A novel formulation for the numerical computation of magnetization modes in complex micromagnetic systems

The small oscillation modes in complex micromagnetic systems around an equilibrium are numerically evaluated in the frequency domain by using a novel formulation, which naturally preserves the main physical properties of the problem. The Landau–Lifshitz–Gilbert (LLG) equation, which describes magnetization dynamics, is linearized around a stable equilibrium configuration and the stability of micromagnetic equilibria is discussed. Special attention is paid to take into account the property of conservation of magnetization magnitude in the continuum as well as discrete model. The linear equation is recast in the frequency domain as a generalized eigenvalue problem for suitable self-adjoint operators connected to the micromagnetic effective field. This allows one to determine the normal oscillation modes and natural frequencies circumventing the difficulties arising in time-domain analysis. The generalized eigenvalue problem may be conveniently discretized by finite difference or finite element methods depending on the geometry of the magnetic system. The spectral properties of the eigenvalue problem are derived in the lossless limit. Perturbation analysis is developed in order to compute the changes in the natural frequencies and oscillation modes arising from the dissipative effects. It is shown that the discrete approximation of the eigenvalue problem obtained either by finite difference or finite element methods has a structure which preserves relevant properties of the continuum formulation. Finally, the generalized eigenvalue problem is solved for a rectangular magnetic thin-film by using the finite differences and for a linear chain of magnetic nanospheres by using the finite elements. The natural frequencies and the spatial distribution of the natural modes are numerically computed.     

Spin-wave resonance and magnetic anisotropy in FePt thin films

Ferromagnetic Resonance (FMR) measurements at room temperature and X-band microwave frequency were performed on polycrystalline FePt thin films with face-centered cubic structure. With the external field perpendicular to the film plane, the absorption fields H_n of the odd and even spin-wave resonance modes n detected for the Fe_0.44Pt_0.56(45 nm)/Si(1 0 0) and Fe_0.51Pt_0.49(105 nm)/Pt(55 nm)/MgO(1 0 0) films, were found to obey the well-known H_n×n² ratio, giving for these films the exchange stiffness constant values of 3.9×10⁻⁸ and 4.4×10^–7 erg/cm, respectively. The study of the out-of-plane angular dependence of the absorption field of the uniform FMR mode allowed the measurement of the effective magnetic anisotropy constants of 5.3×10⁶ , 6.4×10⁶ , and 6.7×10⁶ erg/cm³, related, respectively, to the [1 1 1], [1 0 0], and [1 1 0] textures present in the films.     

FMR studies in the TbFeCo/GdFeCoSi bilayers

Bilayers, TbFeCo/GdFeCoSi, made by sputtering on glass substrate with buffer and capping layers were studied by measuring the hysteresis loop and by ferromagnetic resonance (FMR). When the field H was applied along the film normal, a double H_C hysteresis loop related to the two sublayers was observed. In ferromagnetic resonance measurements, a peculiar out-of-plane angular dependence of FMR spectrum was obtained. When scanning field H was 0-637 kA/m less than the anisotropy field of TbFeCo sublayer, two FMR peaks were observed. One peak was characteristic of uniaxial and unidirectional anisotropy. The anisotropy constants were obtained by fitting the data with the theory of FMR, and this peak was considered to be related to the low anisotropy GdFeCoSi layer. The second peak appeared only when the dc field H was orientated in a limited angular range around 180°. This peak was considered to be related to an uncoupled interfacial GdFeCoSi sublayer near Al capping layer. However, when H was scanned between 0-1114 kA/m, only one peak is observed due to magnetization reversal of TbFeCo layer with uniaxial anisotropy.     

