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Constructing first-principles phase diagrams of amorphous LixSi using machine-learning-assisted sampling with an evolutionary algorithm

Nongnuch Artrith, Alexander Urban, Gerbrand Ceder

arXiv:1802.03548v1cond-mat.dis-nnphysics.comp-ph

TL;DR

Amorphous materials are difficult to model with first-principles methods because the required structure sizes and sampling statistics are challenging to attain. This work combines a genetic algorithm with a specialized machine-learning potential to sample amorphous LixSi; its resulting phase diagram differs from molecular-dynamics sampling and is closer to the experimental average voltage profile, though hysteresis precludes quantitative comparison.

  • Problem

    First-principles modeling of amorphous materials is challenged by the structure sizes and sampling statistics needed to represent their complex atomic configurations.

    Source: §I · p. 1

  • Method

    The approach combines a genetic algorithm with a specialized machine-learning potential to sample low-energy structures in a limited domain, then evaluates relevant structures with DFT.

    Source: §I · p. 1, §V · p. 8

  • Results

    The GA-derived formation energies yield a qualitatively and quantitatively different 0 K phase diagram from MD sampling, and its voltage profile is closer to the experimental average despite hysteresis preventing quantitative comparison.

    Source: §V · p. 8

  • Takeaways & Limitations

    A specialized machine-learning potential can be sufficient for structural sampling within a limited domain, and the approach is not restricted to delithiation or genetic algorithms.

    Source: §V · p. 8, §VI · p. 9

    GA sampling does not represent real-time dynamics and assumes quasi-equilibration at each composition.

    Source: §V · p. 8

Abstract

from arXiv · show

The atomistic modeling of amorphous materials requires structure sizes and sampling statistics that are challenging to achieve with first-principles methods. Here, we propose a methodology to speed up the sampling of amorphous and disordered materials using a combination of a genetic algorithm and a specialized machine-learning potential based on artificial neural networks (ANN). We show for the example of the amorphous LiSi alloy that around 1,000 first-principles calculations are sufficient for the ANN potential assisted sampling of low-energy atomic configurations in the entire amorphous LixSi phase space. The obtained phase diagram is validated by comparison with the results from an extensive sampling of LixSi configurations using molecular dynamics simulations and a general ANN potential trained to ~45,000 first-principles calculations. This demonstrates the utility of the approach for the first-principles modeling of amorphous materials.

I. INTRODUCTION

Amorphous structures are difficult to characterize and costly to sample with first-principles methods, while their varied local motifs complicate complete training databases. The paper proposes specialized machine-learning-assisted sampling with a genetic algorithm, followed by DFT evaluation, and applies it to larger amorphous LixSi models.

I. INTRODUCTION

  • Amorphous structures lack long-range order, limiting conventional diffraction, while DFT's cost restricts model size and sampling.Pair-distribution measurements provide indirect structural information, and the high cost of DFT makes many amorphous or disordered phases challenging to model.Source: §I · p. 1
  • Amorphous phases' varied local motifs make the extensive reference databases needed for reliable machine-learning potentials especially challenging to compile.Machine-learning potentials can interpolate first-principles energies and forces efficiently, but may require thousands to tens of thousands of reference points.Source: §I · p. 1
  • The proposed approach uses a specialized MLP trained on around one thousand DFT calculations to guide a genetic algorithm toward near-ground-state structures, then recomputes relevant structures with DFT.A general MLP and molecular-dynamics simulations are used to validate the approximate sampling approach.Source: §I · p. 1
  • The study applies the approach to LixSi using structure models with more than 600 atoms to examine amorphous LiSi energetics.The paper identifies LixSi as a prototypical amorphous electrode material.Source: §I · p. 1

A. Density-functional theory calculations

  • DFT calculations use PBE and PAW pseudopotentials, generally with a 520 eV plane-wave cutoff and energy convergence of 0.05 meV/atom.The stated force convergence threshold is 50 meV/Å.Source: A. Density-functional theory… · p. 2

B. Machine-learning potentials with Chebyshev descriptor

The paper's machine-learning potentials interpolate first-principles energies and forces using atomic-energy neural networks. Their energy is assembled from local-environment contributions, for which a constant-size descriptor is needed despite variation in neighborhood size.

