A lattice kinetic Monte Carlo study of void morphological evolution during silicon crystal growth

Analysis of the lattice kinetic Monte Carlo method in systems with external fields

The lattice kinetic Monte Carlo (LKMC) method is studied in the context of Brownian particles subjected to drift forces, here principally represented by external fluid flow. LKMC rate expressions for particle hopping are derived that satisfy detailed balance at equilibrium while also providing correct dynamical trajectories in advective-diffusive situations. Error analyses are performed for systems in which collections of particles undergo Brownian motion while also being advected by plug and parabolic flows. We demonstrate how the flow intensity, and its associated drift force, as well as its gradient, each impact the accuracy of the method in relation to reference analytical solutions and Brownian dynamics simulations. Finally, we show how a non-uniform grid that everywhere retains full microscopic detail may be employed to increase the computational efficiency of lattice kinetic Monte Carlo simulations of particles subjected to drift forces arising from the presence of external fields.

Effects of vacancy cluster defects on electrical and thermodynamic properties of silicon crystals

A first-principle plane-wave pseudopotential method based on the density function theory (DFT) was employed to investigate the effects of vacancy cluster (VC) defects on the band structure and thermoelectric properties of silicon (Si) crystals. Simulation results showed that various VC defects changed the energy band and localized electron density distribution of Si crystals and caused the band gap to decrease with increasing VC size. The results can be ascribed to the formation of a defect level produced by the dangling bonds, floating bonds, or high-strain atoms surrounding the VC defects. The appearance of imaginary frequencies in the phonon spectrum of defective Si crystals indicates that the defect-region structure is dynamically unstable and demonstrates phase changes. The phonon dispersion relation and phonon density of state were also investigated using density functional perturbation theory. The obtained Debye temperature (θ_D) for a perfect Si crystal had a minimum value of 448 K at T = 42 K and a maximum value of 671 K at the high-temperature limit, which is consistent with the experimental results reported by Flubacher. Moreover, the Debye temperature decreased with increases in the VC size. VC defects had minimal effects on the heat capacity (C_v) value when temperatures were below 150 K. As the temperature was higher than 150 K, the heat capacity gradually increased with increasing temperature until it achieved a constant value of 11.8 cal/cell·K. The heat capacity significantly decreased as the VC size increased. For a 2 × 2 × 2 superlattice Si crystal containing a hexagonal ring VC (HRVC10), the heat capacity decreased by approximately 17%.

Simulation of aggregating particles in complex flows by the lattice kinetic Monte Carlo method

We develop and validate an efficient lattice kinetic Monte Carlo (LKMC) method for simulating particle aggregation in laminar flows with spatially varying shear rate, such as parabolic flow or flows with standing vortices. A contact time model was developed to describe the particle-particle collision efficiency as a function of the local shear rate, G, and approach angle, θ. This model effectively accounts for the hydrodynamic interactions between approaching particles, which is not explicitly considered in the LKMC framework. For imperfect collisions, the derived collision efficiency [ɛ=1 - ∫(0)(π/2) sinθ exp(-2cotθΓ(agg)/G)dθ] was found to depend only on Γ(agg)∕G, where Γ(agg) is the specified aggregation rate. For aggregating platelets in tube flow, Γ(agg)=0.683 s(-1) predicts the experimentally measured ε across a physiological range (G = 40-1000 s(-1)) and is consistent with α(2b)β(3)-fibrinogen bond dynamics. Aggregation in parabolic flow resulted in the largest aggregates forming near the wall where shear rate and residence time were maximal, however intermediate regions between the wall and the center exhibited the highest aggregation rate due to depletion of reactants nearest the wall. Then, motivated by stenotic or valvular flows, we employed the LKMC simulation developed here for baffled geometries that exhibit regions of squeezing flow and standing recirculation zones. In these calculations, the largest aggregates were formed within the vortices (maximal residence time), while squeezing flow regions corresponded to zones of highest aggregation rate.

Temporal coarse-graining of microscopic-lattice kinetic Monte Carlo simulations via tau leaping

A coarse-time-step method is presented that enables the execution of multiple events at each time increment of microscopic-lattice kinetic Monte Carlo simulations. The method employs the n-fold method to create groups of reactions in which the tau-leap algorithm of Gillespie, originally proposed for well-mixed systems, is applied. Creation of groups of reactions is an essential step to avoid violation of the leap condition that arises when the tau-leap algorithm is applied to a single site. The method is general, very easy to implement, and can result in substantial computational savings when global updating is employed. An illustrative example from crystal growth of a simple cubic lattice with the solid-on-solid approximation is presented.

Coarse-grained lattice kinetic Monte Carlo simulation of systems of strongly interacting particles

A general approach is presented for spatially coarse-graining lattice kinetic Monte Carlo (LKMC) simulations of systems containing strongly interacting particles. While previous work has relied on approximations that are valid in the limit of weak interactions, here we show that it is possible to compute coarse-grained transition rates for strongly interacting systems without a large computational burden. A two-dimensional square lattice is employed on which a collection of (supersaturated) strongly interacting particles is allowed to reversibly evolve into clusters. A detailed analysis is presented of the various approximations applied in LKMC coarse graining, and a number of numerical closure rules are contrasted and compared. In each case, the overall cluster size distribution and individual cluster structures are used to assess the accuracy of the coarse-graining approach. The resulting closure approach is shown to provide an excellent coarse-grained representation of the systems considered in this study.