Origins of structural and electronic transitions in disordered silicon

Abstract

Structurally disordered materials pose fundamental questions1,2,3,4, including how different disordered phases (‘polyamorphs’) can coexist and transform from one phase to another5,6,7,8,9. Amorphous silicon has been extensively studied; it forms a fourfold-coordinated, covalent network at ambient conditions and much-higher-coordinated, metallic phases under pressure10,11,12. However, a detailed mechanistic understanding of the structural transitions in disordered silicon has been lacking, owing to the intrinsic limitations of even the most advanced experimental and computational techniques, for example, in terms of the system sizes accessible via simulation. Here we show how atomistic machine learning models trained on accurate quantum mechanical computations can help to describe liquid–amorphous and amorphous–amorphous transitions for a system of 100,000 atoms (ten-nanometre length scale), predicting structure, stability and electronic properties. Our simulations reveal a three-step transformation sequence for amorphous silicon under increasing external pressure. First, polyamorphic low- and high-density amorphous regions are found to coexist, rather than appearing sequentially. Then, we observe a structural collapse into a distinct very-high-density amorphous (VHDA) phase. Finally, our simulations indicate the transient nature of this VHDA phase: it rapidly nucleates crystallites, ultimately leading to the formation of a polycrystalline structure, consistent with experiments13,14,15 but not seen in earlier simulations11,16,17,18. A machine learning model for the electronic density of states confirms the onset of metallicity during VHDA formation and the subsequent crystallization. These results shed light on the liquid and amorphous states of silicon, and, in a wider context, they exemplify a machine learning-driven approach to predictive materials modelling.

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Fig. 1: Vitrification of supercooled liquid silicon.
Fig. 2: a-Si at high and very-high pressure.
Fig. 3: Electronic fingerprints of structural transitions.

Data availability

Original data supporting this work, including coordinates for all reported structural models, are openly available in the Zenodo repository (https://doi.org/10.5281/zenodo.4174139).

Code availability

The QUIP code, which provides the interface for carrying out GAP-driven simulations with LAMMPS, is publicly available at https://github.com/libAtoms/QUIP; additional information may be found there. The GAP code is available freely for non-commercial research at http://www.libatoms.org/gap/gap_download.html.

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Acknowledgements

V.L.D. acknowledges a Leverhulme Early Career Fellowship and support from the Isaac Newton Trust. Parts of the simulations reported here were carried out during his previous affiliation with the University of Cambridge (until August 2019). N.B. acknowledges support from the Office of Naval Research through the US Naval Research Laboratory’s core basic research programme, and computer time through the US DOD HPCMPO at the AFRL DSRC. D.A.D. acknowledges support from the US NSF under award DMR 1506836. M.C. and C.B.M. acknowledge support by the Swiss National Science Foundation (project no. 200021-182057), and by the NCCR MARVEL, funded by the Swiss National Science Foundation. This work used the ARCHER UK National Supercomputing Service via a Resource Allocation Panel award (project e599) and the UKCP consortium (EPSRC grant EP/P022596/1). All structural drawings were created using OVITO48. We thank A. P. Bartók for technical help.

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Authors

Contributions

V.L.D., G.C. and S.R.E. initiated the project. V.L.D. and N.B. performed the ambient-pressure simulations; V.L.D. performed the high-pressure simulations; V.L.D., N.B. and G.C. carried out further validation with other machine learning potentials. D.A.D. performed the tight-binding electronic-structure computations. C.B.M. and M.C. performed the electronic DOS machine learning predictions and the KPCovR analysis. V.L.D., M.W., D.A.D. and S.R.E. analysed the data and developed the main conclusions regarding high-pressure phases. All authors contributed to discussions. V.L.D. drafted the paper, and all authors contributed to its final version.

Corresponding author

Correspondence to Volker L. Deringer.

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Competing interests

G.C. is listed as an inventor on a patent filed by Cambridge Enterprise Ltd related to SOAP and GAP (US patent 8843509, filed on 5 June 2009 and published on 23 September 2014). The other authors declare no competing interests.

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Peer review information Nature thanks Davide Donadio, Paul McMillan and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Extended data figures and tables

Extended Data Fig. 1 Machine learning-driven modelling beyond the nanometric length scale.

The fully relaxed a-Si structure with 100,000 atoms is shown. The smaller boxes on the left show the size of a 512-atom system from a recent study26, marking the limit of current DFT methods for simulations over several nanoseconds, and that of a 4,096-atom system in our pilot GAP-MD studies26. All boxes are drawn to scale.

