Second-order Anisotropy Constants Research Articles

Magnetic properties of ferromagnetic thin films with three spin layers as described by fourth-order perturbed Heisenberg Hamiltonian

The magnetic properties of ferromagnetic thin films with three spin layers were investigated for the first time using fourth-order perturbed Heisenberg Hamiltonian. Total magnetic energy was determined for ferromagnetic materials with simple cubic lattice. The simulation was carried out for Heisenberg Hamiltonian with different second-order magnetic anisotropy constants in different spin layers. The total magnetic energy periodically varies with spin-exchange interaction, azimuthal angle of spin, and the second-order magnetic anisotropy constant in each spin layer. The peaks along the axis of the angle are closely packed except in the 3-D plot of energy versus angle and second-order magnetic anisotropy constant of the middle spin layer. 2-D plots were perceived by rotating 3-D plots in MATLAB. According to the 3-D plot, the angle between magnetic easy and hard directions is 90 degrees. The minimum energy range was found in the 3-D plot of energy versus angle and the second-order anisotropy constant of the middle spin layer. The samples with the minimum magnetic anisotropy energies have potential application in transformer cores, magnetic switching, and magnetic amplifiers.

Magnon Modes for a Magnetic Disc in a Cone Vortex State

A cone phase appears as the ground state of uniaxial magnets with comparable second-order and fourth-order anisotropy constants. In such systems, specific vortex states can exist. The authors have studied magnetization oscillations in small cylindrical particles in a cone vortex state. The study has determined the frequencies of magnon excitations in the presence of a vortex in the cone phase of a ferromagnet. It has been established that there is a significant splitting of the doublets of modes with opposite values ​​of the azimuthal number m, including modes with |m| > 1.

Enhancement of magnetization damping coefficient of permalloy thin films with dilute Nd dopants

For spintronics application, which requires fast field switching, it is important to have a kind of soft magnetic material with large damping coefficient. Here, we present the studies of the Nd dopant-level-dependent damping coefficient of Nd${}_{x}$-Py${}_{(1\ensuremath{-}x)}$ thin films (30 nm) in a dilute region utilizing ferromagnetic resonance (FMR). With the Nd content increasing, the film structure was found to be changing from polycrystalline to amorphous when the Nd content is around 3.4%. Meanwhile, the magnetization decreases linearly. Interestingly, we find that along the easy axis, both low coercivity and high hysteresis squareness are simultaneously maintained in the system; i.e., the magnetic softness has been well kept. By theoretical fitting of the angular dependence of the FMR field, the first- and second-order magnetic anisotropy constants, ${K}_{1}$ and ${K}_{2}$, and the Lande $g$ factor are obtained and discussed quantitatively. The measurements of angular and frequency dependence of the ferromagnetic resonance linewidth, as well as the theoretical fitting by considering the contributions of Gilbert damping, two-magnon scattering, and inhomogeneous broadening, show that the damping coefficient $\ensuremath{\alpha}$ increases rapidly (about 25-fold) as the Nd content increases to 11.6%, which is mainly due to the enhanced spin-orbit coupling by the Nd additives, supported by x-ray magnetic circular dichroism measurements.

Field-induced lattice deformation contribution to the magnetic anisotropy

Elastic and magnetoelastic effects are important to understand physical properties of many magnetic systems. Particularly, the influence of field-induced lattice deformation in the magnetic anisotropy and non-linear terms are commonly needed to describe the magnetic behavior of nanosystems. In this work, we exploit the contribution of field-induced lattice deformation to magnetic anisotropy for systems with hexagonal and cubic structures. Analytical expressions for changes in the first- and second-order magnetic anisotropy constants were obtained for systems which experience general and biaxial strains. Our estimates using bulk parameters available for ferromagnetic materials suggest that field-induced anisotropy contributions can become comparable to magnetocrystalline anisotropy, if the system possesses a strong magnetoelastic response.

Domain size in systems with canted magnetization

The equilibrium domain size in ferromagnetic single-layer films with canted magnetization, as observed in the spin-reorientation transition, is analytically described. It is demonstrated that for canted magnetization in the case of narrow domain walls the size of the magnetic domains depends solely on the second-order anisotropy constant and not on the first-order anisotropy. The single-layer model is extended to the application to multilayers. An approximation for the domain size of multilayers is developed and used for the analysis of data obtained via soft x-ray holography. Our analysis, including the canting angle, allows for the determination of the relative anisotropy constants to second order.

