Abstract
Purpose
Confronting the unveiled sophisticated structural and physical characteristics of permanent magnets, notably the samarium–cobalt (Sm-Co) alloy, This work aims to introduce a simulation scheme that can link physics-based micromagnetics on the nanostructures and magnetostatic homogenization on the mesoscale polycrystalline structures.
Design/methodology/approach
The simulation scheme is arranged in a multiscale fashion. The magnetization behaviors on the nanostructures examined with various orientations are surrogated as the micromagnetic-informed hysterons. The hysteresis behavior of the mesoscale polycrystalline structures with micromagnetic-informed hysterons is then evaluated by computational magnetostatic homogenization.
Findings
The micromagnetic-informed hysterons can emulate the magnetization reversal of the parameterized Sm-Co nanostructures as the local hysteresis behavior on the mesostructures. The simulation results of the mesoscale polycrystal demonstrate that the demagnetization process starts from the grain with the largest orientation angle (a) and then propagates to the surrounding grains.
Research limitations/implications
The presented scheme depicts the demand for integrating data-driven methods, as the parameters of the surrogate hysteron intrinsically depend on the nanostructure and its orientation. Further hysteron parameters that help the surrogate hysteron emulate the micromagnetic-simulated magnetization reversal should be examined.
Originality/value
This work provides a novel multiscale scheme for simulating the polycrystalline permanent magnets’ hysteresis while recapitulating the nanoscale mechanisms, such as the nucleation of domains, and domain wall migration and pinning. This scheme can be further extended to simulate the part-level hysteresis considering the mesoscale features.
Keywords
Citation
Yang, Y., Kühn, P., Fathidoost, M. and Xu, B.-X. (2023), "Micromagnetics and multiscale hysteresis simulations of permanent magnets", COMPEL - The international journal for computation and mathematics in electrical and electronic engineering, Vol. 42 No. 5, pp. 993-1006. https://doi.org/10.1108/COMPEL-12-2022-0424
Publisher
:Emerald Publishing Limited
Copyright © 2023, Yangyiwei Yang, Patrick Kühn, Mozhdeh Fathidoost and Bai-Xiang Xu.
License
Published by Emerald Publishing Limited. This article is published under the Creative Commons Attribution (CC BY 4.0) licence. Anyone may reproduce, distribute, translate and create derivative works of this article (for both commercial & non-commercial purposes), subject to full attribution to the original publication and authors. The full terms of this licence may be seen at http://creativecommons.org/licences/by/4.0/legalcode
1. Motivations
Due to its superior corrosion resistance under complex chemical environment and outstanding stability at high temperature, samarium–cobalt (Sm-Co)-based magnets have promised industries feasibility in various applications, such as high-performance electric motors for automotive and aeronautic application. However, former researches have unveiled the microstructure of such permanent magnets in a sophisticated multiscale fashion. As shown in Figure 1, it has been investigated that commercial Sm-Co magnets present a three-phase composite nanostructure (<1 µm). This nanostructure can be described as the cellular Sm2Co17 phase surrounded by a coherent stripe-shaped SmCo5 phase. This is further subdivided by the Zr-rich platelet-shaped phases (hereinafter referred to as Z-platelets) which develop perpendicular to the crystallographic c-axis of the Sm2Co17 phase (Duerrschnabel et al., 2017; Katter et al., 1996; Song et al., 2020; Gutfleisch, 2009). The final nanostructure depends on the chemical composition and thermal treatments (Wang and Zhu, 2021). Notably, Zhou et al. (2021) reported increases in thickness of Z-platelets from 2.4 nm up to 28.8 nm with only an increase in annealing time. Meanwhile, as one of the commercial permanent magnets that is manufactured by sintering, polycrystalline structure on the mesoscale (1–100 µm) is also observed and examined in Sm-Co alloy (Giron et al., 2022).
Most of the primary mechanisms contributing to the magnetic behavior (i.e. nucleation of the reversed domain, and migration/pinning of the domain wall) occur on the nanoscale with characteristic length around 1 nm, while receiving effects from grain orientation and local thermal history, hysteresis behavior varies locally on the level of the polycrystal. This stresses the importance of scale-bridging on accurate modeling and simulation of the hysteresis behavior. Although there are well-established models for individual scales, scale-bridging strategy is intricate and essential. It is worth noting that the strategy bridging the atomic and nanoscale micromagnetic combining first-principles calculations, atomistic spin model simulations and micromagnetic simulations has been investigated and discussed (Gong, 2022). Nevertheless, the strategy bridging the nanoscale and mesoscale is still missing. On the other hand, macroscopic hysteresis behavior can be directly modeled using proposed phenomenological models, such as the Preisach’s (Preisach, 1935), Takács’s (Takács, 2001) and Jiles-Atherton’s model (Jiles and Atherton, 1984; Zirka et al., 2012). These models, however, fail to deliver the physics information on individual scales and cannot be used in the sense of tailoring hysteresis of permanent magnets.
