High-power Ferromagnetic Resonance Research Articles

Computational micromagnetics based on normal modes: Bridging the gap between macrospin and full spatial discretization

The Landau–Lifshitz equation governing magnetization dynamics is written in terms of the amplitudes of normal modes associated with the micromagnetic system’s appropriate ground state. This results in a system of nonlinear ordinary differential equations (ODEs), the right-hand side of which can be expressed as the sum of a linear term and nonlinear terms with increasing order of nonlinearity (quadratic, cubic, etc.). The application of the method to nanostructured magnetic systems demonstrates that the accurate description of magnetization dynamics requires a limited number of normal modes, which results in a considerable improvement in computational speed. The proposed method can be used to obtain a reduced-order dynamical description of magnetic nanostructures which allows to adjust the accuracy between low-dimensional models, such as macrospin, and micromagnetic models with full spatial discretization. This new paradigm for micromagnetic simulations is tested for three problems relevant to the areas of spintronics and magnonics: directional spin–wave coupling in magnonic waveguides, high power ferromagnetic resonance in a magnetic nanodot, and injection-locking in spin-torque nano-oscillators. The case studies considered demonstrate the validity of the proposed approach to systematically obtain an intermediate order dynamical model based on normal modes for the analysis of magnetic nanosystems. The time-consuming calculation of the normal modes has to be done only one time for the system. These modes can be used to optimize and predict the system response for all possible time-varying external excitations (magnetic fields, spin currents). This is of utmost importance for applications where fast and accurate system simulations are required, such as in electronic circuits including magnetic devices.

A Twenty-Year Review of Time-Delay Feedback Control and Recent Developments

The delayed feedback control (DFC) method has been invented in 1992 (this year is a 20th anniversary). Following the original paper by Pyragas [Phys. Lett. A, 170, 421, 1992], more than 1500 papers devoted or related to the DFC have been published. Many different modifications of the algorithm have been proposed, and significant achievements are attained in the theory of the DFC. Although this theory is non-trivial, currently the mechanism of the DFC action is rather well understood, and the main limitations of the algorithm are established. The DFC has been successfully implemented in a number of experimental systems of different physical nature. The aim of this talk is to present a brief review of important modifications of the DFC algorithm, significant theoretical results and experimental implementations attained during the past twenty years. The recent results concerning adaptive modifications of the DFC and analytical achievements based on phase reduction of time-delay systems will be discussed as well. 1. Brief review of experimental and theoretical results The DFC algorithm [1] is a simple, robust, and efficient method to stabilize unstable periodic orbits (UPOs) in chaotic systems. Nowadays, it becomes one of the most popular methods in the chaos control research [2]. The method allows a noninvasive stabilization of UPOs of dynamical systems in the sense that the control force vanishes when the target state is reached. The DFC algorithm is reference-free and makes use of a control signal K∆s(t) obtained from the difference ∆s(t) = s(t) − s(t − τ) between the current state s(t) of the system and the state of the system s(t−τ) delayed by one-period τ of the target orbit. The UPO may become stable under the appropriate choice of feedback strength K. Note that only the stability properties of the orbit are changed, while the orbit itself and its period remain unaltered. The controlled system can be treated as a black box, since the method does not require any exact knowledge of either the form of the periodic orbit or the system’s equations. The method is particularly appealing for experimentalists, since one does not need to know anything about the target orbit beyond its period τ. The DFC algorithm is notably superior to other control methods in fast dynamical systems, since it does not require any realtime computer processing. Successful implementation of the DFC algorithm has been attained in diverse experimental systems, including electronic chaotic oscillators, mechanical pendulums, lasers, gas discharge systems, a current-driven ion acoustic instability, a chaotic Taylor-Couette flow, chemical systems, high-power ferromagnetic resonance, helicopter rotor blades, and a cardiac system. (cf. [3] for review up to 2006). An important practical application of the DFC algorithm has been recently demonstrated by Yamasue et al. [4]. The authors have successfully implemented the DFC method in an atomic force microscope and managed to stabilize cantilever oscillations. As a result, they remove artifacts on a surface image. Another interesting application of the DFC for the analysis of bifurcations of periodic states in experimental systems has been recently considered in Ref. [5]. A reach variety of modifications of the DFC has been suggested in order to improve its performance (cf. [3]). Here we mention only the most important modification known as an extended DFC (EDFC), which has been introduced in Ref. [6]. The authors improved an original DFC scheme by using an information from many previous states of the system. The EDFC scheme achieves stabilization of UPOs with a greater degree of instability [7, 8]. The theory of DFC is difficult because the delayed feedback induces an infinite number of degrees of freedom. Even linear analysis of such systems is complicated due to the infinite number of Floquet exponents characterizing the stability of controlled orbits. Nevertheless, some analytical approaches have been developed in vicinity to various bifurcations of periodic orbits, such as the period doubling bifurcation [9, 10], the subcritical Hopf bifurcation [11, 12, 13] and the Nejmark-Sacker (discrete Hopf) bifurcation [14]. In 1997 Nakajima [15, 9] proved the so-called odd number limitation, which states that any UPOs with an odd number of real Floquet multipliers greater than unity can never be stabilized by any DFC technique. This limitation has been commonly accepted and intensively discussed in the literature. However, in 2007, Fiedler at al [16] have shown by a simple example that this limitation does not hold in general for autonomous systems (note that for non-autonomous systems it remains valid in general). Recently, a modified (corrected) proof of the limitation for autonomous systems has been presented by Hooton and 2012 International Symposium on Nonlinear Theory and its Applications NOLTA2012, Palma, Majorca, Spain, October 22-26, 2012

