Constrained DFT-based magnetic machine-learning potentials for magnetic alloys: a case study of Fe-Al.

We propose a machine-learning interatomic potential for multi-component magnetic materials. In this potential we consider magnetic moments as degrees of freedom (features) along with atomic positions, atomic types, and lattice vectors. We create a training set with constrained DFT (cDFT) that allows us to calculate energies of configurations with non-equilibrium (excited) magnetic moments and, thus, it is possible to construct the training set in a wide configuration space with great variety of non-equilibrium atomic positions, magnetic moments, and lattice vectors. Such a training set makes possible to fit reliable potentials that will allow us to predict properties of configurations in the excited states (including the ones with non-equilibrium magnetic moments). We verify the trained potentials on the system of bcc Fe-Al with different concentrations of Al and Fe and different ways Al and Fe atoms occupy the supercell sites. Here, we show that the formation energies, the equilibrium lattice parameters, and the total magnetic moments of the unit cell for different Fe-Al structures calculated with machine-learning potentials are in good correspondence with the ones obtained with DFT. We also demonstrate that the theoretical calculations conducted in this study qualitatively reproduce the experimentally-observed anomalous volume-composition dependence in the Fe-Al system.

Atomistic modeling of the mechanical properties: the rise of machine learning interatomic potentials.

Since the birth of the concept of machine learning interatomic potentials (MLIPs) in 2007, a growing interest has been developed in the replacement of empirical interatomic potentials (EIPs) with MLIPs, in order to conduct more accurate and reliable molecular dynamics calculations. As an exciting novel progress, in the last couple of years the applications of MLIPs have been extended towards the analysis of mechanical and failure responses, providing novel opportunities not heretofore efficiently achievable, neither by EIPs nor by density functional theory (DFT) calculations. In this minireview, we first briefly discuss the basic concepts of MLIPs and outline popular strategies for developing a MLIP. Next, by considering several examples of recent studies, the robustness of MLIPs in the analysis of the mechanical properties will be highlighted, and their advantages over EIP and DFT methods will be emphasized. MLIPs furthermore offer astonishing capabilities to combine the robustness of the DFT method with continuum mechanics, enabling the first-principles multiscale modeling of mechanical properties of nanostructures at the continuum level. Last but not least, the common challenges of MLIP-based molecular dynamics simulations of mechanical properties are outlined and suggestions for future investigations are proposed.

Constrained Density Functional Theory: A Potential-Based Self-Consistency Approach.

Chemical reactions, charge transfer reactions, and magnetic materials are notoriously difficult to describe within Kohn-Sham density functional theory, which is strictly a ground-state technique. However, over the last few decades, an approximate method known as constrained density functional theory (cDFT) has been developed to model low-lying excitations linked to charge transfer or spin fluctuations. Nevertheless, despite becoming very popular due to its versatility, low computational cost, and availability in numerous software applications, none of the previous cDFT implementations is strictly similar to the corresponding ground-state self-consistent density functional theory: the target value of constraints (e.g., local magnetization) is not treated equivalently with atomic positions or lattice parameters. In the present work, by considering a potential-based formulation of the self-consistency problem, the cDFT is recast in the same framework as Kohn-Sham DFT: a new functional of the potential that includes the constraints is proposed, where the constraints, the atomic positions, or the lattice parameters are treated all alike, while all other ingredients of the usual potential-based DFT algorithms are unchanged, thanks to the formulation of the adequate residual. Tests of this approach for the case of spin constraints (collinear and noncollinear) and charge constraints are performed. Expressions for the derivatives with respect to constraints (e.g., the spin torque) for the atomic forces and the stress tensor in cDFT are provided. The latter allows one to study striction effects as a function of the angle between spins. We apply this formalism to body-centered cubic iron and first reproduce the well-known magnetization amplitude as a function of the angle between local magnetizations. We also study stress as a function of such an angle. Then, the local collinear magnetization and the local atomic charge are varied together. Since the atomic spin magnetizations, local atomic charges, atomic positions, and lattice parameters are treated on an equal footing, this formalism is an ideal starting point for the generation of model Hamiltonians and machine-learning potentials, computation of second or third derivatives of the energy as delivered from density-functional perturbation theory, or for second-principles approaches.

Highly anisotropic mechanical and optical properties of 2D NbOX2 (X = Cl, Br, I) revealed by first-principle.

