Origins of structural and electronic transitions in disordered silicon.

Structurally disordered materials pose fundamental questions1-4, including how different disordered phases ('polyamorphs') can coexist and transform from one phase to another5-9. Amorphous silicon has been extensively studied; it forms a fourfold-coordinated, covalent network at ambient conditions and much-higher-coordinated, metallic phases under pressure10-12. 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 experiments13-15 but not seen in earlier simulations11,16-18. 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.

High-dimensional order parameters and neural network classifiers applied to amorphous ices.

Amorphous ice phases are key constituents of water's complex structural landscape. This study investigates the polyamorphic nature of water, focusing on the complexities within low-density amorphous ice (LDA), high-density amorphous ice, and the recently discovered medium-density amorphous ice (MDA). We use rotationally invariant, high-dimensional order parameters to capture a wide spectrum of local symmetries for the characterization of local oxygen environments. We train a neural network to classify these local environments and investigate the distinctiveness of MDA within the structural landscape of amorphous ice. Our results highlight the difficulty in accurately differentiating MDA from LDA due to structural similarities. Beyond water, our methodology can be applied to investigate the structural properties and phases of disordered materials.

Understanding Defects in Amorphous Silicon with Million-Atom Simulations and Machine Learning.

The structure of amorphous silicon (a-Si) is widely thought of as a fourfold-connected random network, and yet it is defective atoms, with fewer or more than four bonds, that make it particularly interesting. Despite many attempts to explain such "dangling-bond" and "floating-bond" defects, respectively, a unified understanding is still missing. Here, we use advanced computational chemistry methods to reveal the complex structural and energetic landscape of defects in a-Si. We study an ultra-large-scale, quantum-accurate structural model containing a million atoms, and thousands of individual defects, allowing reliable defect-related statistics to be obtained. We combine structural descriptors and machine-learned atomic energies to develop a classification of the different types of defects in a-Si. The results suggest a revision of the established floating-bond model by showing that fivefold-bonded atoms in a-Si exhibit a wide range of local environments-analogous to fivefold centers in coordination chemistry. Furthermore, it is shown that fivefold (but not threefold) coordination defects tend to cluster together. Our study provides new insights into one of the most widely studied amorphous solids, and has general implications for understanding defects in disordered materials beyond silicon alone.

Gaussian Process Regression for Materials and Molecules.

We provide an introduction to Gaussian process regression (GPR) machine-learning methods in computational materials science and chemistry. The focus of the present review is on the regression of atomistic properties: in particular, on the construction of interatomic potentials, or force fields, in the Gaussian Approximation Potential (GAP) framework; beyond this, we also discuss the fitting of arbitrary scalar, vectorial, and tensorial quantities. Methodological aspects of reference data generation, representation, and regression, as well as the question of how a data-driven model may be validated, are reviewed and critically discussed. A survey of applications to a variety of research questions in chemistry and materials science illustrates the rapid growth in the field. A vision is outlined for the development of the methodology in the years to come.

Cross-platform hyperparameter optimization for machine learning interatomic potentials.

Machine-learning (ML)-based interatomic potentials are increasingly popular in material modeling, enabling highly accurate simulations with thousands and millions of atoms. However, the performance of machine-learned potentials depends strongly on the choice of hyperparameters-that is, of those parameters that are set before the model encounters data. This problem is particularly acute where hyperparameters have no intuitive physical interpretation and where the corresponding optimization space is large. Here, we describe an openly available Python package that facilitates hyperparameter optimization across different ML potential fitting frameworks. We discuss methodological aspects relating to the optimization itself and to the selection of validation data, and we show example applications. We expect this package to become part of a wider computational framework to speed up the mainstream adaptation of ML potentials in the physical sciences.

How to validate machine-learned interatomic potentials.

Machine learning (ML) approaches enable large-scale atomistic simulations with near-quantum-mechanical accuracy. With the growing availability of these methods, there arises a need for careful validation, particularly for physically agnostic models-that is, for potentials that extract the nature of atomic interactions from reference data. Here, we review the basic principles behind ML potentials and their validation for atomic-scale material modeling. We discuss the best practice in defining error metrics based on numerical performance, as well as physically guided validation. We give specific recommendations that we hope will be useful for the wider community, including those researchers who intend to use ML potentials for materials "off the shelf."

