Periodic Boundary Conditions for Finite-Differentiation-Method Fast-Fourier-Transform Micromagnetics

We describe an accurate periodic boundary condition （PBC） called the symmetric PBC in the calculation of the magnetostatie interaction field in the finite-differentiation-method fast-Fourier-transform （FDM-FFT） micromagneties. The micromagnetic cells in the regular mesh used by the FDM-FFT method are finite-sized elements, but not geometrical points. Therefore, the key PBC operations for FDM-FFT methods are splitting and relocating the micromagnetic cell surfaces to stay symmetrically inside the box of half-total sizes with respect to the origin. The properties of the demagnetizing matrix of the split micromagnetic cells are discussed, and the sum rules of demagnetizing matrix are fulfilled by the symmetric PBC.

Creation and annihilation of artificial magnetic skyrmions with the electric field

Recent theory and experiments show that artificial magnetic skyrmions can be stabilized at room temperature without the need for the external magnetic field,casting strong potentials for the device applications.In this work,we study the electric field manipulation of artificial magnetic skyrmions imprinted by Co disks on CoPt multilayers utilizing the micromagnetic simulations.We find that the reversible annihilation and creation of skyrmions can be realized with the electric field via the strain mediated magnetoelastic coupling.In addition,we also demonstrate controllable manipulation of individual skyrmion,which opens a new platform for constructing magnetic field-free and low-energy dissipation skyrmion based media.

Consistency between domain wall oscillation modes and spin wave modes in nanostrips

Investigations on domain wall(DW) and spin wave(SW) modes in a series of nanostrips with different widths and thicknesses have been carried out using micromagnetic simulation. The simulation results show that the frequencies of SW modes and the corresponding DW modes are consistent with each other if they have the same node number along the width direction. This consistency is more pronounced in wide and thin nanostrips, favoring the DW motion driven by SWs.Further analysis of the moving behavior of a DW driven by SWs is also carried out. The average DW speed can reach a larger value of ~ 140 m/s under two different SW sources. We argue that this study is very meaningful for the potential application of DW motion driven by SWs.

HIGH ACCURACY NUMERICAL METHOD OF THIN-FILM PROBLEMS IN MICROMAGNETICS

In this paper, a new high accuracy numerical method for the thin-film problems of micron and submicron size ferromagnetic elements is proposed. For the computation of stray field, we use the finite element method(FEM) by introducing a semi-discrete artificial boundary condition [1, 2]. In our numerical experiments about the domain patterns and their movement, we can see that the results are accordant to that of experiments and other numerical methods. Our method are very convenient to deal with arbitrary shape of thin films such as a polygon with high accuracy.

Micromagnetic study of magnetization reversal in inhomogeneous permanent magnets

Macroscopic magnetic properties of magnets strongly depend on the magnetization process and the microstructure of the magnets.Complex materials such as hard-soft exchange-coupled magnets or just real technical materials with impurities and inhomogeneities exhibit complex magnetization behavior.Here we investigate the effects of size,volume fraction,and surroundings of inhomogeneities on the magnetic properties of an inhomogeneous magnetic material via micromagnetic simulations.The underlying magnetization reversal and coercivity mechanisms are revealed.Three different demagnetization characteristics corresponding to the exchange coupling phase,semi-coupled phase,and decoupled phase are found,depending on the size of inhomogeneities.In addition,the increase in the size of inhomogeneities leads to a transition of the coercivity mechanism from nucleation to pinning.This work could be useful for optimizing the magnetic properties of both exchange-coupled nanomagnets and inhomogeneous single-phase magnets.

Magnonic band-pass and band-stop filters with structurally modulated waveguides

Magnonics is a fascinating and emerging field, which mainly studies processing information with spin waves.Magnonic devices with in-plane magnetization have recently been realized. Because of the isotropic propagation, magnonic devices based on perpendicular magnetization are attracting extensive interest. Here, we numerically demonstrate two magnonic filters with out-of-plane magnetization using micromagnetic simulations. The band-pass and the band-stop functions have been realized in two structurally modulated waveguides, respectively. The intensity of spin waves is manipulated when they arrive at the uniformly/non-uniformly magnetized modulators, which results in the variation of transmission coefficients. It is found that the proposed filters can work at multiple frequencies, which can be further adjusted by the external magnetic field. Our designed magnonic devices with Néel-type skyrmion could promote the development of spin wave computing using spin textures.