At the end of the day, modeling and simulating the hysteresis of the permanent magnets demands a multiscale scenario bridging mesoscale phenomena and nanoscale mechanisms, which becomes the objective of this work. We propose a novel multiscale scheme for simulating the polycrystalline permanent magnets’ hysteresis combining the merits of both micromagnetics and computational magnetostatic homogenization. This scheme is also extendable to simulate the part-level hysteresis and is capable of integrating machine-learning-based data-driven methods. It is hoped that the present work can serve a new viewpoint/methodology in the field of electromagnetic engineering in the hysteresis behavior of magnetic materials and components and provide a computational toolkit that is practicable and physics-rooted.
2. Models and methods
Figure 2 presents the workflow of proposed multiscale hysteresis simulation for the polycrystalline permanent magnets. We start with performing a series of micromagnetic simulations on distinct parameterized nanostructures. Micromagnetics has a sound physics foundation and thus is suitable for investigating local magnetization switching mechanisms. To have acceptable computational cost-efficiency with fine spatial discretization for resolving physical processes (e.g. domain nucleation, and domain wall migration and pinning), micromagnetic simulations are implemented and performed by the finite difference method (FDM). Next, the surrogate hysteresis unit (or “hysteron,” adopted from its pseudo-particle behavior) parameterized by the results of micromagnetics is used to replace micromagnetic calculations on the mesoscale polycrystalline structures. This hysteron should preserve the physical characteristics unveiled by the micromagnetics, such as the local magnetic coercivity and magnetization rotation. Finally, the hysteresis behavior of the polycrystalline structure with micromagnetics-informed hysterons is evaluated by computational magnetostatic homogenization. Due to the need for geometrically complicated polycrystalline structures without compromising in numerical accuracy, magnetostatic homogenizations are implemented and performed by the finite element method (FEM).
It is worth noting that the magnetization reversal on each level of the proposed scheme is modeled and simulated under the equilibrium conditions, which is sufficient for evaluating the material- and structural-based hysteresis behavior without loss. It also helps reduce the 3D magnetization dynamics (a.k.a. Landau–Lifshitz–Gilbert dynamics) to 2D rotational one (a.k.a. Stoner–Wohlfarth reversal, where magnetization is also in the plane defined by the magnetic field H and the easy axis u).
2.1 Micromagnetics
Below the Curie temperature, the magnetization of most of magnetic materials saturates with constant magnitude (Msat). Therefore, in the micromagnetics, it is important to have the normalized magnetization vector which is position dependent, i.e. m(r). This vector field can be physically interpreted as the mean field of the local atomic magnetic moments, but yet sufficiently small in scale to resolve the magnetization transition across the domain wall. In this regard, we consider the free energy density functional of a micromagnetic system (with a volume V) as the functional of m(r), i.e.:
The term fani represents the contribution due to the magnetocrystalline anisotropy. It provides the energetically preferred orientation to local magnetizations, which is related to the defined easy axis u mostly of the material. Sm-Co permanent magnets generally possess crystalline structures with uniaxial anisotropy with u parallel to the crystallographic c-axis, which is perpendicular to the Zr-rich platelet phase, as shown in Figure 1 (Gutfleisch, 2009). In this regard, fani is formulated as:
It is worth noting that most of the investigations only use the lowest order (i = 1) with the characteristic parameter Ku1. It can be shown that the parameters Aex and Ku1 are related to the Bloch domain wall energy σdw and width ldw of the materials at the equilibrium as:
Besides fex and fani which are material-dependent, the terms fms, fzm and fcp provide the contributions due to the interaction among magnetization and distinct intrinsic/extrinsic fields and thereby change the local thermodynamic stability. The magnetostatic term fms takes account of the energy of each local magnetization (or a magnetic moment) under the demagnetizing field created by the surrounding magnetization (or by all the other magnetic moments). It is generally formulated as:
The exact formulation of the coupling field Hcp depends on the choice of coupled physical effects, such as magnetostriction (Kronmüller, 2003), thermal fluctuation (Brown, 1963) and spin-current interactions (Sun, 2000).
