lennard-Jones, square-well, and simple point chargewater systems. The CG-MMC predicts vapor-liquid phase envelopes, as well as the particle density distributions in both the liquid and vapor phases, in excellent agreement with full-resolution Monte Carlo simulations, at a fraction of the computational cost