Nanocomposite permanent magnets are an important research focus in the field of permanent magnetic materials for the potential of obtaining extraordinary magnetic properties.^[1–4] After decades of development, a series of successes^[1–22] has been achieved in both experiment and micromagnetic simulations for nanocomposite magnets. Firstly, the researchers have predicted the high theoretical energy product of the nanocomposite magnet by micromagnetic simulation. Giant theoretical energy products of 137 MGOe, 90 MGOe, 65 MGOe, and 25 MGOe were acquired from the Sm₂Fe₁₇ N₃/Fe₆₅Co₃₅,^[1] Nd₂Fe₁₄B/Fe₆₅Co₃₅,^[2] SmCo₅/Fe,^[3] and MnBi/FeCo^[4] nanocomposite magnets, respectively. Besides, excellent magnetic properties exceeding the corresponding single hard phase magnets have been achieved from experimental preparation nanocomposite multilayers. Cui et al.^[5] reported that the high energy product of 486 kJ/m³ (61 MGOe) based on the Nd–Fe–B/FeCo nanocomposite multilayers is higher than the maximum experimental energy product of single-phase Nd–Fe–B magnet. Moreover, Nev et al.^[6] obtained the energy product exceeding 50 MGOe based on SmCo5/α Fe multilayers, which is much higher than the theorical energy product (30 MGOe) of single SmCo₅ magnet. However, magnetic film is difficult to practically use massively due to the limitation of their sizes. Instead, the bulk magnets are always the research hotspot in the field of permanent magnet for their application prospects. A lot of experimental and theoretical research progress has been reported for the bulk nanocomposite magnets.^[7–9] Recently, Zhang et al.^[10] reported inspiring results that the bulk SmCo/FeCo nanocomposite magnets have acquired the (BH)_max of 28 MGOe which is 46% enhancement in energy product compared with the corresponding pure rare-earth magnets. One of the pivotal problems to obtain high magnetic properties for bulk nanocomposite magnets is that it is hard to obtain fine soft magnetic phase by using the current technology. Fortunately, the micromagnetic simulation analysis method can be used to explore the magnetic properties, coercivity mechanism and magnetic reversal process for nanocomposite magnet. Leineweber and Kronmüller^[2,13] calculated the thickness-dependent remanence, nucleation field and energy product of the soft phase for the Nd₂Fe₁₄B/α–Fe multilayers. Zhang et al.^[14] simulated the effects of grain size and texture on the magnetic properties and the magnetic reversal behavior of three-dimensional (3D) nanocomposite magnet by micromagnetic finite element. It was also found in our previous work^[15] that all the demagnetization curves with thickness in a range of 2 nm–30 nm of the perpendicular orientation trilayers exhibit “single phase” behavior, while noticeable kinks are present in the demagnetization curves of the parallel orientation Nd₂Fe₁₄B/Fe₆₅Co₃₅/Nd₂Fe₁₄B trilayers with the soft phase thickness equal to or larger than 12 nm. The same phenomenon was also found in the micromagnetic simulation results of 3D Nd₂Fe₁₄B/α–Fe multilayers.^[16] Cui et al.^[17] reported that coupling effects between the soft and hard phase of the perpendicular orientation multilayers due to the exchange interaction and magnetostatic interaction, while the coupling effects in the parallel orientation multilayers only result from the exchange interaction. For 3D bulk magnet, there is a great difference in complex microstructure where the two coupling effects of perpendicular orientation and parallel orientation coexist. So, the critical size of the soft phase is also different from that obtained from the classical one-dimensional (1D) model. In this study, the influence of soft phase size (S) on magnetic properties and magnetic reversal of the simple 3D Nd₂Fe₁₄B/α–Fe nanocomposite magnet with soft phase imbedded in hard phase can be analyzed by micromagnetic simulation. On this account, the theoretical basis can be provided for the selection/control of the size of soft phase in the experimental preparation of the bulk nanocomposite magnets.