Machine-learned acceleration for molecular dynamics in CASTEP.

Machine learning (ML) methods are of rapidly growing interest for materials modeling, and yet, the use of ML interatomic potentials for new systems is often more demanding than that of established density-functional theory (DFT) packages. Here, we describe computational methodology to combine the CASTEP first-principles simulation software with the on-the-fly fitting and evaluation of ML interatomic potential models. Our approach is based on regular checking against DFT reference data, which provides a direct measure of the accuracy of the evolving ML model. We discuss the general framework and the specific solutions implemented, and we present an example application to high-temperature molecular-dynamics simulations of carbon nanostructures. The code is freely available for academic research.

Structure and Bonding in Amorphous Red Phosphorus.

Amorphous red phosphorus (a-P) is one of the remaining puzzling cases in the structural chemistry of the elements. Here, we elucidate the structure, stability, and chemical bonding in a-P from first principles, combining machine-learning and density-functional theory (DFT) methods. We show that a-P structures exist with a range of energies slightly higher than those of phosphorus nanorods, to which they are closely related, and that the stability of a-P is linked to the degree of structural relaxation and medium-range order. We thus complete the stability range of phosphorus allotropes [Angew. Chem. Int. Ed. 2014, 53, 11629] by now including the previously poorly understood amorphous phase, and we quantify the covalent and van der Waals interactions in all main phases of phosphorus. We also study the electronic densities of states, including those of hydrogenated a-P. Beyond the present study, our structural models are expected to enable wider-ranging first-principles investigations-for example, of a-P-based battery materials.

Indirect learning and physically guided validation of interatomic potential models.

Machine learning (ML) based interatomic potentials are emerging tools for material simulations, but require a trade-off between accuracy and speed. Here, we show how one can use one ML potential model to train another: we use an accurate, but more computationally expensive model to generate reference data (locations and labels) for a series of much faster potentials. Without the need for quantum-mechanical reference computations at the secondary stage, extensive reference datasets can be easily generated, and we find that this improves the quality of fast potentials with less flexible functional forms. We apply the technique to disordered silicon, including a simulation of vitrification and polycrystalline grain formation under pressure with a system size of a million atoms. Our work provides conceptual insight into the ML of interatomic potential models and suggests a route toward accelerated simulations of condensed-phase systems.

Mapping Materials and Molecules.

The visualization of data is indispensable in scientific research, from the early stages when human insight forms to the final step of communicating results. In computational physics, chemistry and materials science, it can be as simple as making a scatter plot or as straightforward as looking through the snapshots of atomic positions manually. However, as a result of the "big data" revolution, these conventional approaches are often inadequate. The widespread adoption of high-throughput computation for materials discovery and the associated community-wide repositories have given rise to data sets that contain an enormous number of compounds and atomic configurations. A typical data set contains thousands to millions of atomic structures, along with a diverse range of properties such as formation energies, band gaps, or bioactivities.It would thus be desirable to have a data-driven and automated framework for visualizing and analyzing such structural data sets. The key idea is to construct a low-dimensional representation of the data, which facilitates navigation, reveals underlying patterns, and helps to identify data points with unusual attributes. Such data-intensive maps, often employing machine learning methods, are appearing more and more frequently in the literature. However, to the wider community, it is not always transparent how these maps are made and how they should be interpreted. Furthermore, while these maps undoubtedly serve a decorative purpose in academic publications, it is not always apparent what extra information can be garnered from reading or making them.This Account attempts to answer such questions. We start with a concise summary of the theory of representing chemical environments, followed by the introduction of a simple yet practical conceptual approach for generating structure maps in a generic and automated manner. Such analysis and mapping is made nearly effortless by employing the newly developed software tool ASAP. To showcase the applicability to a wide variety of systems in chemistry and materials science, we provide several illustrative examples, including crystalline and amorphous materials, interfaces, and organic molecules. In these examples, the maps not only help to sift through large data sets but also reveal hidden patterns that could be easily missed using conventional analyses.The explosion in the amount of computed information in chemistry and materials science has made visualization into a science in itself. Not only have we benefited from exploiting these visualization methods in previous works, we also believe that the automated mapping of data sets will in turn stimulate further creativity and exploration, as well as ultimately feed back into future advances in the respective fields.