Skyrmion-based logic gates controlled by electric currents in synthetic antiferromagnet

Skyrmions in synthetic antiferromagnetic(SAF) systems have attracted much attention in recent years due to their superior stability, high-speed mobility, and completely compensated skyrmion Hall effect. They are promising building blocks for the next generation of magnetic storage and computing devices with ultra-low energy and ultra-high density.Here, we theoretically investigate the motion of a skyrmion in an SAF bilayer racetrack and find the velocity of a skyrmion can be controlled jointly by the edge effect and the driving force induced by the spin current. Furthermore, we propose a logic gate that can realize different logic functions of logic AND, OR, NOT, NAND, NOR, and XOR gates. Several effects including the spin–orbit torque, the skyrmion Hall effect, skyrmion–skyrmion repulsion, and skyrmion–edge interaction are considered in this design. Our work may provide a way to utilize the SAF skyrmion as a versatile information carrier for future energy-efficient logic gates.

In-plane spin excitation of skyrmion bags

Skyrmion bags are spin structures with arbitrary topological charges, each of which is composed of a big skyrmion and several small skyrmions. In this work, by using an in-plane alternating current(AC) magnetic field, we investigate the spinwave modes of skyrmion bags, which behave differently from the clockwise(CW) rotation mode and the counterclockwise(CCW) rotation mode of skyrmions because of their complex spin topological structures. The in-plane excitation power spectral density shows that each skyrmion bag possesses four resonance frequencies. By further studying the spin dynamics of a skyrmion bag at each resonance frequency, the four spin-wave modes, i.e., a CCW-CW mode, two CW-breathing modes with different resonance strengths, and an inner CCW mode, appear as a composition mode of outer skyrmion–inner skyrmions. Our results are helpful in understanding the in-plane spin excitation of skyrmion bags, which may contribute to the characterization and detection of skyrmion bags, as well as the applications in logic devices.

Multi-segmented nanowires for vortex magnetic domain wall racetrack memory

A vortex domain wall's(VW) magnetic racetrack memory's high performance depends on VW structural stability,high speed, low power consumption and high storage density. In this study, these critical parameters were investigated in magnetic multi-segmented nanowires using micromagnetic simulation. Thus, an offset magnetic nanowire with a junction at the center was proposed for this purpose. This junction was implemented by shifting one portion of the magnetic nanowire horizontally in the x-direction(l) and vertically(d) in the y-direction. The VW structure became stable by manipulating magnetic properties, such as magnetic saturation(M_(4)) and magnetic anisotropy energy(K_(u)). In this case, increasing the values of M_(4) ≥ 800 kA/m keeps the VW structure stable during its dynamics and pinning and depinning in offset nanowires,which contributes to maintenance of the storage memory's lifetime for a longer period. It was also found that the VW moved with a speed of 500 m/s, which is desirable for VW racetrack memory devices. Moreover, it was revealed that the VW velocity could be controlled by adjusting the offset area dimensions(l and d), which helps to drive the VW by using low current densities and reducing the thermal-magnetic spin fluctuations. Further, the depinning current density of the VW(J_(d)) over the offset area increases as d increases and l decreases. In addition, magnetic properties, such as the M_(4) and K_(u),can affect the depinning process of the VW through the offset area. For high storage density, magnetic nanowires(multisegmented) with four junctions were designed. In total, six states were found with high VW stability, which means three bits per cell. Herein, we observed that the depinning current density(J_(d)) for moving the VW from one state to another was highly influenced by the offset area geometry(l and d) and the material's magnetic properties, such as the M_(4) and K_(u).

Optimization of the grain boundary diffusion process by doping gallium and zirconium in Nd–Fe–B sintered magnets

As the channel for grain boundary diffusion(GBD)in Nd–Fe–B magnets,grain boundary(GB)phases have a very important effect on GBD.As doping elements that are commonly used to regulate the GB phases in Nd–Fe–B sintered magnets,the influences of Ga and Zr on GBD were investigated in this work.The results show that the Zr-doped magnet has the highest coercivity increment(7.97 kOe)by GBD,which is almost twice that of the Ga-doped magnet(4.32 kOe)and the magnet without Ga and Zr(3.24 kOe).Microstructure analysis shows that ZrB_(2)formed in the Zr-doped magnet plays a key role in increasing the diffusion depth.A continuous diffusion channel in the magnet can form because of the presence of ZrB_(2).ZrB_(2)can also increase the defect concentration in GB phases,which can facilitate GBD.Although Ga can also improve the diffusion depth,its effect is not very obvious.The micromagnetic simulation based on the experimental results also proves that the distribution of Tb in the Zr-doped magnet after GBD is beneficial to coercivity.This study reveals that the doping elements Ga and Zr in Nd–Fe–B play an important role in GBD,and could provide a new perspective for researchers to improve the effects of GBD.