B. Machine-learning potentials with Chebyshev descriptor

  • Machine-learning potentials interpolate first-principles energies and forces, and can approach reference-method accuracy at lower computational cost within their trained structure space.The paper uses artificial neural networks to interpolate atomic energies, with total structure energy given by the sum of atomic energies.Source: B. Machine-learning potentials with… · p. 2
  • The number of atoms in an atom's cutoff neighborhood varies with structure density, motivating a constant-size local-environment descriptor for the ANN input.The local environment includes coordinates and chemical species of atoms within radial cutoff Rc.Source: B. Machine-learning potentials with… · p. 2
  • The total ANN energy is formed from atomic-energy predictions for each atom's chemical species.The passage identifies ANN_ti as the atomic-energy ANN potential for species ti.Source: B. Machine-learning potentials with… · p. 2
  • The ANN has two hidden layers of 15 nodes each, 44 descriptor inputs, and 931 parameters; training uses a held-out 10% test set and selects the best of ten fits.The ten fits use different randomly initialized parameters and are trained with L-BFGS.Source: B. Machine-learning potentials with… · p. 2

1. Descriptor of the local atomic environment

The local-environment descriptor represents radial and angular structure with an expanded, constant-size feature set. Its construction separates structural features from chemical-species information and is designed to satisfy the invariances of atomic energy.

1. Descriptor of the local atomic environment

  • The descriptor must have constant dimension and remain invariant to rotation, translation, and exchange of equivalent atoms.Raw coordinates are unsuitable as direct ANN inputs because the number of atoms within the interaction range varies with structure and density.Source: §1 · p. 2
  • Expanding local radial and angular distribution functions in an orthonormal basis yields an approximate constant-size representation, with descriptor complexity that does not increase with species count.The approach uses expansions of RDF and ADF within cutoff radius Rc.Source: §1 · pp. 2–3
  • Separate coefficient sets encode structure alone and structure plus species, using equal species weights for the former and distinct weights for the latter.For LiSi, the selected weights are wLi = -1 and wSi = +1; 11 Chebyshev polynomials per radial and angular expansion give a combined descriptor dimension of 44.Source: §1 · p. 3

A SPECIALIZED ANN POTENTIAL

The study focuses on amorphous LixSi, which forms during room-temperature lithiation and delithiation rather than following the stable crystalline phases. To model battery-anode amorphization, it simulates lithium extraction from crystalline Li15Si4 with relaxation at intermediate compositions.

A SPECIALIZED ANN POTENTIAL

  • During room-temperature lithiation and delithiation, metastable amorphous a-LixSi structures form instead of the known stable crystalline LiSi phases.The amorphous structures crystallize into metastable c-Li15Si4 when the Li potential falls to approximately 50 mV versus Li+/Li.Source: A SPECIALIZED ANN POTENTIAL · p. 3
  • The modeled amorphous phase is a metastable state associated with battery cycling, not the stable crystalline ground-state phases.The passage distinguishes the stable crystalline phases from the metastable amorphous structures formed during room-temperature lithiation and delithiation.Source: A SPECIALIZED ANN POTENTIAL · p. 3
  • The sampling setup models battery-anode amorphization by extracting Li computationally from fully lithiated crystalline Li15Si4 and relaxing positions and cell parameters at intermediate compositions.Source: A SPECIALIZED ANN POTENTIAL · p. 3

A. A specialized ANN potential for the sampling of near-ground-state LixSi structures

The paper uses a specialized ANN potential to guide genetic-algorithm sampling of low-energy Li/vacancy arrangements in amorphous LixSi. Its training is limited to structures related to crystalline LixSi, so its suitability is specific to this sampling task.

A. A specialized ANN potential for the sampling of near-ground-state LixSi structures

  • The specialized ANN potential guides an evolutionary algorithm in sampling Li/vacancy orderings for metastable amorphous LixSi compositions.The algorithm is used to determine the structures at different compositions.Source: A. A specialized ANN potential for the… · p. 3
  • Its training set contains DFT calculations for structures related to crystalline LixSi, including distorted crystals and structures with random Li and Si vacancies.The initial reference set comprised 725 structures.Source: A. A specialized ANN potential for the… · p. 3
  • Because it may not reproduce energetics for motifs unlike crystalline structures, the potential is hypothesized to be suitable specifically for sampling near-ground-state Li/vacancy arrangements.The stated out-of-domain examples include different atomic coordinations and much shorter bond lengths.Source: A. A specialized ANN potential for the… · p. 3

B. Sampling amorphous LixSi structures with a genetic algorithm

The genetic algorithm searches Li/vacancy arrangements at successive delithiation steps, then uses DFT to relax the lowest-energy candidates. The study applies this workflow to two cell sizes and reports both an optimization example and the number of structures selected for DFT evaluation.

