It can be obtained from the volume integral of the square of the stray field outside the magnet, or from the volume integral of the dot product between the demagnetizing field Hd and the internal magnetic flux density B. The maximum energy product, (BH)max is widely used as a value for evaluating the performance of hard magnetic materials. BH)max should be carefully evaluated from the hysteresis loop by considering the exact operating point which is determined by the shape of the magnet. However, many researchers still use (BH)max obtained from the hysteresis loop of a particular shape as a representation of the material, even though the only information BH from a particular shape of the magnet is an energy product in the remanent state.
One of the main reasons for the paradox is known as microstructure, including grain properties, grain boundary, crystalline anisotropy distribution, and so on. The dotted lines on the right side of the graph have no meaning other than the operating points. 35 Table 4.4 The coercivity Hc, the remanent magnetization Mr, and the energy product values from the micromagnetic simulation of the model system.
Introduction
Permanent Magnet
Motivation
Theoretical Development
- Micromagnetic Energy
- Stoner-Wohlfarth Model
- Magnetic Hysteresis
- Energy Product
- Problem of Maximum Energy Product Prediction
Here the demagnetization factor (N) varies depending on the shape of the magnet and the direction of the magnetic moments. The important point is that the M(Happ) curve can also vary depending on the shape of the magnetic body. That is, the energy per unit volume can be expressed by the product of the remanent magnetic flux (Br) and the demagnetization field (Hd) created by the remanent magnetization (Mr).
The demagnetization field is directly related to the value of the demagnetization factor (N), which is determined by the shape of the magnet. This means that even with the same magnetic material the energy product is different depending on the shape and direction of the magnetization. In general, in the case of a strong permanent magnet, the remanent magnetization (Mr) can be assumed to be almost the same as the saturation magnetization (Ms), regardless of the shape and direction of magnetization.
In other words, the theoretical maximum energy product per unit volume a value that depends only on the saturation magnetization (Ms) and the shape of the magnet is N = 1/2. The curve has no relation to the demagnetization factor because it contains no parameters depending on the shape of the magnet.
Research Methods
Micromagnetic Simulation
Effect of Microstructure on Magnetic Properties
- Anisotropy Distribution
- System shape
- Grain Structure
- Model System and Simulation Methods
- Simulation Results
- Conclusion
Where the ratio of the remanent magnetization (Mr) to saturation magnetization (Ms) or by the average angle of deviation from the easy axis represents the degree of alignment (α). Even if the grains in the magnet are not perfectly aligned along the easy axis, the value of the energy product is almost six times greater than that of the randomly aligned structure. Thus, the alignment of the anisotropy direction of the grains within the magnet by the external magnetic field largely affects the performance of the magnet.
In conclusion, the energy product can be controlled by the shape of the magnet in a single-grain model that has sufficient crystalline energy, although the coercivity of each magnet differs slightly. As an extrinsic magnetic property, coercivity has no direct relationship with the energy product of the magnet. To understand the reason for the resulting coercivity and improve the energy product by designing an optimal microstructure, several features of the microstructure need to be studied [36]–.
The exchange length is the smaller value between two kinds of exchange length, which is calculated from the saturation magnetization and the anisotropy constant. For the model microstructure nm3 size cuboid, where the microstructure is composed of many grains with different average size of nm, and schematic view of the microstructure is shown in figure 4.10. The hysteresis curves with different average size of the grains with integran interaction of 0.1 are simulated and plotted in Figure 4.11 (a).
The hysteresis curves have a similar tendency during the turning process and the coercivity of the magnets is not so different. As the residual magnetization decreases, the energy product also decreases as the average grain size increases. The surface area of the interfacial region in a given magnet size is inversely proportional to the average grain size.
To analyze the reversing process, the schematic images of the switching magnet and the hysteresis curves to represent the condition are shown in Figure 4.13. To obtain a high energy BH product, the effect of the microstructure on the magnetic properties must be studied. First of all, the degree of anisotropy alignment is the main factor affecting the energy product, which follows the anomalous angle as a tan-type Gaussian distribution.
Exchange-coupled Cylindrical Core/Shell Structure
Model System and Simulation Method
A cylindrical core/shell nanostructure of length L consisting of a hard magnetic core of diameter D surrounded by a soft magnetic shell of thickness t is adopted. In the finite difference calculation of the micromagnetic properties, a cubic cell of 2 × 2 × 2 nm3 was chosen, which was smaller than the exchange length of both phases. We assumed that the z-axis was the easy magnetization axis of the hard phase with a magnetocrystalline anisotropy constant of 3.3 × 106 J/m3, while the shell of the soft phase exhibited no crystalline anisotropy.
We assumed perfect exchange coupling between the hard and soft phases with a harmonic mean value of Aex/Ms for each phase. To obtain complete hysteresis loops, an external magnetic field ranging from −10 to +10 T was applied along the easy axis. To obtain correct BH values, we calculated the mean value of the dot product of H and B in each cell.
To investigate the effect of the volume fraction of the high-magnetization soft phase on the BH, we calculated the full hysteresis loops of the hard-soft core/shell cylinder with different shell thicknesses, t. To maximize the internal field Hd at remanence, we assumed that the hard core had dimensions of D = 48 nm and L = 24 nm.