Generally, H is implemented as a controllable quantity for the investigator to emulate the magnetic loading/unloading as in the experiments, i.e. an applied magnetic field Hext, and the micromagnetic system should find its equilibrium. This is mathematically determined as follows:
Equation (8) can be physically interpreted as the balance between two components of Landau–Lifshitz torque that, respectively, contribute to the precession (τprec) and the damping (τdamp) actions of the magnetization m (Gilbert, 1955; Coey, 2010), where αd is the damping coefficient. As the precession direction of the m is always perpendicular to both m and the effective field Heff ≡ −δℱ/δm, the term τprec presents thereby no contribution to the magnetization reversal process, when m evolves from antiparallel to parallel with respect to (w.r.t.) Heff. In that sense, only the damping term of equation (8) remains for solving thermodynamically preferred m(r) under an applied Hext, i.e.:
In this work, the FDM-based steepest conjugate gradient (SCG) method is used for solving equation (9) by the merit of its speed, cost-efficiency and capability for GPU-parallel implementation (Leliaert et al., 2018). Another key advantage of using FDM-based SCG method is to obtain the demagnetizing field Hdm directly by magnetostatic convolution of m over the simulation domain without introducing additional degrees of freedom (Vansteenkiste et al., 2014; McMichael et al., 1999). In this regard, BCs of m become significant for correct evaluation of Hdm. Two BCs are majorly used in simulating the magnetization reversal inside magnetic materials: the Neumann BC (∇m|∂V·n = 0 with n the normal vector of the boundary ∂V) represents that the neighboring magnetizations outside of the simulation domain are identical to the ones at the boundary (Donahue and Porter, 2004); the Periodic BC represents that the spatial distribution of the magnetizations inside the simulation domain repeats itself periodically along prescribed directions (Fangohr et al., 2009).
Based on SCG method, the iteration scheme for calculating the magnetization is formulated as follows with a descending direction ℍn, i.e.:
Here,
2.2 Micromagnetics-informed surrogate hysteron
To correctly simulate the local magnetization reversal and associated domain wall migration, sufficiently fine space discretization (mesh) is required to resolve the reversed nucleus formed near grain edges/corners where the demagnetizing field is high (Yi et al., 2016), and the transition profile of magnetization across the domain wall with its thickness characterized by ldw. As most permanent magnets with ldw in the range of several nanometers, e.g. ldw ∼ 2 nm for Nd-Fe-B and Sm-Co magnets and ldw ∼ 50 nm for electrical steels (4.6% Si), it is impractical in both numerical and computational senses for direct micromagnetic calculations on the mesoscale structure of such materials, where the spatial distribution of grains with distinct sizes and orientations (easy axes) is believed to have significant influences as well. In this regard, a surrogate model is required to equivalently replace the direct micromagnetic calculation on every subdomain for the hysteresis simulation on the polycrystal level. Such a surrogate model:
can efficiently describe the local magnetization reversal as an isolated unit but also as the representative component (subsystem) of the polycrystal system. Such local reversal should be also dependent on the given orientation of the subdomain; and
can preserve the important characteristics of the local magnetization reversal by the micromagnetics, e.g. the local magnetic coercivity Hc where the magnetization of the chosen subsystem cannot withstand the applied field and gets reversed.
In this work, we use the vector hysteron as the surrogate model of the micromagnetic simulations, which can well describe the magnetization reversal of the ferromagnetic domain at equilibrium. Each vector hysteron can be regarded as an independent Stoner–Wohlfarth pseudo-particle, as its magnetization can only rotate freely in the plane defined by the magnetic field H and the easy axis u (if H‖u, then the hard axis perpendicular to u should be provided instead), as shown schematically in the inset of Figure 3. A hysteron inside a system (as an assembly of hysterons) can only affect the neighboring one via magnetostatic interactions, in other words, by affecting the local magnetic field.
This vector hysteron consists of two major parameters: the local switching field Hsw and the orientation angle α. Defining the local coordinates by the defined positive direction of the applied magnetic field, i.e. H = Hh‖, the longitudinal magnetization of a single demagnetizing process (simplified as H reversely increasing) is analytically formulated as:
2.3 Computational magnetostatic homogenization
In this work, the overall hysteresis behavior of the polycrystal is examined by performing computational magnetostatic homogenization. The governing equations for magnetostatics are derived by eliminating the time derivative terms in Maxwell’s equations as:
Omitting Ampère’s law in equation (14) by assuming no space current density also disregards the effects of eddy current. The homogenization scheme considering the eddy current losses is in development and will be discussed in upcoming works.
