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Kinetic Activation Relaxation Technique

Laurent Karim Béland, Peter Brommer, Fedwa El-Mellouhi, Jean-François Joly, Normand Mousseau

arXiv:1107.2417v4physics.comp-phcond-mat.mtrl-sci

TL;DR

Standard KMC’s predefined event catalogs and lattice constraints limit its application to complex, off-lattice materials. k-ART combines ART nouveau event searches with topological classification and a reusable catalog. Applied to three materials problems, it handles complex systems faster than molecular dynamics, though more slowly than standard KMC.

  • Problem

    Standard KMC depends on predefined events and fixed lattice positions, limiting its application to materials with off-lattice or disordered configurations.

    Source: §I · p. 1

  • Method

    k-ART combines ART nouveau transition-state searches with nauty-based topological classification and a reusable event catalog.

    Source: B. Topological classification and… · p. 4, §V · p. 11

  • Results

    k-ART handled vacancy-interstitial annihilation in c-Si, interstitial diffusion in Fe, and a-Si relaxation, faster than molecular dynamics but slower than standard KMC.

    Source: §V · p. 11

  • Takeaways & Limitations

    The authors conclude k-ART opens the possibility of studying complex-system problems that have remained out of reach of simulation.

    Source: §V · p. 11

    k-ART is slower than standard KMC.

    Source: §V · p. 11

Abstract

from arXiv · show

We present a detailed description of the kinetic Activation-Relaxation Technique (k-ART), an off-lattice, self-learning kinetic Monte Carlo algorithm with on-the-fly event search. Combining a topological classification for local environments and event generation with ART nouveau, an efficient unbiased sampling method for finding transition states, k-ART can be applied to complex materials with atoms in off-lattice positions or with elastic deformations that cannot be handled with standard KMC approaches. In addition to presenting the various elements of the algorithm, we demonstrate the general character of k-ART by applying the algorithm to three challenging systems: self-defect annihilation in c-Si (crystalline silicon), self-interstitial diffusion in Fe and structural relaxation in a-Si (amorphous silicon).

I. INTRODUCTION

Rare, high-barrier diffusion makes long-timescale atomistic simulation difficult, while standard KMC’s fixed event catalog and lattice constraints limit its use for disordered or off-lattice systems. k-ART combines topology-based classification, on-the-fly event searches, and barrier relaxation to study complex materials.

I. INTRODUCTION

  • Rare, high-barrier diffusion is difficult to reproduce with molecular dynamics, while standard KMC relies on a predefined catalog and fixed lattice positions.Those constraints become problematic for long-range elastic effects, off-lattice motion, and disordered conformations.Source: §I · p. 1
  • Earlier on-the-fly, off-lattice methods are more flexible but costly for complex systems because they rebuild the event catalog at every step.Catalog-based approaches also remain limited to on-lattice positions and can provide ill-controlled corrections for elastic deformation.Source: §I · p. 1
  • k-ART couples ART nouveau event generation and barrier calculation with nauty topological classification, and is demonstrated on three materials problems.The applications are self-defect annihilation in crystalline Si, interstitial diffusion in Fe, and relaxation in amorphous silicon.Source: §I · p. 2

II. OVERVIEW OF KINETIC ART

  • Unlike standard KMC, k-ART classifies local environments by topology, grows its event catalog as environments appear, and relaxes barriers to include elastic rearrangements.This allows atoms to adopt spatial arrangements beyond predefined lattice positions while accounting for short- and long-range elastic deformations.Source: §II · p. 2
  • Atoms sharing a topology are assumed to share an activated-mechanism list; for a single c-Si vacancy, 20 topologies describe local environments regardless of system size.Searches are launched for each topology, events are stored in a catalog, and low-energy barriers are relaxed for specific atoms before KMC selects and executes an event.Source: §II · p. 2

III. ALGORITHMIC AND IMPLEMENTATION DETAILS

ART nouveau searches from a local minimum for a first-order saddle point, then crosses it and relaxes into a new minimum to define an event. Local displacements and iterative saddle-point methods keep the search practical.

A. ART nouveau

  • ART nouveau deforms a local minimum until an unstable direction appears, converges to a first-order saddle, then crosses it and relaxes into a new minimum.The initial minimum, saddle, and final minimum together form an event.Source: A. ART nouveau · p. 2
  • Searches displace a local region in a random direction; in Si vacancy examples this includes atoms within 3.0 Å, with saddle thresholds of 0.5 eV/Å and 0.1 eV/Å for generic and specific events.The displaced region depends on the system and typically includes first and second nearest neighbors in semiconductors.Source: A. ART nouveau · p. 2
  • 600–800 force evaluations are required on average per successful event search, while specific-event relaxation typically takes 1–80 evaluations.ART nouveau avoids direct Hessian computation, using Lanczos and DIIS methods to converge to the saddle point.Source: A. ART nouveau · p. 3

B. Topological classification and event generation

k-ART encodes local environments as connectivity graphs, uses those topologies to search for and classify events, and builds a reusable catalog while relaxing barriers for specific geometries.

