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The hysteresis loops of a soft cylindrical particle with radius R=40–100 nm have been studied by means of numerical simulation in the range of thickness Lz=10–40 nm, close to the exchange length R0. Qualitatively, three different scenarios of magnetization reversal have been obtained depending on the particle thickness and aspect ratio. For a “thick” particle (Lz≫R0) with aspect ratio Lz/R∼1, the magnetization reversal begins via the nucleation of magnetization curling similar to the case of spherical or ellipsoidal particles of a soft magnetic type. On the contrary, for a particle with a small aspect ratio (Lz/R≪1) the magnetization reversal begins via the nucleation of a bending state, whose magnetization is constrained within the particle plane. For a particle with thickness Lz>R0, the bending state can lose its stability at certain value of applied magnetic field, at which one or two vortices nucleate within the particle. The evolution of the vortex pattern in an applied magnetic field determines the ...

Magnetization curves and hysteresis loops for a dense array of soft cylindrical particles are studied by means of numerical simulation. A stray magnetic field of a neighboring particle is approximated by means of a magnetic field of a certain macrodipole located at the neighboring particle. This approach is justified comparing the results of exact numerical simulation for a cluster of two interacting particles with the corresponding data obtained by means of the macrodipole approximation. The well-known Ewald’s method [P. Ewald, Ann. Phys. 64, 250 (1921)] is used to take into account slowly decreasing contributions of distant particles into a total demagnetizing field of an array. It is found that various types of stable nonuniform micromagnetic states can participate in the magnetization reversal process in soft magnetic particles, namely, one-vortex state, multivortex states, and bending type states. The bending state originates due to instability of quasiuniform magnetization when external magnetic field decreases after array saturation. It is found that for a dense array a characteristic field for array saturation changes considerably as a function of the lattice period, and the shape of the hysteresis loop depends significantly on the direction of the in-plane external magnetic field with respect to the array axes.

Three-dimensional numerical simulation of equilibrium micromagnetic configurations existing in thin ferromagnetic films with surface anisotropy is carried out taking into account the strong demagnetization field acting on the film magnetization and the true micromagnetic boundary condition on the film surface. The numerical results are obtained in the simplest Neel approximation for surface anisotropy energy, a surface anisotropy constant Ks being a single phenomenological parameter. It is found that the spin canted state has the lowest total energy as compared to various multi-domain configurations in the intermediate range of thickness, Lz,min Lz,max, nearly uniform in-plane magnetization, or the vortex has been obtained depending on the film in-plane aspect ratio. On the other hand, different labyrin...

The magnetization reversal process in a soft magnetic nanotube with a weak helical magnetic anisotropy is studied by means of numerical simulation. The origin of a helical anisotropy is a small off-diagonal correction to the magneto-elastic energy density. The change of the external magnetic field parallel to the nanotube axis is shown to initiate a magnetic hysteresis associated with the jumps of the circular magnetization component of the nanotube at a critical magnetic field H s. For a uniform nanotube, the critical magnetic field H s is investigated as a function of geometrical and magnetic parameters of the nanotube. Using 2D micromagnetic simulation, we study the behavior of a nanotube having magnetic defects in its middle part. In this case, the jump of the circular magnetization component starts at the defect. As a result, two bamboo domain walls appear near the defect and propagate to the nanotube ends. Similar effect may explain the appearance of the bamboo domain walls in a slightly non uniform amorphous ferromagnetic microwire with negative magnetostriction during magnetization reversal process.

An adequate description of the results of experimental measurement of both diagonal and off-diagonal Giant magneto-impedance (GMI) components has been obtained for Co-rich amorphous microwire at moderate frequencies assuming the existence of a small off-diagonal tensor component of the residual quenching stress. The latter is the origin of a weak helical anisotropy of amorphous microwire. The micromagnetic simulation of the magnetization reversal process in the microwire under the influence of the applied magnetic field and dc bias current has been carried out. It is shown that due to the influence of the magneto–elastic interaction in a wire with a weak helical anisotropy, the behavior of the longitudinal and circular magnetization components is significantly correlated. Namely, the change of the sign of the longitudinal magnetization component under the influence of the axial magnetic field leads to a subsequent jump of the circular magnetization component at some critical value of the applied magnetic field. As a result of the jump of the circular magnetization, the off-diagonal GMI component also changes sign during the wire magnetization reversal. This effect is confirmed experimentally for a Co-rich wire with a small negative magnetostriction. It is also shown that the jump of the circular magnetization can be eliminated by a circular magnetic field of a weak dc bias current flowing along the wire. This effect allows one to design sensitive magnetic field sensor based on the measurement of the off-diagonal GMI component.

Hysteretic properties of dilute assemblies of various types of magnetosome clusters, i.e., linear chains, closed rings, and random three-dimensional (3D) configurations are studied by means of numerical simulation. It is shown that after averaging over random particle positions and random orientations of the particle easy anisotropy axes, there remain only several physical parameters that determine the shape of the assembly hysteresis loop: the cluster topology, the characteristic value of the magnetostatic interaction field, and the number of the nanoparticles within the cluster. The strong magnetostatic interaction between the particles increases significantly the coercive force of an assembly of linear chains or circular rings. On the other hand, for these assemblies, the type of the random anisotropy assumed as well as the number of the particles within the cluster has only minor effect on the hysteresis loop shape. For an assembly of 3D magnetosome clusters, the remanent magnetization shows strong dependence on the volume fractions of magnetic nanoparticles, contrary to the coercive force behavior.