Extended Data Fig. 2 Compression of liquid silicon.

a, Overview of the transition pathways investigated in the present work. The quench (vitrification) and compression runs are discussed in the text. We focus here on additional data that we have collected for validation: namely, the description of the high-temperature liquid. We melted a structure at 1,800 K, above the melting point of diamond-type silicon, and then compressed it by simultaneously adapting the thermostat and barostat settings so as to trace the estimated phase coexistence lines given by Bundy85, in analogy to ref. 86. The temperature was reduced by 41 K GPa−1 to follow the diamond melting line, up to the estimated triple point at 15 GPa, after which the slope was inverted and followed the metallic silicon melting line (+14 K GPa−1)85. The compression rate was 0.5 GPa ps−1. b, Structure factors of liquid silicon during this compression run. Computed values from our simulations (red) are overlaid on experimental reference data by Funamori and Tsuji86 (black) for which the estimated temperatures are at about 50 K above the melting line86, closely mirrored by our computations. In the original experimental work86, arrows indicate the location of the maxima (labelled there as Q1 and Q2) and a shoulder in the first peak (Qh). The height of Qh gradually diminishes at higher pressure, and all these features are correctly described by our simulations. In the context of liquid–liquid transitions, we mention in passing very recent density-functional87 and empirical force-field based studies88; such research questions may become worthwhile targets for future GAP-driven studies as well. Reprinted figure with permission from ref. 86, copyright 2002 by the American Physical Society. Expt., experimental.

Extended Data Fig. 3 Vibrational densities of states (VDOS).

We obtained these by Fourier transformation of the velocity–velocity autocorrelation function, as described in Methods. Two characteristic features associated with the amorphous–amorphous transition under high pressure, observed in previous Raman spectroscopy experiments10,11,12, are reproduced by these simulations. First, the peak at large wavenumbers persists throughout the LDA/HDA coexistence but then disappears entirely. Second, the VHDA formation is associated with the formation of another peak at intermediate wavenumbers. It is noted that this feature appears in both the simulated VHDA and the polycrystalline sh system. The predictions here might be tested, in the future, by combined in situ X-ray diffraction and Raman spectroscopy; the former technique will easily be able to distinguish VHDA silicon from crystalline phases.

Extended Data Fig. 4 Reproducibility of VHDA formation with a separate machine learning potential.

a, Snapshots from a compression simulation using the same starting structure and protocol as for the main result (Fig. 2a–e), but now using a newly fitted GAP machine learning model based on SCAN meta-GGA input data (Methods). This simulation confirms the structural collapse at high pressure, seen in the third panel, and the subsequent crystallization. The SCAN-level machine learning potential initially nucleated β-Sn-like crystallites (N = 6; red colour on the atoms), which is explained in the following. b, Energy–volume curves for relevant crystalline allotropes of silicon, computed using the GAP-18 model (based on PW91 data; top) and the new SCAN-based model (bottom). In both cases, the sequence of dia → β-Sn-type → sh with increasing pressure (decreasing cell volume) is correctly reproduced, consistent with early DFT studies89. With SCAN, the β-Sn-like phase is favoured over a wider range of pressures; the crossover between the two E(V) curves is indicated by arrows in both panels. Note that the absolute energies for both allotropes are very similar, leading to a delicate balance between both. c, Oblique view of the simulation cell from the SCAN simulation after reaching 20 GPa. Initially, β-tin-like crystallites had formed (N = 6; red); then, an sh grain emerged (N = 8; orange). Note that the absolute pressure values at which the subsequent transitions occur are slightly different between the GAP-18 (Fig. 2a–e) and SCAN (Extended Data Fig. 4a) simulations, but the VHDA phases and subsequent formation of polycrystalline phases are clearly observed in both. The same is not the case with an established, empirically fitted interatomic potential, as shown in Extended Data Fig. 6b.

Extended Data Fig. 5 Beyond-DFT modelling with a Δ-GAP machine learning fit.

a, Schematic illustration of the approach, as discussed in Methods. The key ideas are that: (i) the RPA potential-energy surface (PES) can only be sampled at selected points, because of the computational cost, and that: (ii) the difference Δ(RPA–DFT), indicated by red shading, varies more smoothly than the full PES and is therefore more easily amenable to a machine learning fit. b, Example structural snapshots from a GAP-RSS search60. We use such very-small simulation cells to represent large structural diversity in machine learning potential fitting where computational cost is at a premium. c, Quality-of-fit for the difference model, shown in the form of a scatter plot for the training data (blue) and a separate test set (green) of the machine learning prediction (vertical axis) against the ‘ground truth’ to be learned (horizontal axis). The distribution of the target values, σ, is given at the top left, alongside the root mean square error (RMSE) measures for training and testing set. d, Snapshots from a compression simulation using the same starting structure and protocol as for the GAP-18 (Fig. 2a–e) and SCAN (Extended Data Fig. 4) results, but now using the GAP-18 + Δ-GAP(RPA–DFT) difference machine learning potential. The collapse into VHDA is clearly reproduced, as is the subsequent nucleation of crystallites; the result at 20 GPa is a poly-crystalline β-Sn-like phase (compare with Extended Data Fig. 4a).