The software OOMMF^[18] is used in this calculation. OOMMF is one of the micromagnetic calculation software which is based on the Landau–Lifshitz–Gilbert (LLG) dynamic equation^[16]

In this paper, we consider Nd₂Fe₁₄B/α–Fe nanocomposite magnets with the following intrinsic magnetic parameters:^[19,22] for saturated magnetization, J_s(Nd₂Fe₁₄B) = 1.61 T, J_s(Fe) = 2.15 T, for anisotropic constant K₁(Nd₂Fe₁₄B) = 4.331 MJ/m³, K₁(Fe) = 0.046 MJ/m³, and for integral exchange constant A(Nd₂Fe₁₄B) = 0.77 × 10⁻¹¹ J/m, A(Fe) = 2.5 ×10⁻¹¹ J/m.

Figure 1 shows the schematic diagram of simulated model for the soft phase (core) imbedded in hard phase (shell) Nd₂Fe₁₄B/α–Fe nanocomposite magnets. Our calculation model is based on an o-xyz coordinate system. Both the magnetocrystalline anisotropic easy axis (e) and the applied field (H) are assumed to be in the z-axis direction as shown in Fig. 1(a). The simulation model is cubic. The white represents hard phase and the thickness (H) is set to be 8 nm, while the red represents soft phase and the thickness (S) changes from 1 nm to 48 nm as shown in Fig.1(b). For the Nd₂Fe₁₄B/α–Fe nanocomposite magnets, the size of the theoretical Nd₂Fe₁₄B domain wall is about 4 nm. So, in this paper, to ensure accurate calculation, the simulation mesh is 1 nm × 1 nm × 1 nm, except the mesh size of 0.5 nm × 0.5 nm × 0.5 nm that is used for S = 1-nm samples.

Fig. 1.

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	Fig. 1. (color online) Schematic diagram of Nd2Fe14B/α–Fe nanocomposite magnets with soft phase imbedded in hard phase, showing (a) 3D schematic diagram based on the o–xyz coordinate, with e and H representing easy axis and applied field respectively, and (b) green cross section in panel (a), with H being thickness of hard phase and S denoting thickness of soft phase.

Figure 2 shows the hysteresis loops of the Nd₂Fe₁₄B/α–Fe nanocomposite magnets with soft phase imbedded in hard phase. The hysteresis loops of the Nd₂Fe₁₄B/α–Fe nanocomposite magnets show excellent squareness with S from 1 nm to 4 nm. But it is noticeable that the deterioration of quareness appears and turns more and more obvious with the increase of S as shown in Fig. 2. In addition, when S = 48 nm, there is an obvious kink near the coercive field, indicating the non-uniform magnetic reversal.

Fig. 2.

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	Fig. 2. (color online) Simulated hysteresis loops of Nd2Fe14B/α–Fe nanocomposite magnets with soft imbedded in hard phase for various S values.

Figure 3 shows the S-dependent saturation magnetization (J_s), remanence (J_r), and maximum energy product [(BH)_max] of the Nd₂Fe₁₄B/α–Fe nanocomposite magnets. With the increase of S, the J_s (J_s = V_sJ_s + V_hJ_h) increases from 16.1 kGs (S = 1 nm) to 18.4 kGs (S = 48 nm) (1 Gs = 10⁻⁴ T), while the remanence increases from 16.1 kGs (S = 1 nm) to 18.3 kGs (S = 48 nm), respectively. The high remanence ratio (J_r/J_s) in the range of S of 1 nm–48 nm indicates the significant remanence enhancement effect, which results from good exchange coupling interaction and magnetostatic interaction between soft and hard phase.^[17] The (BH)_max of the Nd₂Fe₁₄B/α–Fe nanocomposite magnets rises slowly and then decreases rapidly as the S increases, resulting in peaking at S = 24 nm with maximum (BH)_max of 72.9 MGOe as shown in Fig. 3. Generally, the size of soft phase should not exceed twice the domain wall width of the neighboring hard phase to guarantee the good exchange coupling effect based on classic1D model with parallel orientation.^[23] This means that the size of α-Fe should be smaller than 8.4 nm for Nd₂Fe₁₄B/α–Fe nanocomposite magnets. However, the studies^[17,24] of micromagnetic simulation and experimental work shows that the nanocomposite magnetic films with parallel orientation (the axis is parallel to the film surface) are decoupling for too large soft phase size, while the perpendicular orientation films can be coupled even the size of the soft magnetic phase is large. The 3D model can be regarded as the combination of parallel orientation and perpendicular orientation nanocmposite films. The simulation results in the present paper show that the maximum magnetic energy product can be obtained in the bulk nanocomposite magnets when the size of single soft phase particle is 24 nm.