The magnetostatic homogenization problem can then be defined as:
3. Preliminary results and discussion
3.1 Hysteresis of Sm-Co nanostructure and its orientation dependence
Following Katter et al. (1996), a parameterized nanostructure for Sm-Co is used. The structure parameters of interest are the Sm2Co17 cell size L, thickness of the stripe-shaped SmCo5 phase ws, the distance between Z-platelets d, thickness of the Z-platelets wz and orientation angle α between the field H and the easy axis u, as shown in Figure 4(a). In this work, we take L = 150 nm, d = 50 nm, ws = wZ = 8 nm, while α varies between 0 and π/2. The nanostructure is generated in a 512 × 512 × 4 nm3 finite difference domain. Periodic BC is applied on the two boundaries perpendicular to the z direction, while Neumann BC is applied on other boundaries. A grain boundary layer with the thickness of 2 nm, where magnetocrystalline isotropy is assumed (i.e. Ku1), is also introduced to emulate the effects of the grain boundary in reducing the nucleation field to the system (Yi et al., 2016). To recapture the domain wall behaviors in the micromagnetic simulations without artificial effects related to mesh, the FD cell size is chosen as 0.8 nm, which is smaller than ldw. Micromagnetic parameters of each phase are presented in Table 1.
Figure 4(b) presents a half-cycle hysteresis of a nanostructure with α = π/6 examined over a single demagnetizing process [H from positive to negative with direction denoted in Figure 4(a)]. The magnetization reversal of the nanostructure consists of two steps: the reversed domain is first generated (nucleated) when the magnetic field reaches a certain threshold, denoted as the nucleation contribution Hc. Then, the nucleated reversed domain starts to grow alongside the reversely increasing magnetic field, demonstrated in the form of domain wall migration. When the migrated domain wall front encounters the intersections between different phases where the domain wall energy differences exist, magnetic energy is consumed to compensate such differences and the domain wall front stops migration, i.e. domain wall pinning occurs. The pinning events are reflected on the hysteresis curve as multiple stages where the magnetization is barely changed, as shown in the Figure 4(b). Therefore, the extra magnetic field (denoted as pinning contribution Hp) is required for the magnetization reversal.
We further present that the nucleation and pinning events on the nanostructure vary with the orientation angle, even though the parameters of constituent phases do not change. As shown in Figure 4(c), the half-cycle hysteresis curves present varying staging patterns w.r.t. α, resulting in Hsw as a function of α as shown in the inset of Figure 4(c). We then take this Hsw(α) and feed in the vector hysteron in equation (11) and present its longitudinal magnetization for comparison. We can tell that the hysteron can nicely emulate the magnetization reversal for the coherent case (α = 0). For increasing α to π/2, the hysteron presents an increased deviation in demagnetization compared to the micromagnetic simulation. When H < Hsw, the hysteron shows less longitudinal demagnetization than the micromagnetic one, implying the relatively slower rotation of the surrogate magnetization vector; when H > Hsw, the hysteron shows higher longitudinal demagnetization than the micromagnetic one, implying the relatively faster rotation of the surrogate magnetization vector. This difference in the demagnetization process between the hysteron and the micromagnetics eventually leads to the deviation of the magnetic coercivity for 0 < α < π/2. For α = π/2, both result in zero coercivity, even though the difference in demagnetization process still exists.
3.2 Hysteresis of Sm-Co polycrystal
We apply the surrogate hysterons with micromagnetics-informed Hsw(α) in a 10-grain polycrystal structure with the size 100 × 100 × 100 µm3. It is sufficiently large so that every point inside the polycrystal structure can be conceptually regarded as the homogenized point of the local nanostructure. Figure 5(a) presents the hysteresis loop of the structure with its mesh and orientation histogram shown in the insets. The magnetic coercivity of the polycrystal reads as 1.65 T, which is smaller than the one of 3.17 T from micromagnetic simulation on the coherent nanostructure (H‖u). This is because more than half of the grains inside the examined polycrystal possess orientation angles that are beyond π/4, which significantly affect the overall hysteresis of the structure.