In the latest experimental success, NbOI2two-dimensional (2D) crystals with anisotropic electronic and optical properties have been fabricated (Adv. Mater.33 (2021), 2101505). In this work inspired by the aforementioned accomplishment, we conduct first-principles calculations to explore the mechanical, electronic, and optical properties of NbOX2(X = Cl, Br, I) nanosheets. We show that individual layers in these systems are weakly bonded, with exfoliation energies of 0.22, 0.23, and 0.24 J m-2, for the isolation of the NbOCl2, NbOBr2,and NbOI2monolayers, respectively, distinctly lower than those of the graphene. The optoelectronic properties of the single-layer, bilayer, and bulk NbOCl2, NbOBr2,and NbOI2crystals are investigated via density functional theory calculations with the HSE06 approach. Our results indicate that the layered bulk NbOCl2, NbOBr2,and NbOI2crystals are indirect gap semiconductors, with band gaps of 1.79, 1.69, and 1.60 eV, respectively. We found a slight increase in the electronic gap for the monolayer and bilayer systems due to electron confinement at the nanoscale. Our results show that the monolayer and bilayer of these novel 2D compounds show suitable valence and conduction band edge positions for visible-light-driven water splitting reactions. The first absorption peaks of these novel monolayers along the in-plane polarization are located in the visible range of light which can be a promising feature to design advanced nanoelectronics. We found that the studied 2D systems exhibit highly anisotropic mechanical and optical properties. The presented first-principles results provide a comprehensive vision about direction-dependent mechanical and optical properties of NbOX2(X = Cl, Br, I) nanosheets.

A first-principles and machine-learning investigation on the electronic, photocatalytic, mechanical and heat conduction properties of nanoporous C5N monolayers.

Carbon nitride nanomembranes are currently among the most appealing two-dimensional (2D) materials. As a nonstop endeavor in this field, a novel 2D fused aromatic nanoporous network with a C5N stoichiometry has been most recently synthesized. Inspired by this experimental advance and exciting physics of nanoporous carbon nitrides, herein we conduct extensive density functional theory calculations to explore the electronic, optical and photocatalytic properties of the C5N monolayer. In order to examine the dynamic stability and evaluate the mechanical and heat transport properties under ambient conditions, we employ state of the art methods on the basis of machine-learning interatomic potentials. The C5N monolayer is found to be a direct band gap semiconductor, with a band-gap of 2.63 eV according to the HSE06 method. The obtained results confirm the dynamic stability, remarkable tensile strengths over 10 GPa and a low lattice thermal conductivity of ∼9.5 W m-1 K-1 for the C5N monolayer at room temperature. The first absorption peak of the single-layer C5N along the in-plane polarization is predicted to appear in the visible range of light. With a combination of high carrier mobility, appropriate band edge positions and strong absorption of visible light, the C5N monolayer might be an appealing candidate for photocatalytic water splitting reactions. The presented results provide an extensive understanding concerning the critical physical properties of the C5N nanosheets and also highlight the robustness of machine-learning interatomic potentials in the exploration of complex physical behaviors.

Nanohardness from First Principles with Active Learning on Atomic Environments.

We propose a methodology for the calculation of nanohardness by atomistic simulations of nanoindentation. The methodology is enabled by machine-learning interatomic potentials fitted on the fly to quantum-mechanical calculations of local fragments of the large nanoindentation simulation. We test our methodology by calculating nanohardness, as a function of load and crystallographic orientation of the surface, of diamond, AlN, SiC, BC2N, and Si and comparing it to the calibrated values of the macro- and microhardness. The observed agreement between the computational and experimental results from the literature provides evidence that our method has sufficient predictive power to open up the possibility of designing materials with exceptional hardness directly from first principles. It will be especially valuable at the nanoscale where the experimental measurements are difficult, while empirical models fitted to macrohardness are, as a rule, inapplicable.

First-Principles Multiscale Modeling of Mechanical Properties in Graphene/Borophene Heterostructures Empowered by Machine-Learning Interatomic Potentials.

Density functional theory calculations are robust tools to explore the mechanical properties of pristine structures at their ground state but become exceedingly expensive for large systems at finite temperatures. Classical molecular dynamics (CMD) simulations offer the possibility to study larger systems at elevated temperatures, but they require accurate interatomic potentials. Herein the authors propose the concept of first-principles multiscale modeling of mechanical properties, where ab initio level of accuracy is hierarchically bridged to explore the mechanical/failure response of macroscopic systems. It is demonstrated that machine-learning interatomic potentials (MLIPs) fitted to ab initio datasets play a pivotal role in achieving this goal. To practically illustrate this novel possibility, the mechanical/failure response of graphene/borophene coplanar heterostructures is examined. It is shown that MLIPs conveniently outperform popular CMD models for graphene and borophene and they can evaluate the mechanical properties of pristine and heterostructure phases at room temperature. Based on the information provided by the MLIP-based CMD, continuum models of heterostructures using the finite element method can be constructed. The study highlights that MLIPs were the missing block for conducting first-principles multiscale modeling, and their employment empowers a straightforward route to bridge ab initio level accuracy and flexibility to explore the mechanical/failure response of nanostructures at continuum scale.