Magnetic anisotropy of mesoscale-twinned Ni-Mn-Ga thin films

The ferromagnetic resonance data obtained for the twinned orthorhombic martensitic phase of Ni-Mn-Ga film epitaxially grown on MgO(100) substrate are presented. The reported data prove that the mesoscale twinning reduces the value of in-plane magnetic anisotropy field of the Ni-Mn-Ga film by an order of magnitude. The reduced magnetic anisotropy field corresponds to the tetragonal symmetry, while the unit cells of the film are orthorhombic. The experimentally observed change of the in-plane magnetic anisotropy is explained in the framework of the magnetoelastic model of martensite, and the second-order and fourth-order magnetic anisotropy constants are evaluated. The perpendicular magnetic anisotropy constant proved to be negative and small in the absolute value. Therefore the estimated value of the magnetic domain wall width is comparable with the widths of mesoscale twins observed in the investigated film. This confirms an idea that the magnetic vectors of twin components are strongly coupled by the exchange interaction.

Anisotropy of zigzag chains of palladium

Ab initio calculations of the biaxial anisotropy of infinite Pd zigzag chains are presented. The simulations were performed with the Vienna Ab-Initio Simulation Package, using a tetragonal unit cell with a supercell approach where the atoms are repeated along the z-direction. The anisotropy is determined from the energies along the three principal directions [001], [100], and [010]. The second-order anisotropy constants K1 and K1′ were extracted by fitting the calculated energy values to the phenomenological energy. The easiest magnetization direction is along the wire axis, whereas the hardest direction is perpendicular to the plane of the wire. The calculated anisotropy constants K1 and K1′ are 5.5 MJ/m3 and −13.9 MJ/m3, respectively These anisotropies are large, but they are smaller than those of linear chains, because the zigzag shape leads to a quenched orbital moment.

119Sn Mössbauer spectroscopy study of the magnetic properties of the HfFe 6Ge 6-type compounds: SmMn 6Sn 4Ge 2 and GdMn 6Sn 4Ge 2

The HfFe 6Ge 6-type compound SmMn 6Sn 4Ge 2 has been studied by single-crystal magnetisation and 119Sn Mössbauer spectroscopy. The compound orders ferromagnetically at T c = 420 K and displays an easy-axis anisotropy from T c to T SR = 130 K. Below T SR, both magnetisation and 119Sn Mössbauer spectroscopy measurements indicate a deviation from the [0 0 1] direction and the presence of easy-cone anisotropy. The angle of the moments with respect to the [0 0 1] direction is estimated to 26–31° from Mössbauer spectroscopy results, in good accordance with the magnetisation results. The isotypic compounds GdMn 6Sn 4Ge 2 and GdMn 6Sn 6 studied by 119Sn Mössbauer spectroscopy display easy plane anisotropy in the whole temperature range 300–4.2 K. The anisotropy behaviours of the LMn 6Sn 4Ge 2 compounds are discussed and the coexistence of easy cone anisotropy for both the SmMn 6Sn 4Ge 2 and HoMn 6Sn 4Ge 2 suggests the play of a positive second-order anisotropy constant of the Mn sublattice.

Temperature dependence of the uniaxial magnetic anisotropy of rhombohedral antiferromagnetic crystals with ions in the S state

The contributions of the anisotropic exchange to the firstand second-order uniaxial anisotropy constants have been calculated for hematite at T = 0 K and arbitrary temperatures. The results of the calculations and the most significant mechanisms are taken into account in interpreting the temperature dependence of the anisotropy in rhombohedral antiferromagnetic crystals. The first-order anisotropy constant for hematite is described by the dipole interaction, contributions of the “single-ion” nature, and relatively small contributions of the anisotropic exchange. The second-order constant for hematite includes the “single-ion” contribution and the contribution of the anisotropic exchange. For FeBO3 and MnCO3 crystals, the main contributions to the first-order anisotropy constant come from the dipole and single-ion mechanisms.

