Reduced order model for hard magnetic films Open Access

Permanent magnets can be found in a large variety of applications, e.g., in electronic and medical devices. Especially due to the transition to green technologies, high-performance permanent magnets are needed for windmills and (hybrid) electric cars. The permanent magnets must withstand high opposing magnetic fields at elevated temperatures. The required high coercive fields are generally achieved in rare-earth based magnets like NdFeB-type magnets. However, the energy transition increases the demand for rare-earth elements like Nd drastically, making it a critical material in regards of supply and prices. Therefore, to ensure the success of these new technologies, the content of rare-earth elements in permanent magnets needs to be reduced while maintaining their high performance.

To improve magnet performance, optimizing microstructural properties, like grain size distribution or shape, is crucial. Introducing Dy along grain boundaries reduces heavy rare-earth content in NdFeB grains, while maintaining high coercivity.¹ Another approach is grain size refinement or substituting Nd by less critical elements like La or Ce while maintaining high coercive fields at elevated temperatures.² Tang et al. showed experimentally and by numerical approaches that thermal stability of coercivity can be tuned by changing the aspect ratio of grains or by engineering a nonmagnetic intergranular phase.³ Vial et al. used an optimised heat treatment to create a nonmagnetic rare-earth-rich grain boundary phase displaying increased coercivity.⁴ The influence of the grain boundary phase on the coercivity was investigated by Chen et al.⁵ using micromagnetic simulations with a system of about a micrometer in size.

Only recently, data-driven approaches have been used in the workflow of magnetic materials design.⁶ Reliable and quantitative models for coercivity of permanent magnets require well prepared training data, ideally from experimental and fast simulation results.

Hard magnetic films pose as beneficial model systems since they offer the advantage of controlling microstructural properties by adjusting growth parameters⁷ diverse material compositions.⁸ 

NdFeB-based films typically show grain sizes of up to several micrometers. Therefore, a computational examination of structural properties like grain aspect ratio distribution need geometrical models of at least 1 µm in size. This would result in hundreds of millions of finite elements which requires supercomputing systems and huge amounts of computing resources. To generate a significant amount of data through micromagnetic simulations for large multigrain structures observed in hard magnetic films, it is necessary to reduce the complexity of the simulation models. A reduced order approach based on the embedded Stoner-Wohlfarth model has been established⁹ which allows to investigate granular structures of films at their actual size.

In this work we will give a brief overview of the reduced order model and demonstrate its capabilities by investigating the impact of nonmagnetic grain boundary phase thickness and grain aspect ratio of hard magnetic films. The magnetic systems we investigate consist of realistically shaped grains with a mean grain size of 375 nm and on average, were computed within 6 min.

The reduced order model (ROM) is based on four assumptions: i) the grains are homogeneously magnetised before and after switching. The switching process itself happens by nucleation of a domain wall and immediate propagation through the whole grain.¹⁰ Hence this is not reflected in the ROM and ii) the entire grain is switched in a single step. iii) Magnetisation reversal always starts at the grains’ surface. This has been observed in both micromagnetic simulations and experimental studies.¹¹ In our first implementation of the ROM iv) the grains are decoupled by a nonmagnetic boundary phase.

In classical finite element micromagnetism, the magnetisation of the system is defined by the magnetic moments on the nodes of the mesh. Because of the first assumption the magnetic moments can be reduced to just one magnetic moment per grain. For each grain we have to detect if magnetic reversal happens at a certain applied field. Since reversal starts at the border of the grains, we only need a 2D-mesh close to the surface. This means a substantial reduction of required computation resources compared to full micromagnetic simulations. In Fig. 1 the difference between the two meshes is shown by the sketches of the grains on the left. Manually testing 2D meshes of various sizes for the ROM showed that using 25 evaluation points per grain is sufficient to replicate the reversal curve calculated by the full micromagnetic model.