Fig. 3.

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	Fig. 3. (color online) S-dependent Js, Jr and (BH)max of Nd2Fe14B/α–Fe nanocomposite magnets with soft phase imbedded in hard phase.

Figure 4 shows the S-dependent coercivity (H_ci) and nucleation field (H_n) of the Nd₂Fe₁₄B/α–Fe nanocomposite magnets. The H_ci decreases monotonically from 60.6 kOe (1 Oe = 79.5775 A · m⁻¹) (S = 1 nm) to 5.6 kOe (S = 48 nm). The nucleation field (H_n) is defined as the absolute value of applied field (H) where the magnetic moment deflection begins to occur. With S in a range from 1 nm to 4 nm, the H_ci is almost equal to the H_n, that is to say, the reverse nucleation and reversal occur simultaneously, suggesting that the dominant coercivity mechanism of the Nd₂Fe₁₄B/α–Fe nanocomposite magnets with S ≤ 4 nm is nucleation. Then, as S > 4 nm, the H_ci is always larger than H_n, and the discrepancy between H_ci and H_n, increases with the increase of S. In other words, the reversal of magnetic moments lags behind the nucleation, indicating that the dominant coercivity mechanism of the nanocomposite magnet changes into pinning.

Fig. 4.

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	Fig. 4. (color online) S-dependent Hn, Hci and magnetization reversal modes of Nd2Fe14B/α–Fe nanocomposite magnets with soft phase imbedded in hard phase.

The magnetic moment evolution in magnetic reversal is also influenced by S, besides the effects of S on magnetic properties of nanocomposite magnets. Figure 3 also shows the relationship between the magnetic reversal mode and the size of soft phase particle of the Nd₂Fe₁₄B/α–Fe nanocomposite magnets. By systematically observing the magnetic moment evolutions, the magnetic reversal of the nanocomposite magnets can be divided into three modes with the increase of S: coherent rotation (S < 3 nm), quasi-coherent rotation (3 nm ≤ S < 36 nm), and vortex-like rotation (S ≥ 36 nm) as shown in Fig. 4.

Figure 5 shows the spatial distributions of deflection angles (θ) for the nanocomposite magnets with S = 2 nm (coherent rotation), S = 24 nm (quasi-coherent rotation) and S = 40 nm (vortex-like rotation) when the nucleation and reversal occur. The θ means the deflection angle between magnetic moment and the easy axis. For magnet with S = 2 nm which shows the magnetic reversal mode of coherent rotation, the nucleation and irreversible reversal of the magnet occur simultaneously at H = −59.5 kOe. At the moment just before the nucleation occurs, the maximum θ is less than 7.5°. Figure 6 shows the magnetic moment evolution of the magnet with S = 2 nm (coherent rotation) before and after the nucleation/reversal (in this paper, the observation plane is perpendicular to the applied field and/or easy axis, unless otherwise stated). For the magnets with coherent rotation reversal mode, the magnetic moments have only two states, θ = 0° and θ = 180°. Once the nucleation occurs, the magnetic moments of hard phase and soft phase overturn simultaneously.

Fig. 5.

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	Fig. 5. (color online) Spatial distributions of magnetic moment deflection angles when (a) nucleation and (b) reversal occur, and (c) schematic diagram of corresponding magnetic moment position, with yellow line representing y axis in panels (a) and (b).

Fig. 6.

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	Fig. 6. (color online) Magnetic moment distributions of Nd2Fe14B/α–Fe nanocomposite magnet with S = 2 nm, when (a) H = −59.5 kOe and (b) H = −60.2 kOe before and after reversal. Observation surface is xy plane and z = 9 nm.