Determination of the in-plane anisotropy field in hexagonal systems via rotational magnetization: Theoretical model and Monte Carlo simulations

The magnetic anisotropy field in thin films with in-plane uniaxial anisotropy can be deduced from the VSM magnetization curves measured in magnetic fields of constant magnitudes. This offers a new possibility of applying rotational magnetization curves to determine the first- and second-order anisotropy constant in these films. In this paper we report a theoretical derivation of rotational magnetization curve in hexagonal crystal system with easy-plane anisotropy based on the principle of the minimum total energy. This model is applied to calculate and analyze the rotational magnetization process for magnetic spherical particles with hexagonal easy-plane anisotropy when rotating the external magnetic field in the basal plane. The theoretical calculations are consistent with Monte Carlo simulation results. It is found that to well reproduce experimental curves, the effect of coercive force on the magnetization reversal process should be fully considered when the intensity of the external field is much weaker than that of the anisotropy field. Our research proves that the rotational magnetization curve from VSM measurement provides an effective access to analyze the in-plane anisotropy constant K3 in hexagonal compounds, and the suitable experimental condition to measure K3 is met when the ratio of the magnitude of the external field to that of the anisotropy field is around 0.2.

Perpendicular Ferromagnetic-Resonance Equations Derived by Effective Demagnetizing Factors

Effective demagnetizing factors (Nxe, Nye) of single-crystal films having cubic or uniaxial anisotropy are derived by new Nxe and Nye equations for the use of Kittel's ferromagnetic resonance formula. Nxe and Nye are obtained by one kind of first derivative (∂G/∂θ). Both first-order and second-order magnetocrystalline anisotropy constants are treated. Six arrangements of films with the static field along the film normal are calculated: (1) perpendicular uniaxial anisotropy, (2) in-plane uniaxial anisotropy, (3) cubic-crystal (001) face, (4) cubic-crystal (011) face, (5) cubic-crystal (111) face, and (6) oblique anisotropy.

Magnetic properties of hard magnetic FePt prepared by cold deformation

The second-order anisotropy constant, K/sub 1/(T), in tetragonal L1/sub 0/ FePt is deduced from the analysis of the magnetization versus field curves, between 4.2 K and 700 K. K/sub 1/(T) varies approximately as expected for local moments in the classical limit. The temperature dependence of the coercive field H/sub c/(T) is discussed with reference to the so-called micromagnetic and global models. The observed behavior suggests that coercivity is governed by passage-expansion or de-pinning. Dipolar interactions are shown to increase the coercive field, rather than reduce it, which is usually the case.

Single-ion anisotropy in compounds of rare earths other than Sm: Importance of J-mixing in the room-temperature range

Contrary to the established view, J-mixing is found to strongly affect the magnetic anisotropy of compounds of all light rare earths and also of terbium. The fractional contribution of J-mixing to second-order anisotropy constants in the room-temperature range is proportional to absolute temperature and depends on the rare earth element involved but not on the characteristics of the specific solid. For the light rare earths this contribution is given by 12(2J−1)−1kT/Δso, where Δso is the spin-orbit splitting between the centers of gravity of the ground and first excited multiplets, and amounts to 22% for Pr and Nd and to as much as 83% for Sm at T=400 K. For the heavy rare earths the corresponding expression is 12(2J+3)−1kT/Δso, which is 11% for Tb at T=400 K and significantly less for the rest of the rare earth series. Analytical expressions are obtained which allow for the J-mixing and are accurate in the room-temperature range for all rare earths with the possible exception of Sm.

Orientation transitions in the domain wall structure of orthoferrites

The effect of higher anisotropy constants on the orientational phase transitions in the domain walls (DWs) in orthoferrites of the GxFz type is studied. Equations of the reduced description of the DW structure in the vicinity of the transition point (determined by the equality of two independent second-order crystallographic anisotropy constants) are constructed, which are analogous to the Slonczewski equations for uniaxial ferromagnets. It is established that, in addition to DWs of the known (ab and ac) types, stable DWs with different spin rotation planes may exist depending on the relationships between fourth-order anisotropy constants. In some cases, the DW spectra may contain an antisymmetric local level in addition to the DW translational mode.