Extended Data Fig. 6 Describing VHDA formation and crystallization requires quantum-accurate simulations.

a, The results of our machine learning-driven simulation, with the collapse to VHDA between 12 and 13 GPa, and the crystallization between 15 and 20 GPa (presented in Fig. 2 and shown here for comparison). b, As in a, now using the empirical Stillinger–Weber (SW) potential41, which had been the state of the art for 100,000-atom simulations of silicon so far. Here, neither VHDA formation nor the subsequent crystallization are observed.

Extended Data Fig. 7 The enthalpy landscape of metastable disordered forms of silicon.

a, Computed enthalpy of 100,000-atom systems, ΔH, given relative to the respective most stable crystalline form at any given pressure (diamond-type → β-Sn-type → sh); see Methods. The red line shows the result for snapshots along the 500 K compression trajectory. Square symbols indicate results for snapshots, which have been frozen-in by rapid molecular-dynamics quenching (over 1 ps) and subsequently relaxed with a conjugate-gradient algorithm, all at the given external pressure (relevant structures are visualized below). The shaded area is a guide to the eye and corresponds to the enthalpy difference between the 500 K and fully relaxed a-Si structures at 0 GPa. Relevant structures are shown: note the near-perfect ordering of layers in the polycrystalline (‘pc’) sample. est., estimated; at−1, per atom. b, Enthalpy changes associated with the structural changes during compression. Copies of the 10 GPa structure (LDA/HDA polyamorph) were relaxed with increased external pressure (open symbols); this direct relaxation fixes the structure in place and does not allow it to transform to VHDA. A direct comparison between two competing phases at 13 GPa is therefore possible (labelled as ΔH1) and indicates the preference for VHDA formation. The enthalpy is lowered much further upon crystallization (ΔH2). c, Relaxation of copies of the pc-sh structure with decreased external pressure, mirroring decompression of a sample in experiment. The relative enthalpic stability over a relatively wide pressure range is qualitatively consistent with the observation of a hysteresis upon decompression: for example, in a previous work10, the LDA phase was fully recovered only after decompression to about 4 GPa. A dashed vertical line in b and c emphasizes the change in the crystalline reference, from diamond-type (dia) to β-Sn-type silicon.

Extended Data Fig. 8 Computed structure factors.

a, The static structure factor, S(q), as a probe for medium-range structural order, has been evaluated for the fully relaxed amorphous system. The computed result, including the height of the first sharp diffraction peak (FSDP), is in excellent agreement with previous experimental data90. The inset shows a radial distribution function, g(r), for the same structure, indicating long-range correlations beyond the first nanometre, which our machine learning-driven simulations can access. A dashed line at approximately 11 Å illustrates the limit of DFT modelling (half the cell length of the smallest system sketched in Extended Data Fig. 1). Expt., experimental. b, Computed structure factors during quenching. Left, the evolution of simulated structure factors through the relevant part of the liquid-quenching trajectory in the vicinity of the glass transition, plotted in 1-K temperature increments. The emergence of the FSDP (between 1.5 and 2.0 Å−1), as well as the structuring of the third peak (between 5 and 6 Å−1), are clearly visible. Right, detail view for 1.0 Å−1 ≤ q ≤ 4.0 Å−1 of the evolution of the FSDP with decreasing temperature, using the same colour scale as on the left-hand side.

Extended Data Fig. 9 Tight-binding DOS for an ultralarge system.

Supplementing the tight-binding electronic-structure computations in Fig. 3a–d, this figure shows the electronic DOS computed with the same approach but for a diamond-type crystalline silicon supercell at atmospheric pressure, containing >2 million atoms (see details in Methods). The energy scale is set by ref. 70.

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Deringer, V.L., Bernstein, N., Csányi, G. et al. Origins of structural and electronic transitions in disordered silicon. Nature 589, 59–64 (2021). https://doi.org/10.1038/s41586-020-03072-z

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