The magnet with S = 24 nm shows the magnetic reversal mode of quasi-coherent rotation, and the nucleation and reversal occur at H = −7.7 kOe and H = −13.3 kOe, respectively. At the nucleation state, the maximum θ in the hard phase is about 5°, while the θ in the center of the soft phase is 15.1°. At the reversal state, the θ from the surface of the hard phase to the soft/hard phase interface increases from 0° to 68.2°, while the θ increases to 152.3° in the center of soft phase. Low nucleation field and large magnetic moment rotation at the reversal state lead to poor squareness of demagnetization curve, which is not conducive to obtaining high magnetic energy product. Figure 7 shows the magnetic moment evolution in magnetic reversal of the nanocomposite magnet with S = 24 nm (quasi-coherent rotation) at the nucleation and reversal states. For the nanocomposite magnets with quasi-coherent rotation reversal mode, the magnetic moment rotation occurs in the center of the soft phase, firstly. And then the magnetic moment rotation propagates gradually to the hard/soft interface with the decrease of H. Finally, reversal of the whole magnet occurs as shown in Fig. 7.

Fig. 7.

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	Fig. 7. (color online) Magnetic moment distributions of Nd2Fe14B/α–Fe nanocomposite magnet with S = 24 nm, when (a) H = −7.7 kOe where nucleation occurs, and (b) H = −13.3 kOe before reversal of whole magnet. Observation surface is xy plane and z = 20 nm.

The nanocomposite magnet with S = 40 nm shows the magnetic reversal mode of vortex-like rotation, and the nucleation occurs at H = −2.8 kOe, while the reversal occurs at H = −7.7 kOe. In the nucleation state, the θ from of the surface of hard phase to the soft/hard phase interface increases from 0° to 19.3°. Moreover, the θ increases first, then decreases from the soft/hard interface to the center of soft phase, peaking at y = 16 nm with a maximum θ of 52.5°. At the reversal state, the θ from the surface of the hard phase to the soft/hard phase interface increases from 0° to 48.0°, and the θ in the center of soft phase is about 174.4°. The large θ in both the hard and soft phase leads to low magnetic energy product even though the magnet bears a large J_r = 18 kGs. Figure 8 shows the magnetic moment evolution in magnetic reversal of the nanocomposite magnet with S = 40 nm (vortex-like rotation) at the nucleation and reversal state. As shown in Fig. 8, at the nucleation state, the magnetic moments are distributed like a circle featuring the vortex. Moreover, the θ increases first and then decreases with the increase of the radius of circular magnetic moments. This is the difference between vortex-like rotation in 3D Nd₂Fe₁₄B/α–Fe nanocomposite magnets and vortex rotation^[16] for the perpendicular orientation nanocomposite multilayer. For the 3D Nd₂Fe₁₄B/α–Fe nanocomposite magnets, the soft phase is surrounded by hard phase in the direction both parallel and perpendicular to the easy axis. The magnetic moment of soft phase near the soft/hard interface is restricted by the exchange interaction in the direction parallel to the easy axis. After the nucleation, as the H decreases, the center of the vortex-like state begins to deviate from the center of the soft phase as shown in Fig. 8(b), indicating that the nucleation of vortex-like is unstable. As the H increases continuingly, the vortex-like state disappears and the reverse domains are formed as shown in Figs. 8(c) and 8(d).

Fig. 8.

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Fig. 8. (color online) Magnetic moment distributions of Nd2Fe14B/α–Fe nanocomposite magnet with S = 40 nm, when (a) H = −2.8 kOe where nucleation occurs, (b) H = −4.2 kOe (b), H = −7.7 kOe before reversal of the whole magnet observation plane (c) perpendicular and (d) parallel to the easy axis. Observation surfaces of panels (a), (b) and (c) are xy plane and z = 28 nm while observation surfaces of panel (d) is xz plane and y = 28 nm.

As S increases to 48 nm, the magnetic reversal mode is also vortex-like rotation as shown in Fig. 9(a). The vortex-like nucleation becomes more stable than the magnet with S = 40 nm, because the center of the vortex-like nucleation does not move with the increase of H, only the θ increases continuously as shown in Fig. 9(b). However, it is worth noting that there is much difference in the reversal of the hard phase portion between the magnet with S = 48 nm and the magnet with S = 40 nm. The reversals of hard phase parallel and perpendicular to the easy axis do not occur at the same time. Only the magnetic moments of hard phase perpendicular to the easy axis reverse when the reversal of soft phase happens as shown in Fig. 9(c). Meantime, the magnetic moments of hard phase parallel to the easy axis remain unchanged, indicating the pinning strengths in soft/hard interfaces parallel and perpendicular to the easy axis are different. Therefore, the magnetic reversals of nanocomposite magnets with S = 48 nm can be summarized as follow. Firstly, vortex-like nucleation appears in soft phase. Secondly, the deflection angles of the magnetic moments increase continuously. Thirdly, the magnetic moments of soft phase reverse together with the magnetic moments of the hard phase in the direction perpendicular to the easy axis. Finally, the magnetic moments of hard phase in the direction parallel to the easy axis reverse.