B. Topological classification and event generation

  • A local environment becomes a truncated connectivity graph without geometrical information; nauty classifies it and returns a topology index used to generate a unique hash key.The key and permutation information support identifying an environment and reconstructing its geometry from reference positions.Source: B. Topological classification and… · p. 3
  • The graph is designed to identify a unique relaxed structure in most cases; for crystalline Si, the local sphere has a 5.0 Å radius and atoms link below 2.8 Å.Surrounding atomic positions constrain geometrical reconstruction from the topological graph.Source: B. Topological classification and… · p. 3
  • ART nouveau searches are launched on each new topology, with search counts adjustable to find low-energy events and increased for frequently encountered topologies.New events are compared against known events and classified using the initial, saddle, and final state topologies.Source: B. Topological classification and… · p. 3
  • For simplification, the attempt frequency τ0 is fixed at onset to 10^13 s−1 for all events, although it varies only weakly with the chosen pathway.Source: B. Topological classification and… · pp. 3–4
  • Including atomic-motion direction distinguishes symmetric events with identical initial, saddle, and final topologies, barrier heights, and absolute displacements.Checks on highly symmetric Si and Fe states found that this classification recovers and discriminates between pathways.Source: B. Topological classification and… · p. 4
  • The generic-event and topology catalog can be saved, shared, reused, and merged across simulations to speed up new runs on the same system.The arrays storing events and topologies dominate memory use, requiring a balance between array size, RAM use, and hash collisions.Source: B. Topological classification and… · p. 4

C. Adaptive generic event relaxation into specific events

k-ART refines generic events into atom-specific events selectively, using an adaptive rate-based procedure to include local geometric and elastic effects without refining every event.

C. Adaptive generic event relaxation into specific events

  • Low-barrier generic events are re-converged for each realization as specific events, accounting for local positional variations and long-range elastic interactions in their barriers.Events for a topology are assumed accessible to all atoms sharing it, subject to these barrier adjustments.Source: C. Adaptive generic event relaxation… · p. 4
  • Specific saddle and final-state geometries are reconstructed for each atom by mapping the generic event’s displacement through nauty’s ordered cluster-atom list.Source: C. Adaptive generic event relaxation… · p. 4
  • Unlike the earlier criterion of selecting generic classes with barriers at or below 15 kBT, the procedure adapts relaxation to system kinetics.Source: C. Adaptive generic event relaxation… · p. 4
  • The adaptive procedure bins generic events in 0.1 eV barrier slices, relaxes from the lowest slice upward until 99.9% of total rate is covered, and copies the remainder as unrefined events.The histogram stores events by barrier range and supports cumulative-rate selection.Source: C. Adaptive generic event relaxation… · p. 4, Figure 2 · p. 4

D. Crucial aspects

k-ART addresses topology collisions and low-barrier trapping with topology refinement and basin-based rate acceleration. The latter preserves exit-state probabilities while omitting repeated basin-state visits.

1. Uniqueness in the correspondence between a geometry

  • When one topology maps to multiple geometries, event reconstruction fails; k-ART adjusts the cluster connectivity cutoff to split them into distinct classes.A marker tracks difficult topologies, and associated sub-topologies are tested by mapping events to select the correct catalog; collisions are rare with reasonable cutoffs and usually need one sub-topology.Source: §1 · pp. 4–5

2. Handling low-energy barriers

  • Low-barrier transitions within basins can shorten KMC time increments and consume resources exploring states without much information about effective diffusion.Source: §2 · p. 5
  • Bac-MRM accelerates basin dynamics by averaging jumps among transient states while retaining the correct probability of exit to absorbing states.A basin is identified when both forward and inverse barriers fall below a user-defined threshold; the barriers are measured from the saddle to the initial and final states, respectively.Source: §2 · p. 5
  • Bac-MRM builds basins on the fly, avoids intentionally revisiting states, and gives the exit-state distribution based on internal rates and the basin entry point, while losing internal dynamics.Its basin-operation overhead is negligible compared with exploring one topology.Source: §2 · p. 5

3. Optimizing k-ART for large scale systems

k-ART limits event-generation work to atoms relevant to local mechanisms, while retaining global minimization after each move for elastic effects. Tests on c-Si systems show a weak linear cost component and dominant sub-linear terms at larger sizes.