Investigation on mechanism of magnetization reversal for nanocrystalline Pr-Fe-B permanent magnets by micromagnetic finite element methods

Magnetization configurations were calculated under various magnetic fields for nanocrystalline Pr-Fe-B permanent magnets by micromagnetic finite element method.According to the configurations during demagnetization process, the mechanism of magnetization reversal was analyzed.For the Pr2Fe14B with 10 nm grains or its composite with 10vol.% α-Fe, the coercivity was determined by nucleation of reversed domain that took place at grain boundaries.However, for Pr2Fe14B with 30 nm grains, coercivity was controlled by pinning of the nucle-ated domain.For Pr2Fe14B/α-Fe with 30vol.% α-Fe, the demagnetization behavior was characterized by continuous reversal of α-Fe moment.

Micromagnetic Simulation of Magnetization Reversal in SmCo_5/Sm_2Co_(17) Magnets

A three-dimensional finite element micromagnetic algorithm was developed to study the magnetization reversal of the SmCo 5/Sm 2Co 17 based magnets. The influences of the microstructure and magnetic parameters on the coercivity were studied based on the model consisting of 64 irregular cells according to the experimental microstructure. Numerical results show that the coercivity increases with increasing the 2∶17-type cell size. Large cell boundary thickness leads to small coercivity. The drop of anisotropy constant of 1∶5 phase leads to the coercivity reducing, while the effect of exchange constant of 1∶5 phase on coercivity is contrary to that of exchange constant. The calculated field dependence of coercivity can be predicted by an inhomogeneous domain-wall pinning model. The microstructure parameter was analyzed by comparing the calculated coercivity.

Effect of exchange coupling on magnetic property in Sm–Co/α-Fe layered system

The hysteresis loops as well as the spin distributions of Sm-Co/a-Fe bilayers have been investigated by both three- dimensional （3D） and one-dimensional （1D） micromagnetic calculations, focusing on the effect of the interface exchange coupling under various soft layer thicknesses ts. The exchange coupling coefficient Alas between the hard and soft ,layers varies from 1.8 x10-6 erg/cm to 0.45 x 10-6 erg/cm, while the soft layer thickness increases from 2 nm to 10 nm. As the exchange coupling decreases, the squareness of the loop gradually deteriorates, both pinning and coercive fields rise up monotonically, and the nucleation field goes down. On the other hand, an increment of the soft layer thickness leads to a significant drop of the nucleation field, the deterioration of the hysteresis loop squareness, and an increase of the remanence. The simulated loops based on the 3D and 1D methods are consistent with each other and in good agreement with the measured loops for Sm-Co/a-Fe multilayers.

Computational Design of Rare-Earth Reduced Permanent Magnets

Multiscale simulation is a key research tool in the quest for new permanent magnets.Starting with first principles methods,a sequence of simulation methods can be applied to calculate the maximum possible coercive field and expected energy density product of a magnet made from a novel magnetic material composition.Iron(Fe)-rich magnetic phases suitable for permanent magnets can be found by means of adaptive genetic algorithms.The intrinsic properties computed by ab initio simulations are used as input for micromagnetic simulations of the hysteresis properties of permanent magnets with a realistic structure.Using machine learning techniques,the magnet’s structure can be optimized so that the upper limits for coercivity and energy density product for a given phase can be estimated.Structure property relations of synthetic permanent magnets were computed for several candidate hard magnetic phases.The following pairs(coercive field(T),energy density product(kJ·m^-3))were obtained for iron-tin-antimony(Fe3Sn0.75Sb0.25):(0.49,290),L10-ordered iron-nickel(L10 FeNi):(1,400),cobalt-iron-tantalum(CoFe6Ta):(0.87,425),and manganese-aluminum(MnAl):(0.53,80).

Three-and two-dimensional calculations for the interface anisotropy dependence of magnetic properties of exchange-spring Nd2Fe(14)B/α-Fe multilayers with out-of-plane easy axes

Hysteresis loops,energy products and magnetic moment distributions of perpendicularly oriented Nd2Fe(14)B/α-Fe exchange-spring multilayers are studied systematically based on both three-dimensional(3D)and one-dimensional(1D)micromagnetic methods,focused on the influence of the interface anisotropy.The calculated results are carefully compared with each other.The interface anisotropy effect is very palpable on the nucleation,pinning and coercive fields when the soft layer is very thin.However,as the soft layer thickness increases,the pinning and coercive fields are almost unchanged with the increment of interface anisotropy though the nucleation field still monotonically rises.Negative interface anisotropy decreases the maximum energy products and increases slightly the angles between the magnetization and applied field.The magnetic moment distributions in the thickness direction at various applied fields demonstrate a progress of three-step magnetic reversal,i.e.,nucleation,evolution and irreversible motion of the domain wall.The above results calculated by two models are in good agreement with each other.Moreover,the in-plane magnetic moment orientations based on two models are different.The 3D calculation shows a progress of generation and disappearance of vortex state,however,the magnetization orientations within the film plane calculated by the 1D model are coherent.Simulation results suggest that negative interface anisotropy is necessarily avoided experimentally.