Nonlinear electromagnetic response of ferromagnetic metals: Magnetoimpedance in microwires

Numerical calculations based on simultaneous solution of the Maxwell and Landau–Lifshitz equations were performed, in order to study the voltage response of ferromagnetic conductors carrying ac current. Since no significant approximations are made in the calculations, the model yields both linear and nonlinear giant magnetoimpedance (GMI) behavior and low and high power ferromagnetic resonance. Application to nonlinear GMI in ideal wires, with regions of uniform anisotropy, allows us to understand many aspects of the observed behavior and to predict phenomena such as solitary-wave-like propagation of the magnetization at fairly high current amplitude. Using appropriate magnetic structure, we were able to reproduce, with good agreement, the experimental observations for cobalt rich amorphous microwires. We have also found that even harmonics of GMI signal are very sensitive to the domain structure of the wire, whereas the odd harmonics are not.

Micromagnetic modeling of ferromagnetic resonance assisted switching

We studied the steady state behavior and magnetization switching process of single domain particles subject to ac and dc magnetic fields using analytical and numerical models based on the Landau–Lifshitz–Gilbert equation. We compared the analytical solutions for circularly polarized fields with a numerical single spin model and circularly and linearly polarized ac magnetic fields. It has been found, that the initial conditions and the dynamics of the external fields (field ramps and amplitude changes) strongly determine which precession orbit the magnetization converges to, if the magnetization precession is stable, and if the magnetization switches. We also studied the effects of field amplitudes, field angles, and damping on the switching behavior. The presented results can be applied to high power ferromagnetic resonance experiments and ferromagnetic resonance assisted magnetic recording schemes.

