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Multiscale simulations toward understanding effects of amorphous grain boundary on coercivity in Nd-Fe-B magnets

Applicant Professorin Dr.-Ing. Bai-Xiang Xu, since 4/2019
Subject Area Thermodynamics and Kinetics as well as Properties of Phases and Microstructure of Materials
Glass, Ceramics and Derived Composites
Synthesis and Properties of Functional Materials
Term from 2018 to 2022
Project identifier Deutsche Forschungsgemeinschaft (DFG) - Project number 409656180
 
Final Report Year 2023

Final Report Abstract

1. In summary, ab-initio informed atomistic spin model simulations have been carried out to predict the temperature-dependent intrinsic properties of Nd2Fe14B permanent magnets. The results are relevant for temperature-dependent micromagnetic simulations of Nd-Fe-B magnets. The main 27 conclusions are summarized as follows. (1) The Hamiltonian of the atomistic spin model for Nd2Fe14B includes contributions from the Heisenberg exchange of Fe-Fe and Fe-Nd atomic pairs, the uniaxial single-ion anisotropy energy of Fe atoms, and the crystal-field energy of Nd ions. Specially, the crystal-field Hamiltonian of Nd ions are approximately expanded into an energy formula featured by second, fourth, and sixth-order phenomenological anisotropy constants. (2) Monte Carlo simulations of the atomistic spin model readily capture the Curie temperature Tc of Nd2Fe14B. After applying the temperature rescaling strategy and the fitted rescaling parameter α = 1.802, it is shown that the calculated temperature dependence of saturation magnetization Ms(T) agrees well with the experimental results, and the spin reorientation phenomenon at low temperature is well predicted. (3) Constrained Monte Carlo simulations give the temperature-dependent total internal torque, from which the macroscopically effective second-, fourth-, and sixth-order anisotropy constants are calculated, which match well with the experimental measurements. The culated values at 300 K shows good consistency with literature reports, with K1eff, K2eff, and K2eff as 4.26, 0.15, and −0.10 MJ/m 3, respectively. (4) Mapping the atomistic magnetic moments to the continuum magnetization leads to the domain wall profile, which can be further fitted by hyperbolic functions to evaluate the domain wall width δw. Different domain wall configurations at low temperatures are identified. The calculated δw and its variance increases with temperature, and its value at 300 K is consistent with experimental observation. δ w is found to scale with magnetization as a function of m 2.26. (5) By using a general continuum formula with the exchange stiffness constant Ae(T) as a parameter to describe the domain wall profile, Ae(T) is readily determined. Ae is found to decrease more slowly than K1eff with increasing temperature. The scaling behavior of the exchange stiffness with the normalized magnetization is found to be Ae(T) ∼ m at temperatures below 500 K and Ae(T) ∼ m1.55 at temperatures close to Tc. 2. The exchange anisotropy and its influence on the coercivity of Nd-Fe-B magnets have been identified by using multiscale simulations. The bulk exchange stiffness in Nd2Fe14B phase is found to be intrinsically anisotropic (i.e., depend on the crystallographic axis) and its value along c axis is lower than along a/b axis. The "double anisotropy" phenomenon regarding to AGB is discovered, i.e., in addition to the experimentally determined anisotropy in AGB composition or magnetization, the interface exchange coupling strength between Nd2Fe14B and AGB is also confirmed to be strongly anisotropic. Even when the AGB FexNd1-x has the same composition, Jint for (100) interface is much higher than that for (001) interface. Due to this "double anisotropy", the ferromagnetic exchange coupling for (001) interface is much weaker than that for (100) interface. The coercivity of Nd-Fe-B magnets is demonstrated to be obviously influenced by the exchange anisotropy, suggesting the necessity of including exchange anisotropy in order to realize a reasonable design or prediction by micromagnetic simulations. Overall, these findings in this chapter not only provide comprehensive understanding of exchange in Nd-Fe-B magnets, but also are useful in deciphering coercivity mechanism and inspiring a strategy of tailoring exchange for the design of high-performance Nd-Fe-B permanent magnets. 