SIMULATION OF THE MAGNETIZATION REVERSAL PROCESS OF RECTANGLE-SHAPED NiFe FILM ELEMENTS UNDER AN ORTHOGONAL MAGNETIC FIELD

The magnetization reversal process of nano-size rectangle-shaped NiFe film elements with different aspect ratios have been investigated under the orthogonally applied magnetic fields by micromagnetic simulation. Different magnetization reversal modes can appear depending on whether the bias field is applied or not. When there is no bias field, double “C” state is the initial reversal state. However, when there is a bias field, “S” state is the starting mode. The larger the aspect ratio is, the larger the switching field is. But, when the aspect ratio is larger than 3, the increase of the switching field ceases. These results can provide useful information to the application of the patterned NiFe film with rectangular elements.

Magnetization Reversal for Ni Nanowires Studied by Micromagnetic Simulations

The magnetization reversal mechanisms for Ni nanowires with different diameters were investigated by micromagnetic simulations. The results show that the reversal mechanisms are significantly dependeht on the diameter of wire. For very thin wires, the reversal occurs by pseudo-coherent rotation. With increasing diameter, magnetization reversal takes place via different nucleation （the transverse domain wall and the vortex domain wall） and subsequent propagation. The reason of transition from the transverse domain wall to the vortex domain wall is given by analytical studies. With further increase of the diameter, the reversal nuclear domain wall becomes tundishoshaped form. As the diameter increases, the width of wall becomes larger.

Micromagnetic simulations of reversal magnetization in cerium-containing magnets

Single-grain models with different cerium contents or structural parameters have been introduced to investigate the reversal magnetization behaviors in cerium-containing magnets. All the micromagnetic simulations are carried out via the object oriented micromagnetic framework(OOMMF). As for single(Nd,Ce)_2 Fe_(14)B type grain, the coercivity decreases monotonously with the increase of the cerium content. Four types of grain structure have been compared: single(Nd,Ce)_2 Fe_(14)B type, core((Nd,Ce)_2 Fe_(14)B)-shell(Nd_2 Fe_(14)B) type with 2 nm thick shell, core(Ce_2 Fe_(14)B)-shell(Nd_2 Fe_(14)B) type, and core(Nd_2 Fe_(14)B)-shell(Ce_2 Fe_(14)B) type. It is found that core((Nd,Ce)_2 Fe_(14)B)-shell(Nd_2 Fe_(14)B)type grain with 2 nm thick shell always presents the largest coercivity under the same total cerium content. Furthermore,the relationship between the coercivity and the shell thickness t in core((Nd,Ce)_2 Fe_(14)B)-shell(Nd_2 Fe_(14)B) type grain has been studied. When the total cerium content is kept at 20.51 at.%, the analyzed results show that as t varies from 1 nm to 7 nm, the coercivity gradually ascends at the beginning, then quickly descends after reaching the maximum value when t = 5 nm. From the perspective of the positions of nucleation points, the reasons why t affects the coercivity are discussed in detail.

Magnetic vortex gyration mediated by point-contact position

Micromagnetic simulation is employed to study the gyration motion of magnetic vortices in distinct permalloy nanodisks driven by a spin-polarized current. The critical current density for magnetic vortex gyration, eigenfrequency, trajectory, velocity and the time for a magnetic vortex to obtain the steady gyration are analyzed. Simulation results reveal that the magnetic vortices in larger and thinner nanodisks can achieve a lower-frequency gyration at a lower current density in a shorter time. However, the magnetic vortices in thicker nanodisks need a higher current density and longer time to attain steady gyration but with a higher eigenfrequency. We also find that the point-contact position exerts different influences on these parameters in different nanodisks, which contributes to the control of the magnetic vortex gyration. The conclusions of this paper can serve as a theoretical basis for designing nano-oscillators and microwave frequency modulators.

Micromagnetic studies of perpendicular recording FePt media with controllable grain size distributions

A 3-dimensional (3D) micromagnetic model combined with Fast Fourier Transform (FFT) method was built up to study the writability in the L10 FePt perpendicular medium. The effects of controllable grain size distributions were studied by grain growth simulation. It is found that the cross-track-averaged magnetization changes little between the L10 FePt medium with uniform or non-uniform grain size distribution.