Ferromagnetic resonance saturation and second order Suhl spin wave instability processes in thin Permalloy films

High power ferromagnetic resonance (FMR) loss versus static field H profiles and the corresponding spin wave instability threshold microwave field amplitude hcrit vs H butterfly curves were measured for in-plane magnetized thin Permalloy films of thicknesses 35, 57, 74, 104, and 123nm at a nominal pumping frequency of 9.11GHz. Cavity loading and calibration issues that proved to be problematic in past attempts to obtain accurate resonance saturation data over the full FMR profile in ferrites and metal films were resolved through a careful decoupling of the pump field and a full cavity response calibration. The FMR profiles show a drop in the loss peak, a shift in the peak to lower field, a broadening, and the development of a foldover-like asymmetry as the power is increased. The butterfly curves show a minimum hcrit at the low power FMR field and a smooth rounded increase on either side, except for a small kink on the low field side associated with the shift and asymmetry development. Apart from the kink, the second order Suhl spin wave instability theory, suitably modified for thin films, provided good fits to the butterfly curve data through the use of a single spin wave linewidth ΔHk value for each data set. The ΔHk values ranged from 16to35Oe, with the implied critical mode in-plane wave vectors always directed parallel to the static field. These spin wave linewidths translate into Gilbert damping parameter αk values in the 0.002–0.005 range, the same order as expected for intrinsic magnon-electron scattering losses in metal ferromagnets. These αk values are about a factor of 2 smaller than those implied by the low power FMR linewidths. The FMR in-plane precession cone angles at threshold were on the order of 3°–6°.

Parametric spin wave excitation and cascaded processes during switching in thin films

A theory for calculating the threshold precession angles required to parametrically excite spin waves during large-angle switching of in-plane magnetized films is presented. Analytic expressions for the parametric thresholds are found for finite thickness films when due to three- and four-wave decays. Precession at the threshold angle will produce appreciable spin wave amplitudes only after infinite time. We show that much larger angles are required to experimentally observe nonlinearities within a finite time. This may explain apparent discrepancies reported between ultrafast pulsed reversal and longer time scale high-power ferromagnetic resonance experiments. A comparison of our results to published experimental data for metallic Permalloy films is made. Finally, the film thickness dependence of the threshold is examined and a minimum is found corresponding to when a particular ``cascaded'' spin wave interaction is resonant.

Blowout bifurcations in a model for chaos in spin-wave dynamics

Blowout bifurcations are investigated numerically in a model for chaos in the coincidence regime of high-power ferromagnetic resonance, based on interactions between the uniform mode and two pairs of parametric spin waves. This model possess two orthogonal invariant manifolds corresponding to the excitation of only one spin–wave pair above the first-order Suhl instability threshold. Marginal synchronization of the amplitudes of spin–wave pairs, the exchange of stability of the invariant manifolds, as well as both supercritical and subcritical blowout bifurcations are observed as the system parameters are varied, with the accompanying on–off intermittency, attractor bubbling and intermingled basins of attraction.

High power ferromagnetic resonance and spin wave instability processes in Permalloy thin films

The high power ferromagnetic resonance (FMR) response, as well as butterfly curves of the spin wave instability threshold microwave field amplitude hcrit versus in-plane static field H profiles, have been measured for Permalloy films with thicknesses of 104, 128, and 270nm at a nominal pumping frequency of 9.37GHz. The hcrit values range from about 1 to 7Oe. Both the resonance saturation response at the FMR field and the subsidiary absorption (SA) response for static fields below the FMR field are similar in appearance to those for bulk ferrites. Butterfly curves over the SA response region, while similar to those for ferrites, exhibit a film thickness dependent band edge cutoff effect not found in bulk ferrites. The SA butterfly curve data were analyzed on the basis of a spin wave instability theory adapted to thin films. The observed shift in the SA band edge cutoff with thickness agrees with calculations based on the thin film dispersion response and the assumption of first order instability processes with critical modes at one half the pumping frequency. The fitted SA spin wave linewidths give values consistent with metallic relaxation processes, but indicate critical modes with wave vectors that always make relatively small 0°–20° angles with the static field, very different from the critical modes for bulk ferrites. Three key conclusions from this work are (1) the nonlinear microwave FMR response in Permalloy films is a threshold effect related to well established spin wave instability processes, (2) the details of the SA response are controlled largely by the thin film spin wave dispersion, and (3) these nonlinear processes occur for very small precession angles.

