Modelling Magnetism of hcp Iron under Earth’s Inner Core Conditions : Based on first-principle DFT calculations and Machine Learning

Abstract: The structure of Earth’s core remains largely a mystery. The solid inner core is believed to exist in extreme pressure and temperature conditions comparable to 300 GPa and 6000 K and consists mainly of iron, Fe. Physical samples evidently cannot be collected, thereby making theoretical models increasingly vital in the pursuit of expanding our understanding of the Earth’s core. In this study the hexagonal close-packed (hcp) structure of Fe is investigated at 300 GPa and 6000 K, conditions comparable to Earth’s inner core conditions. Accurately modelling the dynamics of hcp Fe under Earth’s core conditions with careful inclusion of electronic structure magnetism and vibrational effects is a very difficult task and has not been done before. This master thesis provides results derived from a methodology which includes temperature induced magnetic moments, where vibrational effects are appropriately coupled with all magnetic degrees of freedom in a final dynamic simulation, called ASD-AIMD-MLLSF simulation. An interstitial step in this process includes the training of a machine learning (ML) model, utilizing kernel ridge regression (KRR).  Local magnetic moments, their arrangements including long and short range order are the basis for the magnetic properties of most materials. To fully model the dynamics of a magnetic system it is necessary to couple the magnetic degrees of freedom, described in terms of transversal spin fluctuations (TSF) and longitudinal spin fluctuations (LSF), and the vibrational degrees of freedom. In practice, due to the computationally expensive nature of the problem, this work includes a proposed machine learning interstitial step to predict the interaction between LSF and lattice vibrations.  The initial part of this work has demonstrated, via first-principle calculations based on density functional theory (DFT), that if temperature is neglected, iron converges to non- magnetic solutions at extreme pressures similar to Earth’s core conditions. However, this approach does not consider the temperature induced local magnetic moments. After inclusion of such contributions in addition to all other magnetic degrees of freedom coupled with vibrational degrees of freedom the ASD-AIMD-MLLSF simulation reveals a significant nonzero local magnetic moment of roughly 1.54 μB to 1.56 μB in magnitude. Where, based on calculated LSF landscapes for some geometrical snapshots on average, an expected local magnetic moment size of 1.43 μB is estimated. In addition, this work predicts a density of 13.04 g/cm3 for hcp Fe at 300 GPa and 6000 K (based on nonmagnetic ab initio molecular dynamics calculations). This density is close to the expected density in literature for Earth’s inner core, with a difference of around 9.8% higher than the density according to preliminary reference earth model (PREM) at 300 GPa. Furthermore, the ASD-AIMD-MLLSF simulation predicts a density lower than this value, which may be in better agreement with PREM. This result supports that hcp Fe is a relevant candidate in Earth’s inner core. The optimal ML models developed in relation to this thesis work generate a prediction of local magnetic moment magnitudes with a mean absolute error of 0.0667 μB. Finally, the LSF energy landscapes for some geometrical snapshots of hcp Fe under 300 GPa and 6000 K are found to be very similar and also matches in shape with results for bcc Fe at 300 GPa and 6000 K.