B. Sampling amorphous LixSi structures with a genetic algorithm

  • At each delithiation step, the GA identifies stable Li/vacancy configurations, and DFT relaxes at least 30 candidates before the lowest-energy structure seeds the next step.This cycle is repeated for successive compositions.Source: B. Sampling amorphous LixSi structures… · p. 4
  • The GA represents configurations using Li sites, generates trial structures by randomly removing Li, and favors lower-energy trials during selection.Selected trials are crossed and mutated to produce additional configurations.Source: B. Sampling amorphous LixSi structures… · p. 4
  • The study used 32 trials per population, a 10% mutation rate, and at least 30 DFT-optimized configurations at each composition.These settings specify the population and candidate counts for the reported GA workflow.Source: B. Sampling amorphous LixSi structures… · p. 4
  • The workflow delithiated 76-atom Li60Si16 cells in 2-Li intervals and 608-atom Li480Si128 cells in 8-Li steps.Both supercells began from crystalline Li15Si4.Source: B. Sampling amorphous LixSi structures… · p. 4
  • After around 500 GA steps for Li360Si128, the energy changed by only around 13 meV/LixSi, indicating reasonable convergence.Figure 2 plots the best, average, and maximum energies of the GA population during optimization.Source: B. Sampling amorphous LixSi structures… · p. 5, Figure 2 · p. 5
  • 1,263 structures from GA sampling were selected for DFT evaluation and geometry optimization, forming the basis of the first-principles phase diagram with the initial reference set.This is the reported total selected from the GA sampling.Source: B. Sampling amorphous LixSi structures… · p. 5

IV. MOLECULAR DYNAMICS SAMPLING WITH A GENERAL ANN POTENTIAL

The paper notes that GA sampling omits structural relaxation during the search and uses an ANN potential specialized for that task. It checks whether the resulting structures are low-energy metastable amorphous phases by comparing the phase diagram with heat-quench molecular-dynamics sampling.

IV. MOLECULAR DYNAMICS SAMPLING WITH A GENERAL ANN POTENTIAL

  • GA sampling omits structural relaxation during its search, although the final 30 or more low-energy configurations are fully optimized.This is one of the two approximations identified for the GA methodology.Source: §IV · p. 5
  • The specialized ANN potential used for GA sampling is not considered suitable for other applications.The study compares the resulting phase diagram with heat-quench MD sampling to check whether the GA structures are genuinely low-energy metastable amorphous structures.Source: §IV · p. 5
  • The reported MD simulations used a 2 fs time step, the Verlet algorithm, and a Parrinello-Bussi thermostat in the NVT ensemble.These are the stated MD simulation settings.Source: §IV · p. 5

A. ANN potential construction and molecular dynamics simulations

The paper constructs a general ANN potential from a diverse set of LixSi structures and iteratively augments its reference data with MD-generated structures. It reports low test-set errors and agreement with DFT formation-energy features across the reference structures.

A. ANN potential construction and molecular dynamics simulations

  • Training included bulk, slab, and cluster structures so the MD potential would encounter unusual coordinations, unphysical bonding situations, and lattice parameters.Clusters up to about 200 atoms and structures up to 608 atoms were considered.Source: A. ANN potential construction and… · p. 5
  • Short high-temperature MD runs amorphized input structures, followed by 2 ns simulations at 400–1200 K; 400 snapshots per trajectory were recomputed with DFT.The snapshots were spaced every 5 ps along each trajectory and added to the reference data set.Source: A. ANN potential construction and… · p. 5
  • MD and ANN retraining were repeated until the energy RMSE was low and ANN and DFT ground states agreed.The potential training was iteratively based on MD-generated structures.Source: A. ANN potential construction and… · pp. 5–6
  • The general ANN potential was trained on about 45,000 reference structures, with 4,500 held out for testing and the best of ten trained potentials selected.The selection criterion was the smallest overall error relative to DFT reference energies.Source: A. ANN potential construction and… · p. 6
  • The selected potential achieved test-set RMSE 7.7 meV/atom and MAE 5.9 meV/atom.Its training-set RMSE and MAE were 6.3 and 5.7 meV/atom, respectively.Source: A. ANN potential construction and… · p. 6
  • The general potential reproduced the features of DFT formation energies across the reference structures, as compared in Figure 3a.The general potential was used here to validate approximate GA sampling, while its broader use for other LiSi properties was left to future work.Source: A. ANN potential construction and… · p. 6, Figure 3 · p. 6