Magnetic resonance in crystalline La0.9Ca0.1MnO3: Comparative study of bulk and nanometer-sized samples

X-band electron magnetic resonance (EMR) measurements were done at 115?T?600 K on bulk and nanometer size-grain powder single-crystalline samples of La_0.9Ca_0.1MnO₃, in order to study an impact of structural inhomogeneity on magnetic ordering. For the nano-crystal sample, two superimposed EMR lines are observed below 240 K, while for bulk-crystal one, a second line emerges in narrow temperature interval below 130 K. Temperature dependences of resonance field and line width of the main and the secondary line are drastically different. EMR data and complementary magnetic measurements of bulk-crystal sample reveal mixed-magnetic phase, which agrees with the published phase diagram of bulk La_1−xCa_xMnO₃. In a marked contrast, the same analysis for nano-crystal sample shows two phases one of which is definitely ferromagnetic (FM) and other is likely such, or super paramagnetic. The data obtained are interpreted in terms of very different magnetic ground states in the two samples, that is attributed to different randomness of the indirect FM exchange interactions mediated by bound holes.     

Magnetic anisotropy in colossal magnetoresistive FeCr2S4 single crystals

The magnetic properties of spinel FeCr₂S₄ single crystals were investigated by ferromagnetic resonance (FMR). The FMR spectrum displays a single absorption line in the whole temperature range measured for both H∥(111) and H⊥(111). With decreasing temperature, the line with H∥(111) shifts to lower fields, while that with H⊥(111) shifts to higher fields. By superposing all the FMR spectra measured in different directions at 110 K, a double-peak is obtained, which clarifies the origin of the FMR double-peak in polycrystalline sample. By taking account of magnetic anisotropy and demagnetizing effect, the orientation dependence of resonance field is well fitted. It is found that the magnetic anisotropy strengthens with decreasing temperature; however, it has no evident influence on transport and colossal magnetoresistance behavior.     

An effective method for improving diffraction performance of magnetostatic backward volume wave-based magneto-optic Bragg cells by using an appropriately tilted bias magnetic field in YIG film plane

The universal magneto-optic (MO) coupled-mode equations for magnetostatic waves (MSWs) and guided optical waves (GOWs) under arbitrarily tilted bias magnetic fields are presented for the first time and, as an example, applied to the noncollinear Stokes interaction between the incident TE₀-mode light and magnetostatic backward volume wave (MSBVW) excited by single-element microstrip line transducer in yttrium–iron–garnet (YIG) film. Our calculation indicates that, for the case of magnetization parallel to the MSBVW propagation direction, the diffraction efficiency (DE) is equal to the mode-conversion efficiency of the diffracted lights (MCDE) and the calculated curve of relative DE for the MSBVW-based MO Bragg cell in pure YIG waveguide is in good agreement with the experimental data. In contrast, the diffraction performance can be greatly improved by optimizing the bias magnetic field and the DE gain can be increased by 6.3 dB in the tangentially magnetized film. The angular dependences of the DE and the corresponding Bragg angle upon the magnetization direction are also discussed in the paper.     

Determination of anisotropy constants of protein encapsulated iron oxide nanoparticles by electron magnetic resonance

Angle-dependent electron magnetic resonance was performed on 4.9, 8.0, and 19 nm iron oxide nanoparticles encapsulated within protein capsids and suspended in water. Measurements were taken at liquid nitrogen temperature after cooling in a 1 T field to partially align the particles. The angle dependence of the shifts in the resonance field for the iron oxide nanoparticles (synthesized within Listeria-Dps, horse spleen ferritin, and cowpea chlorotic mottle virus) all show evidence of a uniaxial anisotropy. Using a Boltzmann distribution for the particles’ easy-axis direction, we are able to use the resonance field shifts to extract a value for the anisotropy energy, showing that the anisotropy energy density increases with decreasing particle size. This suggests that surface anisotropy plays a significant role in magnetic nanoparticles of this size.     

Transverse susceptibility as the low-frequency limit of ferromagnetic resonance

A new theory of transverse susceptibility (TS) based on magnetization vector dynamics, as described by the Landau–Lifshitz equation of motion, is given. It is shown that the traditional TS experiment is, in fact, the zero-frequency limit of the ferromagnetic resonance (FMR). The importance of these results resides in the generality of the approach which allows one to find the TS for virtually any magnetic system if an expression for the magnetic free-energy density is known. Moreover, the effect of the frequency of excitatory AC field on the TS experiments and the effect of energy dissipation through the imaginary part of TS emerge coherently from the new TS model.     