LOBSTER: Local orbital projections, atomic charges, and chemical-bonding analysis from projector-augmented-wave-based density-functional theory.

We present an update on recently developed methodology and functionality in the computer program Local Orbital Basis Suite Toward Electronic-Structure Reconstruction (LOBSTER) for chemical-bonding analysis in periodic systems. LOBSTER is based on an analytic projection from projector-augmented wave (PAW) density-functional theory (DFT) computations (Maintz et al., J. Comput. Chem. 2013, 34, 2557), reconstructing chemical information in terms of local, auxiliary atomic orbitals and thereby opening the output of PAW-based DFT codes to chemical interpretation. We demonstrate how LOBSTER has been improved by taking into account time-reversal symmetry, thereby speeding up the DFT and LOBSTER calculations by a factor of 2. Over the recent years, the functionalities have also been continually expanded, including accurate projected densities of states (DOSs), crystal orbital Hamilton population (COHP) analysis, atomic and orbital charges, gross populations, and the recently introduced k-dependent COHP. The software is offered free-of-charge for non-commercial research.

Phonon Spectroscopy in Antimony and Tellurium Oxides.

α-Sb2O3 (senarmontite), ß-Sb2O3 (valentinite), and α-TeO2 (paratellurite) are compounds with pronounced stereochemically active Sb and Te lone pairs. The vibrational and lattice properties of each have been previously studied but often lead to incomplete or unreliable results due to modes being inactive in infrared or Raman spectroscopy. Here, we present a study of the relationship between bonding and lattice dynamics of these compounds. Mössbauer spectroscopy is used to study the structure of Sb in α-Sb2O3 and ß-Sb2O3, whereas the vibrational modes of Sb and Te for each oxide are investigated using nuclear inelastic scattering, and further information on O vibrational modes is obtained using inelastic neutron scattering. Additionally, vibrational frequencies obtained by density functional theory (DFT) calculations are compared with experimental results in order to assess the validity of the utilized functional. Good agreement was found between DFT-calculated and experimental density of phonon states with a 7% scaling factor. The Sb-O-Sb wagging mode of α-Sb2O3 whose frequency was not clear in most previous studies is experimentally observed for the first time at ∼340 cm-1. Softer lattice vibrational modes occur in orthorhombic ß-Sb2O3 compared to cubic α-Sb2O3, indicating that the antimony bonds are weakened upon transforming from the molecular α phase to the layer-chained ß structure. The resulting vibrational entropy increase of 0.45 ± 0.1 kB/Sb2O3 at 880 K accounts for about half of the α-ß transition entropy. The comparison of experimental and theoretical approaches presented here provides a detailed picture of the lattice dynamics in these oxides beyond the zone center and shows that the accuracy of DFT is sufficient for future calculations of similar material structures.

Combining phonon accuracy with high transferability in Gaussian approximation potential models.

Machine learning driven interatomic potentials, including Gaussian approximation potential (GAP) models, are emerging tools for atomistic simulations. Here, we address the methodological question of how one can fit GAP models that accurately predict vibrational properties in specific regions of configuration space while retaining flexibility and transferability to others. We use an adaptive regularization of the GAP fit that scales with the absolute force magnitude on any given atom, thereby exploring the Bayesian interpretation of GAP regularization as an "expected error" and its impact on the prediction of physical properties for a material of interest. The approach enables excellent predictions of phonon modes (to within 0.1 THz-0.2 THz) for structurally diverse silicon allotropes, and it can be coupled with existing fitting databases for high transferability across different regions of configuration space, which we demonstrate for liquid and amorphous silicon. These findings and workflows are expected to be useful for GAP-driven materials modeling more generally.

An accurate and transferable machine learning potential for carbon.

