We performed a large-scale micromagnetics simulation on a supercomputing system to investigate the properties of isotropic nanocrystalline permanent magnets consisting of cubic grains. In the simulation, we solved the Landau–Lifshitz–Gilbert equation under a periodic boundary condition for accurate calculation of the magnetization dynamics inside the nanocrystalline isotropic magnet. We reduced the inter-grain exchange interaction perpendicular and parallel to the external field independently. Propagation of the magnetization reversal process is inhibited by reducing the inter-grain exchange interaction perpendicular to the external field, and the coercivity is enhanced by this restraint. In contrast, when we reduce the inter-grain exchange interaction parallel to the external field, the coercivity decreases because the magnetization reversal process propagates owing to dipole interaction. These behaviors show that the coercivity of an isotropic permanent magnet depends on the direction of the inter-grain exchange interaction.

The magnetization dynamics governs the properties of permanent magnets, which are applied in various devices such as the actuators in hard disk drives and high-power motors for electric vehicles.1,2 A nanocrystalline permanent magnet consists of many grains whose diameters range between nano and submicron scale.3–8 The grains are separated from each other by grain boundary phases. The magnetization dynamics in nanocrystalline permanent magnets has been investigated for a long time. However, it is still not clear how the magnetization reversal process propagates in a permanent magnet.

Micromagnetics simulation based on the Landau–Lifshitz–Gilbert (LLG) equation are widely used in magnetic research to investigate the magnetization dynamics because the magnetization reversal process can be observed directly.9–13 Although micromagnetics simulations have been applied for investigating permanent magnets, it is difficult to simulate the magnetization dynamics inside the magnet accurately.

In our previous studies, we implemented a new type of fast Fourier transform algorithm that enables performing micromagnetics simulation using more than 0.1 billion calculation cells in a super computing system.14–16 Thus, we can accurately simulate the propagation of the magnetization reversal process inside the magnet.

In this study, we performed micromagnetics simulations using a model of the isotropic nanocrystalline permanent magnet and demonstrated how the coercivity and remanent magnetization are changed by inter-grain exchange interactions. Propagation of the magnetization reversal process depends on not only the strength but also the direction of the inter-grain exchange interaction.

A schematic illustration of the isotropic nanocrystalline permanent magnet is shown in Fig. 1. The simulation system (512 nm × 512 nm × 512 nm) consists of many cubic grains (16 nm × 16 nm × 16 nm). We discretized the simulation model into 2.0 nm × 2.0 nm × 2.0 nm cells. The easy axes of the grains are oriented at random. The magnetization dynamics is simulated by solving the LLG equation, which is defined as

dM(𝒙)dt=γ𝑴(x)×𝑯eff(𝑥)+α𝑴×d𝑴(𝒙)dt,
(1)

where M(𝒙) is the magnetization at x, γ is the gyromagnetic ratio, α is the Gilbert damping constant, and 𝑯eff(𝒙) is the effective field, which consists of anisotropy, dipole, external, and exchange fields. The exchange field 𝑯exc(𝒙), which is defined as

𝑯exc(𝒙)=μ2Ms2[A(𝒙)Mμ(𝒙)]𝒆^μ,
(2)

where Ms is the saturation magnetization and A(x) is the exchange stiffness coefficient. The exchange stiffness coefficient has different values for intra- and inter-grain exchange interactions. In this paper, the exchange stiffness constant for the inter-grain interaction is defined as

A(𝒙)={Ain gainβgAon interface,
(3)

where βg takes a value from 0 to 1. In particular, βg and βg represent βg perpendicular and parallel to the external field, respectively. We consider three types of inter-grain exchange interactions called types A, B, and O. In type A (B), the inter-grain exchange interaction works on the xy plane (along the z-direction) and this interaction works along all direction in type O. We assume the following Nd–Fe–B material parameters: Ms = 1281.2 emu/cm3, the uniaxial constant K1u=4.5×107 erg/cm3, α=1.0, |γ|=1.76×107 s−1G−1, and A=12.5×107 erg/cm. We performed the micromagnetics simulation while varying the strength of the external field applied in the z-direction.

FIG. 1.

Schematic of the isotropic nanocrystalline permanent magnet model. Inset shows a cubic grain.

FIG. 1.

Schematic of the isotropic nanocrystalline permanent magnet model. Inset shows a cubic grain.