Magnetic excitations in ferromagnetic semiconductors

Magnetic excitations in a series of GaMnAs ferromagnetic semiconductor films were studied by ferromagnetic resonance (FMR). Using the FMR approach, multi-mode spin wave resonance spectra have been observed, whose analysis provides information on magnetic anisotropy (including surface anisotropy), distribution of magnetization precession within the GaMnAs film, dynamic surface spin pinning (derived from surface anisotropy), and the value of exchange stiffness constant D. These studies illustrate a combination of magnetism and semiconductor physics that is unique to magnetic semiconductors.     

Anisotropy and dynamic properties of Co2MnGe Heusler thin films

Co₂MnGe films of 30 and 50 nm in thickness were grown by RF-sputtering. Their magnetic anisotropies, dynamic properties and the different excited spin wave modes have been studied using conventional ferromagnetic resonance (FMR) and Microstrip line FMR (MS-FMR). From the in-plane and the out-of-plane resonance field values, the effective magnetization (4πM_eff) and the g-factor are deduced. These values are then used to fit the in-plane angular-dependence of the uniform precession mode and the field-dependence of the resonance frequency of the uniform mode and the first perpendicular standing spin wave to determine the in-plane uniaxial, the four-fold anisotropy fields, the exchange stiffness constant and the magnetization at saturation. The samples exhibit a clear predominant four-fold magnetic anisotropy besides a smaller uniaxial anisotropy. This uniaxial anisotropy is most probably induced by the growth conditions.     

Ferromagnetic resonance studies on (Co40Fe40B20)x(SiO2)1−x granular magnetic films

Magnetic properties of granular (Co₄₀Fe₄₀B₂₀)_x(SiO₂)_1−x   thin films (x=0.37-0.53$x=0.37-0.53$) have been studied by ferromagnetic resonance (FMR) technique. Samples have been prepared by ion-beam deposition of Co–Fe–B particles and SiO₂ on sitall ceramic substrate. The FMR measurements have been done for different orientations of DC magnetic field with respect to the sample plane. It was found that the deduced value of effective magnetization from FMR data of the thin granular film is reduced by the volume-filling factor of the bulk saturation magnetization. The overall magnetization changes from 152 to 515 G depending on the ratio of the magnetic nanoparticles in the SiO₂ matrix. From angular measurements an induced in-plane uniaxial anisotropy has been obtained due to the preparation of the film conditions as well.     

Yafet-Kittel-type magnetic ordering in Ni0.35Zn0.65Fe2O4 ferrite detected by magnetosensitive microwave absorption measurements

Magnetosensitive microwave absorption measurements of polycrystalline ferrite Ni_0.35Zn_0.65Fe₂O₄ was carried out at 9.4 GHz (X-band) as a function of temperature. Temperature dependence of the total linewidth (ΔH_pp) deduced from the resonance spectra showed the passage through the Curie point (T_c~430 K). Additionally, the plot ΔH_pp vs. T also indicated the existence of another magnetic phase transition at ~240 K, which can be associated with a Yafet-Kittel-type canting of the magnetic moments. Low-field microwave absorption (LFMA) and the magnetically modulated microwave absorption spectroscopy (MAMMAS) were used to give a further knowledge on this material. For low temperature, these techniques give evidence of a Yafet-Kittel-type canting of the magnetic moments.     

Frequency dependence of the ferromagnetic resonance width in magneto-impedance measurements

Giant magneto-impedance has shown large sensitivities that are of great interest in practical applications. Above certain frequencies the ferromagnetic resonance controls the magneto-impedance behavior, and the resonance width limits the maximum achievable magneto-impedance ratio. In this work we present the evolution of the resonance width as a function of the frequency, determined through magneto-impedance and microwave absorption measurements on a NiFe–Au multilayer thin film. The width of the resonance can be fitted to a curve in the form of Δf∼f⁻¹. This is explained by means of a simple model taking into account all inhomogeneities in the sample through a Gaussian distribution of anisotropy fields, as suggested by the shape of the hysteresis loop.     