3. Optimizing k-ART for large scale systems

  • Event generation and event relaxation are local, with forces computed on involved atoms and first and second neighbors; a global minimization follows each KMC move to include elastic effects.Atoms exceeding the 0.01 eV/Å force threshold in c-Si are added with their first and second neighbors.Source: §3 · pp. 5–6
  • 0.03 s per atom is the weak linear component of new-topology exploration, while roughly 600 s of sub-linear work dominates for systems of several tens of thousands of atoms.The tests used 2,744 to 27,000 c-Si atoms; the sub-linear terms reflect 15 saddle-point searches per topology.Source: §3 · p. 6, Figure 3 · p. 6

IV. APPLICATION OF K-ART

The applications test k-ART on crystalline silicon defects, iron interstitial diffusion, and amorphous-silicon relaxation. The examples span defect motion and a disordered material.

IV. APPLICATION OF K-ART

  • The application set comprises vacancies and interstitials in c-Si, self-interstitial diffusion in Fe, and structural relaxation in a-Si.Source: §IV · p. 6
  • The initial c-Si configuration contains eight vacancies and eight interstitials in an 8,000-atom box; the figure distinguishes over-coordinated from under-coordinated atoms by sphere size.Source: Figure 4 · p. 6
  • The Fe system tests sampling of a complex diffusion landscape, while a-Si tests whether topology-based k-ART handles disordered materials.Source: §IV · p. 6

A. Vacancies and interstitials in Si

In crystalline silicon, k-ART captures interstitial-vacancy recombination and the changing local environments that shape defect motion. The simulations show that nearby vacancies can alter recombination pathways, while the event catalog grows as new topologies are visited.

  • Five large energy drops each mark an interstitial-vacancy pair annihilation during the 2000-step simulation.The simulation annealed eight vacancy-interstitial pairs in an 8000-atom c-Si box at 500 K.Source: A. Vacancies and interstitials in Si · p. 6
  • An isolated interstitial-vacancy pair visits several metastable bound states before recombination.This behavior is reported to agree with earlier findings by Tang et al. and Marqu5s et al..Source: A. Vacancies and interstitials in Si · p. 7
  • Nearby vacancies about 1 nm away introduce an oscillatory intermediate lasting nearly 45 ns before recombination.This case differs from the other recombinations, and the intermediate highlights the importance of long-range elastic and topological effects.Source: A. Vacancies and interstitials in Si · p. 7
  • Two pathways for a nearly isolated pair have barriers of 0.45, 0.39, 0.19, and 0.20 eV, versus 0.36, 0.19, 0.22, and 0.51 eV.The first pathway is reported in agreement with Gilmer et al..Source: A. Vacancies and interstitials in Si · p. 7
  • As recombination proceeds, sampled conformations grow from a few hundred to almost 20 000 topologies, whose stored events reduce CPU cost in later simulations.New topologies are encountered mainly as recombination begins, after few are seen during the first 20 ns of isolated diffusion.Source: A. Vacancies and interstitials in Si · pp. 7–8

B. Self-interstitial cluster diffusion in iron

Iron self-interstitial clusters offer many migration pathways and low-barrier state sequences that challenge catalogs assembled in advance. k-ART searches events on the fly and applies basin acceleration, recovering Johnson migration mechanisms and basin dynamics in the studied systems.

  • In a 1024-atom supercell, a single self-interstitial forms a ground-state [110] dumbbell; its lowest-barrier event is Johnson’s nearest-neighbor translation-rotation mechanism, with activation barrier 0.3 eV.Source: B. Self-interstitial cluster diffusion… · p. 8
  • For two clustered interstitials, k-ART recovers Johnson migration through nearest-neighbor translation-rotation jumps, either in one step or two sequential moves.The states and barriers agree with Marinica et al..Source: B. Self-interstitial cluster diffusion… · p. 8
  • The di-interstitial basin lies 0.25 eV above the ground state, with separating barriers below 0.1 eV and a nearly stalled simulation clock.The trajectory detail shows basin states separated by low barriers, with the system clock hardly moving during basin motion.Source: B. Self-interstitial cluster diffusion… · p. 8, Figure 9 · p. 9
  • A four-interstitial cluster has a basin of about 20 configurations formed by small dumbbell reorientations, separated by barriers below 0.1 eV.Source: B. Self-interstitial cluster diffusion… · p. 8
  • At 300 K, the 1024-atom di-interstitial cluster propagates mainly by the two-step Johnson process with an activation energy around 0.33 eV.The simulation used a pre-built catalog containing information about 9074 topologies and stalled when many new topologies had to be explored.Source: Figure 8 · p. 9
  • Low-barrier basins call for acceleration; with bac-MRM, the iron system's rich configuration and event space becomes tractable in KMC.The study notes that an a priori catalog would be virtually impossible to assemble for the SIA-cluster pathways.Source: B. Self-interstitial cluster diffusion… · p. 9