B. Comparison of the amorphous phase space sampled by GA and MD

Across the considered LixSi compositions, GA sampling produced consistently lower-energy structures than MD, with GA structures within around 100 meV/atom of the lowest energy for each composition. This supports using the approach to identify low-energy metastable amorphous phases without a fully converged general ANN potential when that is the sole objective.

B. Comparison of the amorphous phase space sampled by GA and MD

  • GA sampling yielded consistently lower-energy structures than MD sampling at all considered LixSi compositions.The comparison used independently sampled structure sets.Source: B. Comparison of the amorphous phase… · p. 6
  • Around 100 meV/atom above the lowest energy for each composition, GA structures remained within a small energy range.The comparison concerns all a-LixSi structures considered.Source: B. Comparison of the amorphous phase… · p. 6
  • ANN-potential-assisted GA sampling identified low-energy metastable amorphous phases without requiring a fully converged general ANN potential for that objective.Source: B. Comparison of the amorphous phase… · p. 6
  • Figure 4 superposes RDFs from 1,000 randomly selected structures for each sampling approach, with GA above MD.Source: Figure 4 · p. 7

V. DISCUSSION

GA and MD sampling produce distinct amorphous LixSi structures and phase diagrams: GA finds lower-energy structures, while its computed voltage profile is closer to the experimental average. Which sampling strategy is appropriate depends on the application, and GA sampling represents neither real-time dynamics nor every amorphization mechanism.

V. DISCUSSION

  • In this electrochemical amorphization example, simulated delithiation more closely resembles the actual mechanism than heat-quenching a melt.The comparison also depends on sampling conditions: heat-quench outcomes vary with cooling rate and simulated timescale, and rapid quenching can trap high-energy states.Source: §V · p. 7
  • MD heat-quench structures mostly lose correlations beyond the first coordination shell, while GA structures retain stronger long-range correlations resembling crystalline LiSi.At 8 Å, MD-sampled RDFs approach 1; the contrast is attributed likely to rapid cooling versus lithium extraction without temperature effects.Source: §V · p. 7
  • The computed T = 0 K conversion voltage uses first-principles energy differences relative to Li metal, with phase energies taken from the lower convex hull.The composition axis is normalized to Li per Si, directly reflecting battery capacity per Si atom; voltage steps are expected to smooth at higher temperatures.Source: §V · p. 7, Figure 5 · p. 8
  • MD-sampled structures have higher energies at low Li contents and a steeper voltage profile than GA structures, whose profile is closer to the experimental average.Experimental hysteresis prevents a quantitative comparison with the computed voltages.Source: Figure 5 · p. 8, §V · p. 8
  • The sampling approach is not restricted to delithiation or genetic algorithms; off-stoichiometry amorphization and Monte Carlo sampling are also identified as alternatives.GA sampling does not describe real-time dynamics and assumes quasi-equilibration at each composition.Source: §V · pp. 8–9

VI. CONCLUSIONS

For amorphous LiSi, the authors combine a genetic algorithm with an artificial neural network potential to speed first-principles sampling. They report successful identification of low-energy amorphous structures and greater computational efficiency than constructing a converged general model.

VI. CONCLUSIONS

  • The genetic algorithm and artificial neural network potential together successfully identify low-energy amorphous structures.Source: §VI · p. 9
  • The approach combines a genetic algorithm with an artificial neural network potential to speed first-principles sampling of complex structure spaces.Source: §VI · p. 9
  • The authors report greater computational efficiency than constructing a converged general model.Source: §VI · p. 9
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