The demagnetisation curve of a multigrain system is computed by applying an external field H_ext in the same direction as the system is magnetically saturated. H_ext is reversed step by step in the opposite direction until the system reverses as well. For each field step, H_total is evaluated in all embedded SW particles of all grains. If a grain includes a particle which shows H_SW < H_total, it gets reversed. If multiple grains include particles which meet this requirement, the one with the smallest difference between H_SW and H_total is switched. Sequentially, H_total of all particles need to be recomputed. For each field step, the reversal of grains and recomputation of the entire system is done iteratively until a magnetic equilibrium state is reached. After that the external field is further reduced by one step.

In full micromagnetics for permanent magnets we are interested in the local configuration of the magnetization as a response to an externally applied field. For this purpose time-dependent magnetization dynamics is irrelevant and it is sufficient to compute stable or meta-stable equilibrium states by minimizing the total Gibb’s free energy. For micromagnetic minimization, the magnitude of the magnetization has to be constant over the entire system and depends on the temperature, which is constant in time and space. To validate our ROM we use a micromagnetic finite-element code for energy minimization which takes advantage of massively parallized graphical processing units.¹⁷ We compute the demagnetisation curve of a Nd₂Fe₁₄B cube with 150 nm edge length with both models. Typical intrinsic properties for this materials are used with μ₀M_s = 1.61 T, K₁=4.9 × 10⁶ J m⁻³ and A_exch=8 × 10⁻¹² J m⁻¹. The magnetocrystalline anisotropic easy axes of the 36 grains are uniformly distributed with a maximal deviation angle of 30° with respect to the applied field. The grains, column-shaped as observed in thin hard magnetic films, are assumed to be separated by a nonmagnetic boundary phase. The microstructure, intrinsic properties, and simulation setup are the same for both the micromagnetic and the ROM model. We begin by saturating the cube magnetically in the z-direction and then apply an increasing external field in the opposite direction in 5 mT increments from 0 to −6 T. The reversible part of the curve, up to −3.5 T, is tracked perfectly by the ROM (see Fig. 2). Each small step in the curve represents a switch of a single grain. The overall conformity of both curves is good and only small deviations can be observed (see third snapshot). The computation time is drastically reduced and depends on the number of grains and the embedded Stoner-Wohlfarth particles per grain. For the full micromagnetic model the computation time is 202 minutes (GPU). For the ROM an average of 25 evaluation points per grain were calculated, resulting in a computation time of 6 minutes (CPU). This is a nearly 34-fold speed up.

After describing the model and demonstrating its accuracy for the calculation of coercivity, we increase the grains to an average size of 375 nm and investigate two microstructural features: i) the impact of the thickness of a nonmagnetic grain boundary phase between the grains, and ii) the influence of the aspect ratio of the grains on the coercivity of a magnetic system. For both investigations we start by generating cuboid models with one micron edge-length, tesselated into 36 grains by the software Neper.¹⁸ The geometric models are then modified to introduce the grain boundary phase and meshed by Salome.¹⁹ The demagnetization curves are computed by the ROM and the coercive field, defined as the field which is needed to reduce the magnetisation from saturation to zero in the applied field’s direction, is recorded for each variation and parameter set.

At first we investigate the impact of the grain boundary phase thickness on the coercivity. Experimentally, heat treatments have been proposed to introduce non-magnetic grain boundaries by creating Nd-rich phases.⁴ The ROM enables us to quickly explore the possibilities of this approach for large multigrain systems.

Figure 3 shows the coercivity as a function of the grain boundary thickness. Each dot represents the mean coercivity of ten simulated random structure variations. The software Neper offers the possibility to create random granular structures by Voronoi tessellation. While the structures vary in the size, shape, and position of the individual grains, the quantity, average grain size, and aspect ratio of the grains are maintained. This is done to mitigate the finite size effect. The grain boundary phase thickness is increased in 1 nm-steps from 2 nm to 20 nm. The coercivity is rising from approximately 3.65 T to 4 T in this range. The dependency is not linear, it is starting with a steep gradient and getting more and more flat as the grain boundary phase gets thicker. This trend was also observed by Chen et al.⁵ A possible explanation is the reduced influence of the stray field with increasing distance between grains. It should be noted, that an increasing grain boundary phase thickness comes with two implications: i) the aspect ratio of the grains is increased, ii) the magnetic volume of the grains is reduced. Both parameters are contributing to the increased coercivity. These effects have also been seen in experiments, where an increasing Nd-content in permanent magnets increases the coercivity but decreases the magnetic remanence due to the loss of magnetic volume.²⁰ 