C. Relaxation dynamics in amorphous silicon

The section uses k-ART’s off-lattice, self-learning catalog to study long-timescale relaxation in a-Si, where defect roles remain debated. Over a 12 µs run, the system relaxes by more than 6 eV while coordination defects feature prominently in low-barrier events.

C. Relaxation dynamics in amorphous silicon

  • In a well-relaxed 1000-atom a-Si model, no two atoms share a topology and repeated topologies remain rare after many thousands of events, so a meaningful catalog requires multiple independent trajectories.The trajectories are started from various initial configurations.Source: C. Relaxation dynamics in amorphous… · p. 10
  • Small-displacement flickers usually generate fewer than 10 new topologies, while diffusive events can generate up to 140 in one step.The number of new topologies per step depends strongly on the preceding displacement amplitude.Source: C. Relaxation dynamics in amorphous… · p. 10
  • The mean-rate method uses a 0.3 eV basin threshold at 300 K, averaging events associated with timescales of 16 ns or less and ignoring their internal dynamics.The authors consider this acceptable for simulations targeting microseconds or longer.Source: C. Relaxation dynamics in amorphous… · p. 11
  • Over 12 µs and 3360 k-ART steps, a-Si relaxed by more than 6 eV while averaging 0.8 at.% defects, with most low-barrier events involving coordination defects.The defect concentration remained relatively constant throughout the simulation.Source: C. Relaxation dynamics in amorphous… · p. 11
  • A 4 eV drop at 10 µs began with a 0.28 eV bond-switching event that lowered energy by 0.84 eV, followed by 84 smaller relaxation events.The average defect population rose from 0.8 to 1.4 at.% during the process.Source: C. Relaxation dynamics in amorphous… · p. 11
  • In a 10 ns comparison at 300 K, molecular dynamics and k-ART showed atomic displacements of the same amplitude.The comparison found almost no deformation on this timescale.Source: C. Relaxation dynamics in amorphous… · p. 11
  • The a-Si simulation’s computational time was significant on a single CPU, and parallelization with a larger catalog was expected to reduce the computational effort.The reported run started from a preconstructed catalog on a single 2.66 GHz Intel Xeon X5550 CPU.Source: C. Relaxation dynamics in amorphous… · p. 11

V. CONCLUSION

Kinetic ART combines ART nouveau event generation and barrier calculation with nauty-based topological classification, alongside catalog and rate-handling improvements. The authors report that it handles complex systems with many tens of thousands of topologies more accurately than standard KMC and much faster than molecular dynamics.

V. CONCLUSION

  • Kinetic ART couples ART nouveau for event generation and barrier calculation with nauty-based topological classification of events.ART nouveau is described as an efficient, non-biased, open-ended method for finding transition states.Source: §V · p. 11
  • The reusable catalog stores generic events by topology, while low-energy barriers are separately relaxed to account for geometric and elastic deformations.K-ART also includes local force calculations for sub-linear scaling, exact flicker handling, and parallel event relaxation and generation.Source: §V · p. 11
  • K-ART handles complex systems with many tens of thousands of topologies more accurately than standard KMC and much faster than molecular dynamics.The paper demonstrates it on vacancy-interstitial annihilation in c-Si, interstitial diffusion in Fe, and a-Si relaxation.Source: §V · p. 11

Appendix: The Mean Rate Method

The mean-rate method models transitions among transient basin states and absorbing exit states, then uses basin-state residence times to modify exit rates for ordinary KMC.

Appendix: The Mean Rate Method

  • The transition probability matrix T represents transitions among basin and exit states, with rates normalized over all rates leaving each basin state.The method uses this matrix to determine the probability of exiting the basin to a given state.Source: Appendix: The Mean Rate Method · p. 11
  • Repeated application of T from the starting-state occupation vector gives basin-state occupation probabilities, whose sum over jumps yields average visits and residence times.The mean-rate calculation requires a single matrix inversion to obtain modified rates.Source: Appendix: The Mean Rate Method · pp. 11–12
  • The resulting residence times accelerate rates from basin states to exit states, after which standard KMC rules select the next step.Source: Appendix: The Mean Rate Method · p. 12
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