Theoretical and experimental aspects of chaos control by time-delayed feedback.

We review recent developments for the control of chaos by time-delayed feedback methods. While such methods are easily applied even in quite complex experimental context the theoretical analysis yields infinite-dimensional differential-difference systems which are hard to tackle. The essential ideas for a general theoretical approach are sketched and the results are compared to electronic circuits and to high power ferromagnetic resonance experiments. Our results show that the control performance can be understood on the basis of experimentally accessible quantities without resort to any model for the internal dynamics.

Synchronization Among Spin-Wave Amplitudes in Chaotic Nonlinear Ferromagnetic Resonance

A model of chaos in high-power ferromagnetic resonance in coincidence regime, based on three-magnon interactions of the uniform mode with a group of pairs of parametric spin waves, is investigated numerically. The re- sults are interpreted from the point of view of chaotic synchronization theory. If all spin waves are identical, all of them are excited above the first-order Suhl instability threshold and in the chaotic regime their amplitudes show marginal synchronization, i.e. they differ only by a multiplicative factor, con- stant in time. If spin waves have slightly different instability thresholds, only one or few of them are excited. In the latter case, addition of weak thermal noise changes the results qualitatively. For low rf field amplitude, but above the threshold for chaos, still only few spin-wave pairs are excited above the thermal level. For higher rf field amplitude all spin waves in the group are excited, and their amplitudes are not synchronized. These results sug- gest that low correlation dimension of chaotic attractors, observed often in nonlinear ferromagnetic resonance, can be connected with chaotic synchronization among spin-wave amplitudes, in particular just above the threshold for chaos.

Characterization of spin-wave dynamics near homoclinic orbits through one-dimensional mapping

A description is given of pulsed high-power ferromagnetic resonance experiments in a YIG sphere at room temperature. Beyond the Suhl instability threshold, evidences were found for the alternating periodic-chaotic sequence. The chaotic spin-wave self-oscillations were characterized through return maps obtained from the local maxima in the observed transient-free time series. As a result, a simple one-dimensional mapping scheme that exhibits the basic features of the observed transition is presented.

Observation of mixed-mode oscillations in spin-wave experiments

High-power ferromagnetic resonance experiments in a 1-mm-diam yttrium iron garnet sphere, driven at 8.9 GHz at room temperature, reveal evidence for a universal scenario not yet observed in this system, namely, the periodic–chaotic sequence. Well above the first-order Suhl instability threshold (P/PC∼6 dB), low-frequency (f∼200 kHz) auto-oscillations consisting of large-amplitude peaks followed by n small undulations (mixed-mode oscillations) have been observed. Both periodic and aperiodic trajectories were detected with n=1, 2, 3, and 4, which are reminiscent of the so-called homoclinic chaos. A discussion is made on the basis of a standard two-mode model.

Bifurcation to chaos in auto-oscillations in circular yttrium-ion-garnet films

One of the most intriguing features of any nonlinear system is its dynamic evolution. Starting from the steady state the system evolves to a linear periodic motion and then experiences a series of “phase transitions” known as bifurcation which lead to a final chaotic state. In order to better understand the nonlinear behavior in magnetic systems, an experimental and numerical study of bifurcation to chaos in auto-oscillations at ferromagnetic resonance is presented. Experiments have been performed with thin circular iron garnet films using perpendicular high-power ferromagnetic resonance. Taking the magnetic field as a variable parameter the bifurcation route to chaos has been investigated. The system has been modeled by the usual three magnetostatic mode interaction model derived from the macroscopic Hamiltonian. Along with the modeled signal, the time dependence of the complex Lyapunov exponent has been investigated for the three magnetostatic modes. The possibilities for controlling chaos with periodic perturbation have been studied and the reason for desynchronization bursts is discussed.