3. A multiscale computational scheme integrating atomistic spin model (ASM) and micromagnetic simulations is proposed to calculate the coercivity of Nd-Fe-B permanent magnets at high temperatures. Using the ASM Hamiltonian constructed for Nd2Fe14B, ASM simulations are carried out to obtain the temperature-dependent saturated magnetization Ms(T), magnetocrystalline anisotropy K1(T), and exchange stiffness constant Ae(T) at high temperatures. The calculated Ms(T), K1(T), and Ae(T) are demonstrated to coincide with the experimental measurement. Taking the ASM results as input, finite-temperature micromagnetic simulations using the stochastic LL equation are performed to calculate the magnetic reversal, thermal activation volume v, thermal fluctuations induced coercivity reduction Hth and its ratio ∆hth, and coercivity Hc and its temperature coefficient β in pure Nd2Fe14B and Nd2Fe14B grain with surface defect layer as AGB phase or Dy-rich hard shell. Specifically, the stepwise external field and the step time for calculating the magnetic reversal curves are optimized. It is found that apart from the anisotropy field decreasing with temperature, the thermal fluctuations further reduce H c by 5–10% and β by 0.02–0.1%/K. The defect layer with strong magnetization (e.g. 1 T) is demonstrated to result in a remarkably increased v (which can be reduced by adding the 28 Dy-rich hard shell) and significantly decrease Hc , while suppress the influence of thermal fluctuations and thus reduce Hth and ∆hth. It is also revealed that even though the presence of Dy-rich hard shell cannot fully cancel out the reduction of coercivity from the defect layer, a 4.5- nm-thick (Nd0.53Dy0.47)2Fe14B shell enhances Hc by 0.5 T and considerably improves the thermal stability. Both Hc and β are found to saturate at a Dy-rich shell thickness of 6–8 nm. An even thicker shell or Dy alloying into the core prior to grain boundary diffusion is not essential. The multiscale scheme and the calculation results in this chapter are useful for the design of highperformance Nd-Fe-B permanent magnets used at high temperatures in terms of microstructure engineering. 4. As an outlook, the atomisctic spin model of Nd2Fe14B has to be extended for its usage in other rare-earth permanent magnet system. In this project, the total magnetization is assumed to be collinear to the Fe magnetization in Nd2Fe14B. This assumption is correct for Nd2Fe14B, but it raises a serious error in evaluating magnetocrystalline anisotropy of other R2Fe14B systems in which R magnetization is highly non-collinear to Fe magnetization. For instance, Dy2Fe14B possesses a strong non-collinearity effect which remarkably influences the temperature dependence of its magnetocrystalline anisotropy. Therefore, some modifications of the atomisctic spin model including the non-collinearity effect are essential. In addition, the nonmonotonic variation of magnetocrystalline anisotropy with temperature is observed experimentally for some R2Fe14B -type magnets (e.g. Y2Fe14B). It is recently proposed that atomistic spin model simulations that include contributions from competing two-ion and singleion anisotropies are able to reproduce the observed non-monotonic behavior. In addition, the idea of tuning interface exchange has already been proposed to design exchange-spring nanocomposite magnets, but its application to sintered and hot-pressed rare-earth permanent magnets is less-focused (except for the control of AGB for an exchange decoupling) and the related theoretical guidance (especially in terms of nanostructure and atomic-level engineering) is still not well established. An alternative strategy based on the tuning of exchange parameters via the nanostructure and atomic-level engineering will be a potentially feasible avenue for the design of high-performance rare-earth permanent magnets. Theoretical foundations of the tuning-exchange strategy should be developed. However, the accurate and efficient theoretical calculation of interatomic exchange parameters, especially the exchange among atoms in the interfacial vicinity, is still of difficulty.

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