Young's Modulus and Tensile Strength of Ti3C2 MXene Nanosheets As Revealed by In Situ TEM Probing, AFM Nanomechanical Mapping, and Theoretical Calculations.

Two-dimensional transition metal carbides, that is, MXenes and especially Ti3C2, attract attention due to their excellent combination of properties. Ti3C2 nanosheets could be the material of choice for future flexible electronics, energy storage, and electromechanical nanodevices. There has been limited information available on the mechanical properties of Ti3C2, which is essential for their utilization. We have fabricated Ti3C2 nanosheets and studied their mechanical properties using direct in situ tensile tests inside a transmission electron microscope, quantitative nanomechanical mapping, and theoretical calculations employing machine-learning derived potentials. Young's modulus in the direction perpendicular to the Ti3C2 basal plane was found to be 80-100 GPa. The tensile strength of Ti3C2 nanosheets reached up to 670 MPa for ∼40 nm thin nanoflakes, while a strong dependence of tensile strength on nanosheet thickness was demonstrated. Theoretical calculations allowed us to study mechanical characteristics of Ti3C2 as a function of nanosheet geometrical parameters and structural defect concentration.

Performance and Cost Assessment of Machine Learning Interatomic Potentials.

Machine learning of the quantitative relationship between local environment descriptors and the potential energy surface of a system of atoms has emerged as a new frontier in the development of interatomic potentials (IAPs). Here, we present a comprehensive evaluation of machine learning IAPs (ML-IAPs) based on four local environment descriptors-atom-centered symmetry functions (ACSF), smooth overlap of atomic positions (SOAP), the spectral neighbor analysis potential (SNAP) bispectrum components, and moment tensors-using a diverse data set generated using high-throughput density functional theory (DFT) calculations. The data set comprising bcc (Li, Mo) and fcc (Cu, Ni) metals and diamond group IV semiconductors (Si, Ge) is chosen to span a range of crystal structures and bonding. All descriptors studied show excellent performance in predicting energies and forces far surpassing that of classical IAPs, as well as predicting properties such as elastic constants and phonon dispersion curves. We observe a general trade-off between accuracy and the degrees of freedom of each model and, consequently, computational cost. We will discuss these trade-offs in the context of model selection for molecular dynamics and other applications.

Ring polymer molecular dynamics and active learning of moment tensor potential for gas-phase barrierless reactions: Application to S + H2.

Ring polymer molecular dynamics (RPMD) has proven to be an accurate approach for calculating thermal rate coefficients of various chemical reactions. For wider application of this methodology, efficient ways to generate the underlying full-dimensional potential energy surfaces (PESs) and the corresponding energy gradients are required. Recently, we have proposed a fully automated procedure based on combining the original RPMDrate code with active learning for PES on-the-fly using moment tensor potential and successfully applied it to two representative thermally activated chemical reactions [I. S. Novikov et al., Phys. Chem. Chem. Phys. 20, 29503-29512 (2018)]. In this work, using a prototype insertion chemical reaction S + H2, we show that this procedure works equally well for another class of chemical reactions. We find that the corresponding PES can be generated by fitting to less than 1500 automatically generated structures, while the RPMD rate coefficients show deviation from the reference values within the typical convergence error of the RPMDrate. We note that more structures are accumulated during the real-time propagation of the dynamic factor (the recrossing factor) as opposed to the previous study. We also observe that a relatively flat free energy profile along the reaction coordinate before entering the complex-formation well can cause issues with locating the maximum of the free energy surface for less converged PESs. However, the final RPMD rate coefficient is independent of the position of the dividing surface that makes it invulnerable to this problem, keeping the total number of necessary structures within a few thousand. Our work concludes that, in the future, the proposed methodology can be applied to realistic complex chemical reactions with various energy profiles.

Machine learning of molecular properties: Locality and active learning.

In recent years, the machine learning techniques have shown great potent1ial in various problems from a multitude of disciplines, including materials design and drug discovery. The high computational speed on the one hand and the accuracy comparable to that of density functional theory on another hand make machine learning algorithms efficient for high-throughput screening through chemical and configurational space. However, the machine learning algorithms available in the literature require large training datasets to reach the chemical accuracy and also show large errors for the so-called outliers-the out-of-sample molecules, not well-represented in the training set. In the present paper, we propose a new machine learning algorithm for predicting molecular properties that addresses these two issues: it is based on a local model of interatomic interactions providing high accuracy when trained on relatively small training sets and an active learning algorithm of optimally choosing the training set that significantly reduces the errors for the outliers. We compare our model to the other state-of-the-art algorithms from the literature on the widely used benchmark tests.