To deliver insight into how those grains with relatively larger α affect the demagnetization of the structure, we sample 10 × 10 × 10 points and visualize their on-site hysterons as oriented cones. With the applied magnetic field reduced to zero, we can tell that the local hysterons inside certain grains [denoted in Figure 5(b1)] point to the direction almost π/2 w.r.t. the magnetic field direction, which is also the easy axis of the grain. This grain is regarded as the “soft” grain as the local coercivity inside is nearly zero. The surrounding hysterons receive the influence of the grain and deviate slightly from their stable directions. It is worth noting that the hysterons inside the grain with π/4 < α < π/2 would be affected relatively easier, presenting a trend of reversal propagation toward those directions. When the field starts to reversely increase (H < 0), the hysterons inside the “soft” grain already present the reversed magnetization and continue propagating the effect to the surrounded grains with relatively large α, as shown from Figure 5(b2) to Figure 5(b3). Meanwhile, the hysterons inside grains with 0 ≤ α < π/4 (“hard” grains) receive less effects from already reversed ones, until the local field is large enough to suddenly reverse all of them, as denoted in Figure 5(b4). This is due to the cut-off switching of those hysterons as demonstrated in Figure 3, where the local field reaches the Hsw.
4. Conclusions
We present in this work a novel multiscale simulation scheme for permanent magnets recapitulating its structural and physical characteristics from the nanoscale to the mesoscale. We perform the micromagnetic simulations on the parameterized Sm-Co nanostructures and investigate the mechanisms that are tightly related to the local magnetic coercivity, including the nucleation of the reversed domain, domain wall migration and pinning and unveil the orientation dependence of the demagnetization processes via half-cycle hysteresis curves. This information is then carried by micromagnetics-informed surrogate hysterons in the magnetostatic homogenization of a 10-grain polycrystalline structure with assigned orientations. The simulation results of the polycrystal demonstrate that the grains with largest α (“soft” grains) influence the overall demagnetization process significantly by their early reversing and further propagate such effect to the grains with π/4 < α < π/2 via affecting the local field. This is believed to result in a magnetic coercivity of 1.65 T, which is smaller than the one of 3.17 T examined from coherent nanostructures.
The presented work also showcases the demand of integrating data-driven methods, as the parameters of the surrogate hysteron intrinsically depend on the nanostructure and its orientation. In other words, effects of the nanostructural parameters (here L, d, ws and wZ) on the behavior of surrogate hysterons and the sensitivity analysis should be addressed in the upcoming works. More hysteron parameters that further help the surrogate hysteron to emulate the micromagnetic-simulated magnetization reversal, e.g. an orientation-dependent offset that adjusts the demagnetization (or rotation) of the hysteron, should be also discussed and examined.
Figures
Figure 4.
(a) Parameterized nanostructure of Sm-Co magnets. The positive direction of the magnetic field is also denoted; (b) The micromagnetic simulated half-cycle hysteresis of the single demagnetizing process of Sm-Co nanostructure with corresponding domain configuration denoted; (c) Average half-cycle hysteresis of five demagnetizing processes by micromagnetic simulation (MM) for each orientation angle (α), which is compared with parameterized vector hysteron (VH). Inset: orientation dependence of the switching field Hsw
Figure 5.
(a) The simulated hysteresis loop of the polycrystal structure. Insets: the FE mesh and the orientation histogram of the structure. Local magnetization reversal of sampled points (visualized as rotating cones) is also visualized under different applied field: (b1) 0.0 T; (b2) −1.0 T; (b3) −2.5 T; (b4) −4.0 T
Micromagnetic parameters for the phases appearing in this work
| Parameters | Unit | Sm2Co17 | SmCo5 | Z-plateletsa |
|---|---|---|---|---|
| Aex | pJ m−1 | 19.6 | 8.6 | 0.7 |
| Ku1 | MJ m−3 | 3.9 | 18.3 | 1.4 |
| Msat | kA m−1 | 987.7 | 810.8 | 310.4 |
| ldw | nm | 7.0 | 2.2 | 2.2 |
| σdw | mJ m−2 | 35.0 | 50.2 | 4.0 |
aRescaled Aex and Ku1 according to equation (4) by taking the σdw from literature and the ldw assumed to be coherent with one in SmCo5 phase
Source: Table courtesy of Katter et al. (1996)
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Acknowledgements
The authors acknowledge the financial support of German Science Foundation (DFG) in the framework of the Collaborative Research Centre Transregio 270 (CRC-TRR 270, project number 405553726, subprojects A06, B13) and 361 (CRC-TRR 361, project number 492661287, subprojects A05). The authors also greatly appreciate their access to the Lichtenberg High-Performance Computer and the technical supports from the HHLR, Technische Universität Darmstadt and the GPU Cluster from the CRC-TRR 270 subproject Z-INF.