Fig. 9.

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Fig. 9. (color online) Magnetic moment distributions of Nd2Fe14B/α–Fe nanocomposite magnet with 4S = 48 nm, when (a) H = −1.4 kOe where the nucleation occurs, (b) H = −4.9 kOe, (c) H = −5.6 kOe after the reversal of hard phase perpendicular to the easy axis and (d) H = −7 kOe before the reversal of hard phase parallel to the easy axis, the observation surfaces of panels (a) and (b) are xy plane and z = 32 nm while the observation surfaces of panels (c) and (d) are xz plane and y = 32 nm.

In order to meet the requirements for practical applications, the magnetic properties of the permanent magnet must be guaranteed that H_ci ≥ J_r. For the Nd₂Fe₁₄B/α–Fe nanocomposite magnets simulated in this paper, the critical S to meet requirements for practical application is 19.3 nm, falling in the region of quasi-coherent rotation. The corresponding Nd₂Fe₁₄B/α–Fe nanocomposite magnet shows the magnetic properties of J_r = 17 kGs, H_ci = 17 kOe, and (BH)_max = 71.7 MGOe. When S = 24 nm, the Nd₂Fe₁₄B/α–Fe nanocomposite magnet with soft phase imbedded in hard phase reaches a maximum (BH)_max of 72.9 MGOe, which is just between the maximum calculated (BH)_max values of perpendicular (67.9 MGOe for 1D, 68.5 MGOe for 3D) and parallel (79.1 MGOe for 1D, 75.1 MGOe for 3D) orientation 1D and 3D Nd₂Fe₁₄B/α–Fe multilayers.^[16,25]

In this paper, the hysteresis loop of Nd₂Fe₁₄B/α–Fe nanocomposite magnet still maintains the characteristics of single phase permanent magnet, even if the size of soft phase (α-Fe) is larger than twice the width of domain wall for hard magnetic phase (Nd₂Fe₁₄B). In addition, when S = 48 nm, there are obvious kinks in the hysteresis loop due to the stepwise reversal of the hard phase.

For the 3D Nd₂Fe₁₄B/α–Fe nanocomposite magnets with the soft phase imbedded in the hard phase, the magnetic reversal can be summarized in the following three processes: reverse domain nucleation in the center of soft phase, propagation of reverse domain walls, and pinning at the soft/hard interface, irreversible reversal of the whole sample, which is the same as the results in Refs. [16], [19], and [25] for the nanocomposite multilayers. With the increase of S, the mode of the soft magnetic nucleation is changed. In this paper, the reversal mode of the magnet is defined as the kind of nucleation mode. In addition, with the increase of S, the deflection of the soft magnetic phase becomes more and more serious, leading to the deterioration of the squareness of the hysteresis loop.

(i) The magnetic energy product [(BH)_max] of the Nd₂Fe₁₄B/α–Fe nanocomposite magnets with soft phase imbedded in hard phase peaks at S = 24 nm with a (BH)_max of 72.9 MGOe.

(ii) As S increases, the dominant coercivity mechanism of the Nd₂Fe₁₄B/α–Fe nanocomposite magnets with soft phase imbedded in hard phase changes from nucleation into pinning.

(iii) The hysteresis loops of Nd₂Fe₁₄B/α–Fe nanocomposite magnets with S from 1 to 40 nm show the characteristics of single phase permanent magnet. For the Nd₂Fe₁₄B/α–Fe nanocomposite magnet with S = 48 nm, the reversal of hard phase is divided into two portions due to the different pinning strengths of the soft/hard interface parallel and perpendicular to the easy axis.

(iv) The magnetic reversal mode of Nd₂Fe₁₄B/α–Fe nanocomposite magnet with soft phase imbedded in hard phase can be divided into three modes with the increase of S: coherent rotation (S < 3 nm), quasi-coherent rotation (3 nm ≤ S < 36 nm), and the vortex-like rotation (S ≥ 36 nm).