Second-order spin-reorientation transition via magnetoelastic coupling inCoxPd1−xalloy films

We report an observation of smooth spin-reorientation transitions from in-plane to normal-to-plane direction with increasing Pd overlayer thickness in ${\mathrm{Co}}_{x}{\mathrm{Pd}}_{1\ensuremath{-}x}$ alloy films, showing characteristics of a second-order transition (SOT). The magnetoelastic coupling term $\ensuremath{\Gamma}$ plays a crucial role in such behavior via its contribution to the fourth- as well as second-order anisotropy constants. The nature of the SOT is characterized by $\ensuremath{-}{\ensuremath{\Gamma}}_{4}/{\ensuremath{\Gamma}}_{2}=0.12--0.47$ determined for $x=0.25--0.40$ in conjunction with lattice-misfit strains. The controlling spin orientation or anisotropy axis via magnetoelastic coupling without applying a magnetic field may be applicable to miniatured memory and sensor devices.

Effect of the second-order anistropy constant on the transverse susceptibility of uniaxial ferromagnets

A generalized theoretical approach of the transverse susceptibility, χT, in the case of uniaxial ferromagnets is presented. This advances the classical model of transverse susceptibility where only the first-order anisotropy constant, K1, is taken into account and makes possible the study of the influence of the second-order anisotropy constant, K2 on the χT curves. It is shown that additional Barkhausen jumps driven by higher-order anisotropy constants emerge more evidently in the field dependence of χT making their detection much easier than from the hysteresis loop. Based on the obtained results, we propose a simple method to determine both, K1 and K2 independently from χT experiments.

Magnetism and overlayer-induced reorientation of magnetization in thin itinerant-electron films

The magnetic properties and overlayer-induced reorientation of magnetization in thin ferromagnetic films are investigated within the single-band Hubbard model. A generalization of modified alloy analogy (MAA) is applied to deal with modifications due to the reduced translational symmetry. Within the MAA the actual type of alloy analogy is determined self-consistently. The MAA allows, in particular, the investigation of quasiparticle lifetime effects in the paramagnetic as well as the ferromagnetic phase. The layer magnetizations are calculated as a function of temperature and film thickness and are microscopically analyzed in detail with help of the spin- and layer-dependent quasiparticle density of states. By calculating the overlayers dependence of the second-order anisotropy constants, we find that very thin overlayers can induce a reorientation of magnetization from in-plane to out-of-plane directions, however, thick overlayers rotate the magnetization from out-of-plane to in-plane directions again. In addition, the temperature and thickness induced reorientation of magnetization can also be understood in our theory.

Micromagnetic study of reversible transverse susceptibility

A generalized theoretical approach of the transverse susceptibility in the case of uniaxial ferromagnets is presented. This advances the classical model of transverse susceptibility where only the first-order anisotropy constant is taken into account, and makes possible the study of the influence of the second-order anisotropy constant on the transverse susceptibility curves. The complicated switching behavior occurring in certain cases was analyzed and discussed from the dynamical point of view, comparing the results with those obtained with a Landau–Lifshitz–Gilbert approach.

A computer simulation study on a two-sublattice magnetic system

We report on the magnetization, its first- and second-field derivatives, and torque on a PrCo5 single crystal using the two-sublattice model in the mean field approximation. The anisotropy energy of the compound is described by first- and second-order anisotropy constants of the Co and Pr sublattices. The structure of the magnetization, its field derivatives and torque fairly supports the working of this model. Comparison with the magnetization behavior, however, in the absence of canting shows some variance with the two-sublattice model especially as the magnetic saturation is approached.

Comparison among the second-order anisotropy constants of RE (Pr, Nd, Sm) and Fe sublattices in the RE 2Fe 17 rhombohedral structure. Effects of Ti substitution for Fe

The temperature and composition dependence of the anisotropy field H A of the planar Pr 2(FeTi) 17 compound has been measured. From the deduced overall second-order anisotropy constant K 1=− H A M s/2, the Pr contribution has been obtained and compared with those of Nd and Sm in the same structure and composition. All RE as well as Fe contributions to the anisotropy are found to be planar (K RE 1, K Fe 1<0) in the considered temperature and composition ranges. Ti substitution markedly reduces the Sm planar anisotropy while it reinforces the negative contribution of Nd (slightly), Pr and Fe. The effect of Ti on crystal field parameters has been analyzed.