We present an accurate machine learning (ML) model for atomistic simulations of carbon, constructed using the Gaussian approximation potential (GAP) methodology. The potential, named GAP-20, describes the properties of the bulk crystalline and amorphous phases, crystal surfaces, and defect structures with an accuracy approaching that of direct ab initio simulation, but at a significantly reduced cost. We combine structural databases for amorphous carbon and graphene, which we extend substantially by adding suitable configurations, for example, for defects in graphene and other nanostructures. The final potential is fitted to reference data computed using the optB88-vdW density functional theory (DFT) functional. Dispersion interactions, which are crucial to describe multilayer carbonaceous materials, are therefore implicitly included. We additionally account for long-range dispersion interactions using a semianalytical two-body term and show that an improved model can be obtained through an optimization of the many-body smooth overlap of atomic positions descriptor. We rigorously test the potential on lattice parameters, bond lengths, formation energies, and phonon dispersions of numerous carbon allotropes. We compare the formation energies of an extensive set of defect structures, surfaces, and surface reconstructions to DFT reference calculations. The present work demonstrates the ability to combine, in the same ML model, the previously attained flexibility required for amorphous carbon [V. L. Deringer and G. Csányi, Phys. Rev. B 95, 094203 (2017)] with the high numerical accuracy necessary for crystalline graphene [Rowe et al., Phys. Rev. B 97, 054303 (2018)], thereby providing an interatomic potential that will be applicable to a wide range of applications concerning diverse forms of bulk and nanostructured carbon.

Hierarchically Structured Allotropes of Phosphorus from Data-Driven Exploration.

The discovery of materials is increasingly guided by quantum-mechanical crystal-structure prediction, but the structural complexity in bulk and nanoscale materials remains a bottleneck. Here we demonstrate how data-driven approaches can vastly accelerate the search for complex structures, combining a machine-learning (ML) model for the potential-energy surface with efficient, fragment-based searching. We use the characteristic building units observed in Hittorf's and fibrous phosphorus to seed stochastic ("random") structure searches over hundreds of thousands of runs. Our study identifies a family of hierarchically structured allotropes based on a P8 cage as principal building unit, including one-dimensional (1D) single and double helix structures, nanowires, and two-dimensional (2D) phosphorene allotropes with square-lattice and kagome topologies. These findings yield new insight into the intriguingly diverse structural chemistry of phosphorus, and they provide an example for how ML methods may, in the long run, be expected to accelerate the discovery of hierarchical nanostructures.

Fast Lithium Ion Conduction in Lithium Phosphidoaluminates.

Solid electrolyte materials are crucial for the development of high-energy-density all-solid-state batteries (ASSB) using a nonflammable electrolyte. In order to retain a low lithium-ion transfer resistance, fast lithium ion conducting solid electrolytes are required. We report on the novel superionic conductor Li9 AlP4 which is easily synthesised from the elements via ball-milling and subsequent annealing at moderate temperatures and which is characterized by single-crystal and powder X-ray diffraction. This representative of the novel compound class of lithium phosphidoaluminates has, as an undoped material, a remarkable fast ionic conductivity of 3 mS cm-1 and a low activation energy of 29 kJ mol-1 as determined by impedance spectroscopy. Temperature-dependent 7 Li NMR spectroscopy supports the fast lithium motion. In addition, Li9 AlP4 combines a very high lithium content with a very low theoretical density of 1.703 g cm-3 . The distribution of the Li atoms over the diverse crystallographic positions between the [AlP4 ]9- tetrahedra is analyzed by means of DFT calculations.

Fast Ionic Conductivity in the Most Lithium-Rich Phosphidosilicate Li14SiP6.

Solid electrolytes with superionic conductivity are required as a main component for all-solid-state batteries. Here we present a novel solid electrolyte with three-dimensional conducting pathways based on "lithium-rich" phosphidosilicates with ionic conductivity of σ > 10-3 S cm-1 at room temperature and activation energy of 30-32 kJ mol-1 expanding the recently introduced family of lithium phosphidotetrelates. Aiming toward higher lithium ion conductivities, systematic investigations of lithium phosphidosilicates gave access to the so far lithium-richest compound within this class of materials. The crystalline material (space group Fm3m), which shows reversible thermal phase transitions, can be readily obtained by ball mill synthesis from the elements followed by moderate thermal treatment of the mixture. Lithium diffusion pathways via both tetrahedral and octahedral voids are analyzed by temperature-dependent powder neutron diffraction measurements in combination with maximum entropy method and DFT calculations. Moreover, the lithium ion mobility structurally indicated by a disordered Li/Si occupancy in the tetrahedral voids plus partially filled octahedral voids is studied by temperature-dependent impedance and 7Li NMR spectroscopy.