Magnetic and high-pressure magnetotransport properties of cesium-substituted lanthanum calcium manganites

The nature of the double-exchange (DE) interaction in lanthanum manganites is studied through chemical substitutions, Cs for La, and high-pressure measurements. Static and high-frequency magnetic measurements and high-pressure electrical transport studies were carried out on bulk polycrystalline and radio-frequency sputtered thin films of La_0.7-xCs_xCa_0.3MnO₃ for x=0-0.1. The samples are found to be cubic. Curie temperature T_c measurements provide evidence for bond-length-related weakening of DE as x is increased from 0 to 0.03. For higher x, the bond-angle-related changes lead to an increase in the strength of DE. High-pressure mangetoresistance data indicate both bond length and bond-angle-related increase of 10–20 K/GPa in T_c with pressure, with the largest increase measured for x=0.03. The rate of increase in the Curie temperature with pressure decreases with increasing T_c. Anomalies are observed in the magnetic parameters for x=0.03. The Cs-concentration dependence of the low-temperature saturation magnetization shows a minimum close to x=0.03. Ferromagnetic resonance studies at x-band reveal a 5% decrease in the g-value for x=0.03 relative to the end members (x=0 and 0.1). The low-field magnetostriction for x=0.03 indicates a relatively strong electron–phonon spin coupling compared to neighboring compositions. Received: 15 May 2000 / Accepted: 24 July 2000 / Published online: 9 November 2000     

Magnetic study of CaMn0.96Mo0.04O3, canting vs. phase separation

CaMn_0.96Mo_0.04O₃ is an example of Mn⁴⁺ rich perovskite manganites, which exhibits a net ferromagnetic component at low temperature, observed by dc magnetization and ac susceptibility. To characterize the magnetic state of this compound, neutron powder diffraction was carried out in the 2-400 K temperature range, showing that it is necessary to use three components (ferromagnetic and G- and A-type antiferromagnetic) to describe it. This particular state is in agreement with the unusual magnetic behaviour observed by macroscopic measurements and is compared to the one observed for manganites with similar Mn valence but obtained by A-site substitution.     

Nonlinear response of uniaxial ferromagnets and applications to surface magnetostatic waves

The nonlinear susceptibilities of uniaxially anisotropic ferromagnets are obtained analytically. The expressions show that the anisotropy effect on the first- and second-order components means just an increased H_1a (the first-order anisotropy field) of the dc field H₀ along the anisotropy axis, but the third-order components are complicated and new terms appear. Applying the above results to surface magnetostatic waves in the films, we find new magnetostatic modes from the joint effect of the anisotropy and nonlinearity since higher powers of frequency are introduced in the dispersion equation by the nonlinearity and anisotropy. Very obvious non-reciprocity is seen from the dispersion curves.     

The peculiarities of coupled electromagnetic and magnetoelastic waves in antiferromagnets

A spectrum of coupled electromagnetic, spin and elastic waves in a two-sublattice antiferromagnet with weak ferromagnetism is theoretically investigated. The influence of the g-factor anisotropy and the transverse and longitudinal relaxation in magnetic subsystem on the spectrum of coupled waves is considered. The most changes of dispersion laws occurred in long-wavelength approximation and near the spin reorientation point, then the vectors of ferromagnetism and antiferromagnetism reoriented onto another crystallographic axis. It is shown that the magnetoelastic, the Dzyaloshinsky and the dipole interactions, the anisotropy of g-factor, the external magnetic field and the longitudinal susceptibility determine the activation of quasiferromagnetic waves. The dispersion laws of quasielectromagnetic and quasielastic waves can change from linear dependence to square. At large damping in magnetic subsystem, one from these modes can become the pure relaxation one.     

Brillouin-zone mapping of the spin waves in a ferromagnetic bilayer system

The eigenproblems of spin waves in a heterogeneous ferromagnetic bilayer system with periodic boundary conditions are solved using the interface-rescaling approach. Brillouin zone mapping and the eigenmodes of the system are investigated. We find three types of spin waves may exist in the system: the bulk mode, the interface mode, and the perfect confined mode. The fine structure of the energy band in the heterogeneous bilayer system is first given for the whole two-dimensional Brillouin zone. Conditions for the existence of the interface mode are discussed. Finally, we analyze the resonant-confined spin waves in bulk modes and their oscillating behavior.