The development of permanent magnets containing less or no rareearth elements is linked to profound knowledge of the coercivity mechanism. Prerequisites for a promising permanent magnet material are a high spontaneous magnetization and a sufficiently high magnetic anisotropy. In addition to the intrinsic magnetic properties the microstructure of the magnet plays a significant role in establishing coercivity. The influence of the microstructure on coercivity, remanence, and energy density product can be understood by using micromagnetic simulations. With advances in computer hardware and numerical methods, hysteresis curves of magnets can be computed quickly so that the simulations can readily provide guidance for the development of permanent magnets. The potential of rare-earth reduced and free permanent magnets is investigated using micromagnetic simulations. The results show excellent hard magnetic properties can be achieved in grain boundary engineered NdFeB, rare-earth magnets with a ThMn 12 structure, Co-based nano-wires, and L1 0 -FeNi provided that the magnet's microstructure is optimized.

International audienceThe coercive field and angular dependence of the coercive field of single-grain Nd$_{2}$Fe$_{14}$B permanent magnets are computed using finite element micromagnetics. It is shown that the thickness of surface defects plays a critical role in determining the reversal process. For small defect thicknesses reversal is heavily driven by nucleation, whereas with increasing defect thickness domain wall de-pinning becomes more important. This change results in an observable shift between two well-known behavioral models. A similar trend is observed in experimental measurements of bulk samples, where a Nd-Cu infiltration process has been used to enhance coercivity by modifying the grain boundaries. When account is taken of the imperfect grain alignment of real magnets, the single-grain computed results appears to closely match experimental behaviour

Conjugate gradient methods for energy minimization in micromagnetics are compared. When the step length in the line search is controlled, conjugate gradient techniques are a fast and reliable way to compute the hysteresis properties of permanent magnets. The method is applied to investigate demagnetizing effects in NdFe12 based permanent magnets. The reduction of the coercive field by demagnetizing effects is µ0H = 1.4 T at 450 K. I. INTRODUCTIONThe computation of hysteresis properties of large ferromagnetic systems such as sensor elements or permanent magnets require fast and reliable solvers. Hysteresis simulations are based on the theory of micromagnetics "Brown (1963)". The primary purpose of these simulations is to understand the influence of the microstructure on magnetization reversal. In this work we are focusing on the role of demagnetizing fields in platelet shaped grains of permanent magnets. We also describe the key elements of a micromagnetic solver suitable for simulating large magnetic systems.After discretization of the total Gibbs free energy with finite elements or finite differences the states along the demagnetization curve can be computed by subsequent minimization of the energy for decreasing applied field as outlined in "Kinderlehrer (1997)". The system is in a metastable state. A change of the applied field shifts the position of the local energy minimum. At a critical field, the magnetization becomes unstable. An irreversible switching occurs which is seen as a kink in the demagnetization curve. Then the system either accesses a different metastable state or if fully reversed the magnetization is in a stable state. A reliable numerical method for energy minimization must track all local minima along the demagnetization curve. The resulting algebraic minimization problem is large. Typically the number of unknowns is in the order of 10 to 50 million for a model magnet consisting of around 10 grains. Therefore fast numerical methods are required to obtain results in a Electronic

First order reversal curve (FORC) diagrams are a useful tool to analyze the magnetization processes in magnetic materials. FORC diagrams are computed from measured first order reversal curves on sintered Nd 2 Fe 14 B magnets. It is shown that the FORC diagram simplifies if the first order reversal curves a desheared using the macroscopic demagnetizing field given by the geometry of the sample. Furthermore the resulting FORC diagram is almost identical to the FORC diagram measured for a thin platelet of the same material. This opens the possibility to compare experimental FORC diagrams with FORC diagrams computed by micromagnetic simulations.

We use a machine learning approach to identify the importance of microstructure characteristics in causing magnetization reversal in ideally structured large-grained Nd 2 Fe 14 B permanent magnets. The embedded Stoner-Wohlfarth method is used as a reduced order model for determining local switching field maps which guide the data-driven learning procedure. The predictor model is a random forest classifier which we validate by comparing with full micromagnetic simulations in the case of small granular test structures. In the course of the machine learning microstructure analysis the most important features explaining magnetization reversal were found to be the misorientation and the position of the grain within the magnet. The lowest switching fields occur near the top and bottom edges of the magnet. While the dependence of the local switching field on the grain orientation is known from theory, the influence of the position of the grain on the local coercive field strength is less obvious. As a direct result of our findings of the machine learning analysis we show that edge hardening via Dy-diffusion leads to higher coercive fields.

Deep neural networks are used to model the magnetization dynamics in magnetic thin film elements. The magnetic states of a thin film element can be represented in a low dimensional space. With convolutional autoencoders a compression ratio of 1024:1 was achieved. Time integration can be performed in the latent space with a second network which was trained by solutions of the Landau-Lifshitz-Gilbert equation. Thus the magnetic response to an external field can be computed quickly.

The coercive field of permanent magnets decays with temperature. At non-zero temperature the system can overcome a finite energy barrier through thermal fluctuations. Using finite element micromagnetic simulations, we quantify this effect, which reduces coercivity in addition to the decrease of the coercive field associated with the temperature dependence of the anisotropy field, and validate the method through comparison with existing experimental data.

The maximum coercivity that can be achieved for a given hard magnetic alloy is estimated by computing the energy barrier for the nucleation of a reversed domain in an idealized microstructure without any structural defects and without any soft magnetic secondary phases. For Sm 1−z Zr z (Fe 1−y Co y ) 12−x Ti x based alloys, which are considered an alternative to Nd 2 Fe 14 B magnets with lower rare-earth content, the coercive field of a small magnetic cube is reduced to 60 percent of the anisotropy field at room temperature and to 50 percent of the anisotropy field at elevated temperature (473K). This decrease of the coercive field is caused by misorientation, demagnetizing fields and thermal fluctuations.PACS numbers: 75.50. Ww,75.60 Permanent magnets are an important material for energy conversion in modern technologies. Wind power as well as hybrid and electric vehicles require high performance permanent magnets. In motor applications the magnet should retain a high magnetization and coercive field at an operating temperature around 450 K. At this temperature the magnetization and the anisotropy field of Sm 1−z Zr z (Fe 1−y Co y ) 12−x Ti x are higher than those of Nd 2 Fe 14 B.