Close modal

Figure 2(a) shows the hysteresis curves of type O for various values of βg and βg. When there is no inter-grain exchange interaction (βg=0.0,βg=0.0), the hysteresis curve is almost the same as that of the Stoner-Wohlfarth model. In this case, the coercive force Hc is 30.4 kOe. As the strengths of the inter-grain exchange interactions increase, Hc decreases rapidly. When the strengths of the intra- and inter-grain exchange interactions are same (βg=1.0,βg=1.0), Hc becomes 9.79 kOe. This behavior of the hysteresis curve is explained by propagation of the magnetization reversal process, as argued in the past.17 When there is no inter-grain exchange interaction, the magnetization tends to be reversed independently in each grain. In contrast, if the grains interact strongly through the inter-grain exchange interaction, the magnetization reversal process propagates across the grain boundary, and the magnetization is reversed in a large region. Hence, the magnetization is gradually (quickly) damped when the inter-grain exchange interaction is weak (strong). The remanent magnetization Mr becomes large as the strength of the inter-grain exchange interaction increases, in contrast with Hc.

FIG. 2.

The hysteresis curves of (a) type O and (b) types A and B for various of βg and βg. Inset illustrates the definition of βg and βg.

FIG. 2.

The hysteresis curves of (a) type O and (b) types A and B for various of βg and βg. Inset illustrates the definition of βg and βg.

Close modal

The reduction in Hc depends on the direction in which the inter-grain exchange interaction works. Figure 2(b) shows the hysteresis curves of types A and B. For type A, the magnetization decreases gradually in the demagnetization process even if the strengths of the inter- and intra-grain exchange interactions are the same along the z-direction. The shape of the hysteresis curve of type A is similar to that without the inter-grain exchange interaction. The reduction in Hc is small, and Mr still has a large value. In contrast, the hysteresis curve of type B has similar properties to that with the strong inter-grain exchange interaction, in which the domain wall moves across the grain boundary. The magnetization of type B decreases quickly when the external field approaches Hc, and Mr is larger than that for type A.

We performed micromagnetics simulations while changing the strength of the inter-grain exchange interaction continuously to reveal how the coercive force and remanent magnetization depend on the type of inter-grain exchange interaction. Figure 3(a) shows the Hc values for types A, B, and O as a function of the strength of the inter-grain exchange interaction. For type O, the reduction in Hc is largest because the magnetization reversal process is propagated easily by the strong inter-grain exchange interaction. The most remarkable property is that the reduction in Hc for type A is smaller than that for type B. This property shows that propagation of the magnetization reversal in the xy plane is inhibited even if the grains interact through a strong inter-grain exchange interaction in the z-direction. In contrast with that in type A, the magnetization reversal is propagated in the z-direction by the dipole interaction in type B. Thus, these properties show that it is worked for the high Hc effectively to inhibit propagation of the magnetization reversal in the xy plane. On the other hand, the strong inter-grain exchange interaction enhances Mr as shown in Fig. 3(b). For types B and O, Mr is larger than that for type A.

FIG. 3.

Coercive force Hc and remanent magnetization Mr as a function of the strength of the inter-grain exchange interaction ξ.

FIG. 3.

Coercive force Hc and remanent magnetization Mr as a function of the strength of the inter-grain exchange interaction ξ.

Close modal

We performed a large-scale micromagnetics simulation to investigate the magnetization dynamics inside an isotropic nanocrystalline permanent magnet consisting of nano meter-sized cubic grains. When we performed the micromagnetics simulations, we changed the strength of the inter-grain exchange interaction in the direction parallel and perpendicular to the external field. When the inter-grain exchange interaction worked only in the direction parallel to the external field, the magnetization decreased gradually in the demagnetization process. In contrast, the magnetization decreased quickly even if there is no inter-grain exchange interaction in direction parallel to the external field, because the magnetization reversal process propagated in this direction owing to the dipole interaction. Thus, the coercivity has a large value when the inter-grain exchange interaction works only along the direction parallel to the external field, although the remanent magnetization is largest, when the inter-grain exchange interaction works in all directions. Therefore, it is important for the high coercivity to inhibit the propagation of the magnetization reversal process in perpendicular to the external field.

The authors thank the crew of the Center for Computational Materials Science of the Institute for Materials Research, Tohoku University, for their continuous support of the SR16000 supercomputing facilities. This work was supported by the Large Scale Simulation Program No. 15/16-01 (FY2015/16) of KEK. This work was supported in part by the JST under Collaborative Research Based on Industrial Demand “High Performance Magnets: Towards Innovative Development of Next Generation Magnets.”

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