A machine-learned spin-lattice potential for dynamic simulations of defective magnetic iron

www.nature.com/scientificreports OPEN A machine‑learned spin‑lattice potential for dynamic simulations of defective magnetic iron Jacob B. J. Chapman * & Pui‑Wai Ma A machine-learned spin-lattice interatomic potential (MSLP) for magnetic iron is developed and applied to mesoscopic scale defects. It is achieved by augmenting a spin-lattice Hamiltonian with a neural network term trained to descriptors representing a mix of local atomic configuration and magnetic environments. It reproduces the cohesive energy of BCC and FCC phases with various magnetic states. It predicts the formation energy and complex magnetic structure of point defects in quantitative agreement with density functional theory (DFT) including the reversal and quenching of magnetic moments near the core of defects. The Curie temperature is calculated through spin-lattice dynamics showing good computational stability at high temperature. The potential is applied to study magnetic fluctuations near sizable dislocation loops. The MSLP transcends current treatments using DFT and molecular dynamics, and surpasses other spin-lattice potentials that only treat near-perfect crystal cases. The success of density functional theory (DFT)1,2 has drastically advanced the scientific and technological aspects of materials development due to its unprecedented predictive power at a modest computational cost. However, the order O(n3 ) scalability of DFT calculations, where n is the number of electrons, has severely limited the simulation box size and time scale. Machine-learned potentials have demonstrated their ability to perform scalable atomic scale simulations with DFT accuracy using only a fraction of its computational r equirements3. Since the seminal work of Behler and Parrinello4, who introduced the concept of invariant descriptors to represent local chemical environment, a range of machine-learned potentials based on kernel methods5,6 and network networks7–10 have been developed and applied to investigate real physical problems. Spin-polarized and non-collinear magnetism are well established extensions of DFT for magnetic materials but their results are valid only for the electronic ground state. Attempts to mimic magnetic excitation by coupling spin dynamics to constrained non-collinear calculations have been m ade11,12. However, the limitations of the DFT method on the simulation box size has yet to be overcome. In addition the effects of magnetic excitation and their interaction with atomic trajectories are irreconcilable within the framework of classical molecular dynamics (MD)13. Nevertheless, magnetic effects cannot be ignored in many situations. In magnetic iron, the bcc-fcc and fccbcc phase transitions at 1185K and 1667K, respectively, are due to the competing phonon and magnon free energies14–18. The softening of tetragonal shear modulus C ′ near the Curie temperature TC19,20 and stability of anomalous 110 dumbbell self-interstitial atom (SIA) configurations21–23 are also believed to be magnetically driven. Itinerant ferromagnetism, in the form of increased magnitudes of magnetic moment, have been linked to the stability of grain boundaries and intergranular cohesion24. Spin-lattice dynamics25 was developed to treat both spin (magnetic) and lattice degrees of freedom within a unified framework. Spin-lattice dynamics is a general framework similar to molecular dynamics and applicable to any arbitrary atomic scale Hamiltonian. The latest development on the Langevin spin equation of m otion26 allows simultaneous treatment of both the rotational (direction) and longitudinal fluctuations (magnitude) of magnetic moments. In most other studies the magnitudes of magnetic moments are assumed to be fi xed27–29 or have been performed on a fixed lattice30,31. Whilst spin-lattice dynamics has been used to investigate a variety of microscopic dynamic effects in iron14,25,27–29,32,33, there is still not a spin-lattice potential capable of simultaneously modelling mechanical deformations, magnetic fluctuations and defect p roperties13. The difficulty of developing spin-lattice potentials are two-fold. First, a spin-lattice potential has double the degrees of freedoms (6N) of a conventional MD potential (3N), where N is the number of atoms. A substantial amount of extra data is required for potential fitting for each extra degree of freedom, drastically expending the United Kingdom Atomic Energy Authority, Culham Science Centre, Abingdon, Oxfordshire OX14 3DB, UK. *email: Scientific Reports | (2022) 12:22451 | https://doi.org/10.1038/s41598-022-25682-5 1 Vol.:(0123456789) www.nature.com/scientificreports/ MSLP BCC FCC DFT (VASP) a0 (Å) |M| (µB) FM 2.817 2.16 SL-AFM 2.824 1.54 NM 2.753 0.00 DL-AFM 3.470 SL-AFM E (eV/atom) DFT (OpenMX) a0 (Å) |M| (µB) E (eV/atom) a0 (Å) 2.831 2.19 |M| (µB) 2.842 2.25 0.36 2.800 1.34 0.46 0.42 2.764 0.00 2.08 0.08 3.466 2.04 3.494 0.96 0.16 3.494 FM 3.47 1.03 0.15 NM 3.428 0.00 0.18 E (eV/atom) 0.47 2.766 0.00 0.56 0.08 3.476 2.38 0.10 1.30 0.12 3.435 2.00 0.13 3.50 1.00 0.16 3.648 2.63 0.12 3.456 0.00 0.16 3.462 0.00 0.25 Table 1.  The equilibrium lattice constant a0, the magnitude of spontaneous magnetic moment |M|, and the relative energy difference with respect to the BCC ground state E calculated using our machine-learned spinlattice potential (MSLP) for iron at non-magnetic (NM), ferromagnetic (FM), single layer antiferromagnetic (SL-AFM), and double layer antiferromagnetic (DL-AFM) states in BCC and FCC structures. DFT calculations using VASP and OpenMX are shown for comparison. Details are in Supplementary Materials. representable phase space. Recent data-driven techniques can aid in parameter optimisation for such cases33. Second, potentials that adopt the Heisenberg or Heisenberg-Landau functional form in various s tudies23 are shown to be too restrictive to near-perfect crystal cases. A good functional form that is applicable to both perfect and defective configurations is yet to be derived. Machine-learned potentials for spin-lattice dynamics that go beyond the need of a well defined functional form could be a viable s olution10,34. While the number of machine-learned potentials for iron has rapidly increased over the past d ecade3,35–37, applications including explicit spin degrees of freedoms are very limited. Recently, Nikolov et al.33 produced a machine-learned spectral neighbor analysis potential. Since they kept using the Heisenberg functional form, the potential does not consider the change of the magnitudes of magnetic moments due to thermal excitation or the change of local atomic environment. Novikov et al.38 developed a moment tensor spin-lattice potential that includes longitudinal fluctuation, but they limited their approach to collinear configurations near perfect crystal structures. Domina et al.34 extended the spectral-neighbour representation to be applicable (...truncated)