Another important factor influencing the behaviour of a permanent magnet can be the shape of the magnetic grains. Here, we examine the aspect ratio of the grains in relation to the direction of the applied field and its impact on coercivity. In our previous work²¹ it has been shown, that by varying the deposition temperature of a permanent magnetic film the shape of the grains can be tuned. In the simulation we can easily change the aspect ratio of grains to look at the impact on the coercive field. Figure 4 shows the coercivity as a function of a changing aspect ratio. The aspect ratio was altered from 1.0 to 4.0 in steps of 0.1. Here it is defined as the length of the grain in z-direction, i.e., the direction of the applied field, divided by the width in x-direction. The coercivity is continuously rising with increasing aspect ratio from approximately 3.15 T at equiaxed grains to 3.8 T at an ratio of 3.2. The gradient is slowly decreasing and above the aspect ratio of 3.2 up to 4 no coercivity enhancement was observed. Tang et al.³ could show the same behaviour when increasing the aspect ratio.

Large hard magnetic multi-grain-models, beyond the limits of conventional micromagnetism, can be calculated using the reduced order model based on the embedded Stoner-Wohlfarth method. The presented reduced order model drastically reduces the computational effort in terms of memory and processing time by a) requiring only a 2D surface mesh instead of a volumetric finite element mesh, b) approximate the grains’ reversal by analytic computation of the local switching field, and c) compute the demagnetization field analytically employing hierarchical matrices. We used this model to investigate the influence of the thickness of a nonmagnetic grain boundary phase and the aspect ratio of grains on the magnet’s coercivity. An increased thickness of the boundary phase improves the coercivity of the magnet but the effect weakens with increasing thickness. Compared to equiaxed grains, columnar grains with an aspect ratio of 3 exhibit an increase by 0.65 T (21%) in the direction of the longer axis. These results can be taken as guiding principles to increase coercivity in permanent magnets. The thickness of an ideally nearly non-magnetic grain boundary phase in NdFeB-type magnets can be influenced to some extend by methods such as increasing the Nd content²² or through grain boundary diffusion.²³ The aspect ratio of grains could be modified using manufacturing processes like hot-pressing or annealing. Nevertheless, a satisfactory level of control over this parameter has not been reached yet.²⁴ 

This research was funded in whole or in part by the Austrian Science Fund (FWF) I 6159-N and the French National Research Agency ANR-22-CE91-0008.

The authors have no conflicts to disclose.

H. Moustafa: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Software (equal); Validation (equal); Visualization (equal); Writing – original draft (equal). A. Kovacs: Methodology (equal); Software (equal); Supervision (equal); Visualization (equal); Writing – review & editing (equal). J. Fischbacher: Data curation (equal); Methodology (equal); Software (equal). M. Gusenbauer: Methodology (equal); Software (equal). Q. Ali: Software (equal). L. Breth: Methodology (equal); Software (equal). Y. Hong: Data curation (equal); Investigation (equal); Resources (equal). W. Rigaut: Data curation (equal); Investigation (equal); Resources (equal). T. Devillers: Investigation (equal); Resources (equal). N. M. Dempsey: Project administration (equal); Resources (equal). T. Schrefl: Conceptualization (equal); Methodology (equal); Software (equal). H. Oezelt: Conceptualization (equal); Funding acquisition (equal); Project administration (equal); Software (equal); Supervision (equal); Writing – review & editing (equal).

The data that support the findings of this study are available from the corresponding author upon reasonable request.