We study the influence of the distribution of the particles' aspect ratio on the magnetostatic properties of an assembly of nanoparticles with cubic magnetocrystalline anisotropy, such as iron, nickel, iron oxides, and ferrites. Because of the large values of the dimensionless ratio Ms2/|Kc| for these particles, the shape anisotropy energy makes considerable contribution to the total nanoparticle energy even for relatively small shape distortions, with equivalent ellipsoids having aspect ratios ξ ≥ 1.1. As a result, the magnetostatic properties of a randomly oriented assembly of particles with combined anisotropy at ξ ≥ 1.1 are similar to those for an assembly of particles with purely uniaxial magnetic anisotropy. This conclusion is shown to be valid not only for the assembly hysteresis loops but also for the magnetic relaxation characteristics, at least in the high damping limit.

It is shown that the magnetic dynamics of an assembly of nanoparticles dispersed in a viscous liquid differs significantly from the behavior of the same assembly of nanoparticles immobilized in a solid matrix. For an assembly of magnetic nanoparticles in a liquid two characteristic mode for stationary magnetization oscillations are found that can be called the viscous and magnetic modes, respectively. In the viscous mode, which occurs for small amplitude of the alternating magnetic field H0 as compared to the particle anisotropy field Hk, the particle rotates in the liquid as a whole. In a stationary motion the unit magnetization vector and the director, describing the spatial orientation of the particle, move in unison, but the phase of oscillations of these vectors is shifted relative to that of the alternating magnetic field. Therefore, for the viscous mode the energy absorption is mainly due to viscous losses associated with the particle rotation in the liquid. In the opposite regime, H0 ≥ Hk, the dir...

The results of the numerical simulation of field-cooled and zero field-cooled (ZFC) experiments in a dilute assembly of superparamagnetic nanoparticles with uniaxial anisotropy are presented. The numerical simulation uses a solution of the kinetic rate equations for population numbers of the potential wells. The particle relaxation times are rigorously obtained from the corresponding Fokker–Planck equation. For an assembly of particles with a single diameter a monotonic decrease in a blocking temperature as a function of the applied magnetic field is found, the blocking temperatures of aligned and randomly oriented assemblies being close. For an assembly with lognormal volume distribution the location of the maximum at the ZFC assembly magnetization differs considerably from the blocking temperature of particles with the average diameter. Equating of both quantities may lead to a considerable overestimation of the particle effective anisotropy constant. The effective blocking temperature of the assembly w...

The low frequency hysteresis loops of superparamagnetic nanoparticles with uniaxial anisotropy are calculated as a function of the particle diameter, alternating magnetic field amplitude H0, frequency, and particle magnetic parameters both for oriented and nonoriented assemblies. The magnetic field frequency is assumed so high, f>50–100 kHz, that the mechanical rotation of a particle in surrounding medium is restricted. Thus, only the Neel–Brown magnetization relaxation process is taken into account. Pronounced dependence of the specific loss power on the particle diameter is found in the linear regime pertaining to small magnetic field amplitudes. For an oriented assembly of Co nanoparticles with optimal diameter D≈6 nm the specific power loss can be as high as 1600 W/g for typical values H0=200 Oe and f=500 kHz. It is three times less for the corresponding nonoriented assembly. Nevertheless, using of particles with low anisotropy field, i.e., particles of soft magnetic type, seems to be preferable for t...

A simple kinetic approach based on approximate solution of the Fokker–Planck equation for magnetic moment orientations is developed for the calculation of the hysteresis loop of a superparamagnetic nanoparticle assembly with predominantly uniaxial magnetic anisotropy. The hysteresis loops of the oriented assembly have been obtained in the intermediate to high damping limit as a function of temperature at various angles that the applied magnetic field makes with the particle easy anisotropy axis. An analytic approximation is given for the effective energy barriers separating energy wells. The evolution of the hysteresis loop as a function of temperature is shown to take place between the ultimate Stoner–Wohlfarth loop and the equilibrium magnetization curve. Analytical estimates for the coercive force and the blocking temperature are obtained both for ordered and randomly oriented assemblies of uniaxial particles.

The analytical estimates for the relaxation time of a single-domain particle with a nonaxially symmetric double-well potential are obtained in the limits of low and high dissipation regimes. The expressions derived turn out to be valid also for an important case of spheroidal nanoparticle whose easy anisotropy axis is inclined at an arbitrary angle to a shape anisotropy axis. The analytical estimates are compared with the numerical simulation results obtained for the same particle by means of solution of the stochastic Landau–Lifshitz equation at various values of the dimensionless damping parameter κ. It is shown that in the case of moderate reduced energy barrier the high damping limit occurs at κ≥1.0, whereas the low damping limit corresponds to very small values of the damping parameter, κ≤0.003. For intermediate values of κ, a heuristic interpolation formula for the particle relaxation time is suggested. It describes the numerical simulation results with accuracy of several percent.

We have performed both one-dimensional variational and two-dimensional numerical calculations of the magnetic curling structure near the ends of soft magnetic nanotubes whose axes are magnetically favored, and compared the two. The variational calculation is based on the observation that the numerical simulation results for ending curling structure are similar to those for head-to-head domain walls. Except for magnetic fields opposing the average magnetization and larger than a critical value, the wall centers are outside the tube, and there is an appreciable energy barrier against penetration of the tube by the wall. The two calculations are in reasonable agreement for a range of tube dimensions and material parameters. Both predict critical fields lower than those for infinite tubes.

The magnetization distribution in a single domain particle with appreciable surface anisotropy energy contribution is investigated for particles of cylindrical, spherical, and rectangular shapes. It is shown that the behavior of the particle in applied magnetic field can be described using effective energy functional. The latter determines the direction of average particle magnetization in external magnetic field, as well as the particle coercive force, at least for the case when the uniform rotation mode is the easiest one. The nonuniform correction to average particle magnetization is found to be proportional to a small parameter KsL/C⪡1, where Ks is the surface anisotropy constant, L is the characteristic particle size, and C is the exchange constant.