Investigating magnetic materials at realistic conditions with first-principles methods is a challenging task due to the interplay of vibrational and magnetic degrees of freedom. The most difficult contribution to include in simulations is represented by the longitudinal magnetic degrees of freedom [longitudinal spin fluctuation (LSF)] due to their inherent many-body nature; nonetheless, schemes that enable to take into account this effect on a semiclassical level have been proposed and employed in the investigation of magnetic systems. However, assessment of the effect of vibrations on LSF is lacking in the literature. For this reason, in this work we develop a supercell approach within the framework of constrained density functional theory to calculate self-consistently the size of local-environment-dependent magnetic moments in the paramagnetic, high-temperature state in the presence of lattice vibrations and for liquid Fe in different conditions. First, we consider the case of bcc Fe at the Curie temperature and ambient pressure. Then, we perform a similar analysis on bcc Fe at Earths inner-core conditions, and we find that LSFs stabilize nonzero moments which affect atomic forces and the electronic density of states of the system. Finally, we employ the present scheme on liquid Fe at the melting point at ambient pressure and at Earths outer-core conditions (p approximate to 200 GPa, T approximate to 6000 K). In both cases, we obtain local magnetic moments of sizes comparable to the solid-state counterparts.

We propose a machine learning approach to predict the shapes of the longitudinal spin fluctuation (LSF) energy landscapes for each local magnetic moment. This approach allows the inclusion of the effects of LSFs in, e.g., the simulation of a magnetic material with ab initio molecular dynamics in an effective way. This type of simulation requires knowledge of the reciprocal interaction between atoms and moments, which, in principle, would entail calculating the energy landscape of each atom at every instant in time. The machine learning approach is based on the kernel ridge regression method and developed using bcc Fe at the Curie temperature and ambient pressure as a test case. We apply the trained machine learning models in a combined atomistic spin dynamics and ab initio molecular dynamics (ASD-AIMD) simulation, where they are used to determine the sizes of the magnetic moments of every atom at each time step. In addition to running an ASD-AIMD simulation with the LSF machine learning approach for bcc Fe at the Curie temperature, we also simulate Fe at temperature and pressure comparable to the conditions at the Earth's inner solid core. The latter simulation serves as a critical test of the generality of the method and demonstrates the importance of the magnetic effects in Fe in the Earth's core despite its extreme temperature and pressure.

We develop a technique for predicting the Curie temperature of magnetic materials using density functional theory calculations suitable to include in high-throughput frameworks. We apply four different models, including physically relevant observables, and assess numerical constants by studying 32 ferro- and ferrimagnets. With the best-performing model, the Curie temperature can be predicted with a mean absolute error of approximately 126 K. As predictive factors, the models consider either the energy differences between the magnetic ground state and a magnetically disordered paramagnetic state, or the average constraining fields acting on magnetic moments in a disordered local moments calculation. Additionally, the energy differences are refined by incorporating the magnetic entropy of the paramagnetic state and the number of nearest magnetic neighbors of the magnetic atoms. The most advanced model is found to extend well into Fe1-xCox alloys, indicating the potential efficacy of utilizing our model in designing materials with tailored Curie temperatures by altering alloy compositions. This examination can illuminate the factors influencing magnetic transition temperatures in magnetic materials and provide insights into how they can be employed to make quantitative predictions of Curie temperatures. Our approach is not restricted to specific crystal structures or chemical compositions. It offers a more cost-effective alternative, in terms of human time and need for hands-on oversight, to other density functional theory methods for predicting the Curie temperature. As a result, it provides a practical strategy for conducting high-throughput screening for new technologically applicable magnetic materials. Alternatively, it can complement ML-based screening of magnetic materials by integrating physical principles into such approaches, thereby enhancing their prediction accuracy.

When exploring new magnetic materials, the effect of alloying plays a crucial role for numerous properties. By altering the alloy composition, it is possible to tailor, e.g., the Curie temperature (TC). In this work, TC of various alloys is investigated using a previously developed technique (Brännvall et al., 2024) designed for robust predictions of TC across diverse chemistries and structures. The technique is based on density functional theory calculations and utilizes the energy difference between the magnetic ground state and the magnetically disordered paramagnetic state. It also accounts for the magnetic entropy in the paramagnetic state and the number of nearest magnetic neighbors. The experimentally known systems, Fe_1−xCo_x, Fe_1−xCr_x, Fe_1−xV_x, NiMnSb-based Heusler alloys, Ti_1−xCr_xN, and Co_1−xAl_x are investigated. The experimentally unexplored system Fe_1−xTc_x is also tested to demonstrate the usefulness of the developed method in guiding future experimental efforts. This work demonstrates the broad applicability of the developed method across various systems, requiring less hands-on adjustments compared to other theoretical approaches.