Stochastic Resonance in Chaotic Spin-Wave Dynamics

Institut fu¨r Festk¨orperphysik, Technische Hochschule Darmstadt, Hochschulstrase 6, D–64289 Darmstadt, Germany(January 31, 1997)We report the first experimental observation of noise–freestochastic resonance by utilizing the intrinsic chaotic dynam-ics of the system. To this end we have investigated the effectof an external periodic modulation on intermittent signals ob-served by high power ferromagnetic resonance in yttrium irongarnet spheres. Both the signal–to–noise ratio and the resi-dence time distributions show the characteristic features ofstochastic resonance. The phenomena can be explained bymeans of a one–dimensional intermittent map. We presentanalytical results as well as computer simulations.PACS numbers: 05.45.+b, 05.40.+j, 75.30.DsI. INTRODUCTION

Thickness dependence of auto oscillations in circular yttrium-iron-garnet films

Using high-power ferromagnetic resonance the dependence of the auto-oscillation frequency has been investigated as a function of the thickness of thin circular yttrium-iron-garnet films. For thicknesses over the range from 0.5 to 1.4 μm, the auto-oscillation frequencies are found to be directly proportional to the thickness of the films, even though the linewidth remains nearly unchanged over this range. These results indicate that the auto-oscillation frequency is determined by a magnetostatic mode spacing. The experimental results are compared with numerical calculations, based on a microscopic Hamiltonian and show good agreement (within 10% for first finger) with the experimental data.

Auto-Oscillations and Chaos for Microwave Signals in Ferrite Single Crystals

The recent progress of nonlinear dynamics has stimulated a renaissance of high-power ferromagnetic resonance experiments on ferrites, both under the aspects of basic research and application. We report on resonance experiments in yttrium iron garnet spheres probing chaotic scenarios above the first-order Suhl threshold. Different techniques have been proposed to change the observed irregular auto-oscillations of magnetization into regular motion. The delayed feedback control method of Pyragas is studied both in theory and experimental application.

On-off intermittency in spin-wave instabilities.

A particular case of chaos-to-chaos intermittency identified as on-off intermittency has experimentally been observed in a yttrium iron garnet sphere at high power ferromagnetic resonance. The theoretically predicted power law scaling with the exponent $\ensuremath{-}1$ has been detected in the dependencies of the mean ``laminar'' phase length on both static magnetic field and pumping microwave power. In the distribution of the laminar lengths a $\ensuremath{-}3/2$ power law was observed close to the onset of intermittency.

Transient and crisis-induced intermittencies in high-power ferromagnetic resonance.

Pulsed ferromagnetic resonance experiments at unusually high power levels reveal the existence of long-lived transient forms of type-III Pomeau-Manneville intermittency and crisis-induced bursting in a spin-wave system. The average lifetime of this particular chaotic transient as well as the mean duration between chaotic bursts are shown to obey a scaling law. The results are in good qualitative agreement with numerical simulations based on a two-mode spin-wave model.

Controlling unstable periodic orbits in a spin-wave experiment using the OGY scheme

Dynamic instabilities of magnetization in YIG spheres were studied by high-power ferromagnetic resonance at the first-order Suhl instability. By means of an analog feedback technique implementing the controlling scheme of Ott, Grebogi and Yorke, we suppressed the chaotic FMR signal and stabilized a periodic oscillation by applying a very small time-dependent feedback signal of less than 10−3 of the applied microwave power.

Controlling chaos in ferromagnetic resonance

The chaotic response of yttirum iron garnet (YIG) films in high-power ferromagnetic resonance (FMR) has been successfully controlled in experiments using a time-delayed control method. The ac component of the FMR signal was delayed and used to perturb a system parameter, the static magnetic field. The application of the perturbations initially modified the chaotic oscillations in the FMR absorption signal, converting them to periodic oscillations. Increasing the gain lead to a period halving (a debifurcation) of the controlled periodic oscillation, and with sufficient gain, all oscillations were eliminated and the quiescent state is achieved. Numerical simulations yielded good agreement with the experimental results.