Simulation Results
The fs value at which the plot deviates from the theoretical estimate depends on the degree of magnetism; the larger the scale, the lower the critical value of fs. As mentioned in the previous paragraph, we demonstrated that a positive core field is essential for achieving a stable exchange-coupled nanostructure magnet with a high BH. In this way, by changing the aspect ratio of the elements, shape anisotropy can help strengthen the exchange.
First, we investigated the effect of the aspect ratio (L/D) of a single exchange-coupled nanostructure magnet. As L increases, the easy axis of shape anisotropy matches that of exchange hardening, allowing the soft-magnetic phase to fully couple with the hard-magnetic phase, resulting in the hysteresis loops illustrated in Figure 5.7(a) and (B). In particular, for the case of D = 24 in Figure 5.7 (b), the negative HN value becomes positive as L increases.
All plots show a similar trend with the shape anisotropy energy for an ellipsoid or cylinder, but with different offsets depending on the scale. Interestingly, when D = 48 nm, HN does not become positive even at significantly high values of L/D, although this is expected to be positive based on the coupling of shape anisotropy and exchange hardening. The nucleation field (HN) from the simulation results is plotted as black dots and its color is linearly interpolated in each diameter.
Since the BH depends only on the rest state, the value of the nucleation field itself does not affect the value of the BH, except for the change in the BH when the sign changes. This may be due to the dipole-dipole interactions between the unit cylinders in the array structure, which reduce the shape anisotropy of the unit cylinders and cause non-uniform switching from cylinder to cylinder in the array structure. On the other hand, in the case of an array consisting of a uniform cylinder (D, t, L, the exchange hardening overcomes the dipole-dipole interaction up to n = 40 with a periodic boundary condition, and HN remains positive, resulting in BH following a quadratic function, maximized at Hd = 1/2Ms.
Furthermore, BH can be maximized by designing unit nanostructures with an optimal geometry and arranging them in an optimal array structure, taking into account the sign of the core field resulting from exchange strengthening and shape anisotropy, as well as the anisotropy of shape of the bulk structure due to dipole-dipole interactions between the unit structures.
Conclusion
Summary
Lee, “Breathing Modes of a Magnetic Skyrmion on a Defective Surface,” Korean Magnetics Society Winter Conference 2020, Republic of Korea, Nov. 26. Materials”, Korean Magnetics Society 2020 Winter Conference, Republic of Korea, 26 Nov. Lee, “Practical Energy Product of Magnetic Materials,” Korean Magnetics Society Summer Conference 2020, Republic of Korea, July 19.
Lee, "The Vortex Core Switching by Bloch Point Pair", The Korean Magnetic Society 2019 Winter Conference, Republic of Korea, 21 Nov. Lee, “Profile of potential energy of a geometrically confined skyrmion”, The Korean Magnetics Society 2019 Winter Conference, Republic of Korea, Nov. 21. Lee, "Diode-like manipulation of magnetic skyrmion transmission using an asymmetric modification of edge potential energy surface", The Korean Magnetics Society 2019 Winter Conference, Republic of Korea, 21 Nov.
Lee, "The Energy Product of Hard- and Soft-magnetic Cylindrical Core/Shell", The Korean Magnetics Society 2019 Summer Conference, Republikken Korea, 22. maj. Lee, "Manipulation of Magnetic Skyrmions Motion in Confined Geometries for Potential Neuromorphic Applications", The Korean Magnetics Society 2018 Winter Conference, Republikken Korea, 23. nov. Lee, "Tredimensional Magnetic Vortex Structure Transformation", The Korean Magnetics Society 2018 Winter Conference Konference, Republikken Korea, 22. nov.
Lee, "Skyrmion manipulation by a rotating current", The Korean Magnetics Society 2018 Winter Conference, Republikken Korea, 22. nov. Lee, "Energy Product Enhancement in Soft-and Hard-magnetic Mixtures", The Korean Magnetics Society 2018 Winter Conference, Republikken Korea, 22. nov. Lee, "A Limited Prediction of Maximum Energy Product from the Magnetization Hysteresis Loop", The Korean Magnetics Society 2018 Winter Conference, Republic of Korea, 22. Nov.
Lee, "Non-Parabolic Confining Potential Model of Magnetic Skyrmion", spomladanska konferenca Korean Magnetics Society 2018, Republika Koreja, 29. marec. Lee, "The symmetry breaking during a transformation from a vortex core to an asymmetric Bloch wall", The Pomladna konferenca Korean Magnetics Society 2018, Republika Koreja, 29. marec. Lee, "Symmetry breaking in 3D flux-closure domain structure", Pomladna konferenca Korean Magnetics Society 2018, Republika Koreja, 25. okt.
![Figure 1.1 Brown’s paradox: The experimentally realized coercivities amount to a factor of 3 – 5 smaller than the ideal nucleation field [9]](https://thumb-ap.123doks.com/thumbv2/123dokinfo/10486651.0/15.892.246.661.142.487/figure-brown-paradox-experimentally-realized-coercivities-smaller-nucleation.webp)