Quantifying Chemical Structure and Machine-Learned Atomic Energies in Amorphous and Liquid Silicon.

Amorphous materials are being described by increasingly powerful computer simulations, but new approaches are still needed to fully understand their intricate atomic structures. Here, we show how machine-learning-based techniques can give new, quantitative chemical insight into the atomic-scale structure of amorphous silicon (a-Si). We combine a quantitative description of the nearest- and next-nearest-neighbor structure with a quantitative description of local stability. The analysis is applied to an ensemble of a-Si networks in which we tailor the degree of ordering by varying the quench rates down to 1010  K s-1 . Our approach associates coordination defects in a-Si with distinct stability regions and it has also been applied to liquid Si, where it traces a clear-cut transition in local energies during vitrification. The method is straightforward and inexpensive to apply, and therefore expected to have more general significance for developing a quantitative understanding of liquid and amorphous states of matter.

Plane-Wave Density Functional Theory Meets Molecular Crystals: Thermal Ellipsoids and Intermolecular Interactions.

Molecular compounds, organic and inorganic, crystallize in diverse and complex structures. They continue to inspire synthetic efforts and "crystal engineering", with implications ranging from fundamental questions to pharmaceutical research. The structural complexity of molecular solids is linked with diverse intermolecular interactions: hydrogen bonding with all its facets, halogen bonding, and other secondary bonding mechanisms of recent interest (and debate). Today, high-resolution diffraction experiments allow unprecedented insight into the structures of molecular crystals. Despite their usefulness, however, these experiments also face problems: hydrogen atoms are challenging to locate, and thermal effects may complicate matters. Moreover, even if the structure of a crystal is precisely known, this does not yet reveal the nature and strength of the intermolecular forces that hold it together. In this Account, we show that periodic plane-wave-based density functional theory (DFT) can be a useful, and sometimes unexpected, complement to molecular crystallography. Initially developed in the solid-state physics communities to treat inorganic solids, periodic DFT can be applied to molecular crystals just as well: theoretical structural optimizations "help out" by accurately localizing the elusive hydrogen atoms, reaching neutron-diffraction quality with much less expensive measurement equipment. In addition, phonon computations, again developed by physicists, can quantify the thermal motion of atoms and thus predict anisotropic displacement parameters and ORTEP ellipsoids "from scratch". But the synergy between experiment and theory goes much further than that. Once a structure has been accurately determined, computations give new and detailed insights into the aforementioned intermolecular interactions. For example, it has been debated whether short hydrogen bonds in solids have covalent character, and we have added a new twist to this discussion using an orbital-based theory that once more had been developed for inorganic solids. However, there is more to a crystal structure than a handful of short contacts between neighboring residues. We hence have used dimensionally resolved analyses to dissect crystalline networks in a systematic fashion, one spatial direction at a time. Initially applied to hydrogen bonding, these techniques can be seamlessly extended to halogen, chalcogen, and pnictogen bonding, quantifying bond strength and cooperativity in truly infinite networks. Finally, these methods promise to be useful for (bio)polymers, as we have recently exemplified for α-chitin. At the interface of increasingly accurate and popular DFT methods, ever-improving crystallographic expertise, and new challenging, chemical questions, we believe that combined experimental and theoretical studies of molecular crystals are just beginning to pick up speed.

Data-Driven Learning of Total and Local Energies in Elemental Boron.

The allotropes of boron continue to challenge structural elucidation and solid-state theory. Here we use machine learning combined with random structure searching (RSS) algorithms to systematically construct an interatomic potential for boron. Starting from ensembles of randomized atomic configurations, we use alternating single-point quantum-mechanical energy and force computations, Gaussian approximation potential (GAP) fitting, and GAP-driven RSS to iteratively generate a representation of the element's potential-energy surface. Beyond the total energies of the very different boron allotropes, our model readily provides atom-resolved, local energies and thus deepened insight into the frustrated ß-rhombohedral boron structure. Our results open the door for the efficient and automated generation of GAPs, and other machine-learning-based interatomic potentials, and suggest their usefulness as a tool for materials discovery.