First-order Suhl Research Articles

Suhl instabilities for spin waves in ferromagnetic nanostripes and ultrathin films

A microscopic (or Hamiltonian-based) theory is employed for the spin-wave instability thresholds of nonlinear processes in ultrathin ferromagnetic stripes and films under perpendicular pumping with an intense microwave field. The spatially-quantized linear spin waves in these nanostructures may participate in parametric processes through the three-magnon interactions (the first-order Suhl process) and the four-magnon interactions (the second-order Suhl process) when pumped. By contrast with most previous studies of spin-wave instabilities made for larger samples, where macroscopic (or continuum) theories involving Maxwell's equations for magnetic dipolar effects are used, a discrete lattice of effective spins is employed. Then a dipole-exchange spin Hamiltonian is employed to investigate the behavior of the quantized spin waves under perpendicular pumping, when modifications due to the more extensive spatial confinement and edges effects in these nanostructures become pronounced. The instability thresholds versus applied magnetic field are calculated, with emphasis on the size effects and geometries of the nanostructures and on the different relative strengths of the magnetic dipole-dipole and exchange interactions in materials. Numerical results are presented using parameters for Permalloy, YIG, and EuS.

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.

Parametrically driven first-order Suhl instability and nonlinearities in bismuth-substituted yttrium iron garnet films

We have demonstrated the enhancement of spin-wave nonlinearities in a bismuth-substituted yttrium iron garnet (Bi-YIG) film by parametrically exciting the first-order Suhl instability at the difference frequency of two simultaneous driving frequencies. The results show that parametric excitation of spin waves can cause this instability to grow at low threshold powers, which, in turn, channels power to other nonlinear spin-wave modes. Multiple interactions between spin waves lead to the enhancement of nonlinear processes.

Control of spin-wave instability threshold in YIG sphere

Spin-wave instability threshold under multiple drive excitations is studied in an yttrium iron garnet sphere at 4.2 K . Both a static magnetic field and a parallel RF field are applied along the [1 1 1] direction. Another perpendicular RF field excites magnetostatic modes through the first-order Suhl process. Interactions between spin-waves and magneostatic modes cause a change of instability thresholds.

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.

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.

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.

Delayed bifurcations at the first-order Suhl threshold

In this paper, we present both theoretical and experimental analysis of delayed dynamic bifurcations as could be observed by ferromagnetic resonance (FMR) on an yttrium iron garnet (YIG) sphere. In Sec. 1 the static bifurcation behaviour of the first-order Suhl instability is analyzed in terms of a two-mode model, which represents the simplest possible description of this process. In Sec. 2 our analytical results are extended to the dynamic case by means of numerical simulations, including the influence of noise. We present our experimental results and compare them with the obtained theoretical predictions.

On-Off Intermittency and Peculiar Properties of Attractors in a Simple Model of Chaos in Ferromagnetic Resonance

Some non-trivial phenomena in chaos in ferromagnetic resonance above the first-order Suhl instability threshold are obtained numerically from a simple model of three interacting modes. They include sudden changes of the correlation dimension of the attractor and the largest Lyapunov exponent with the rise of the rf fleld amplitude. Numerical evidence is provided that these effects may result from the on-off intermittency in the system of interacting modes. These results are qualitatively similar to the experimental ones, known from the literature, obtained for in-plane magnetized thin films. PACS numbers: 05.45.+b, 76.50.+g, 75.30.Ds

Critical modes at the first-order Suhl instability in YIG spheres

The FMR fine structure of the first-order Suhl instability has been investigated in YIG spheres at room and helium temperature. Several series of closely spaced resonances were observed and analyzed in terms of spherical modes. Different types of critical modes resulting in contrasting dynamic behaviours, can be attributed to competing damping mechanisms.

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.

Fine structure and critical modes at the first-order Suhl instability in YIG spheres

The FMR fine structure of the first-order Suhl instability in YIG spheres, previously observed by Jantz and Schneider (1975), has been re-investigated both for the conditions of coincidence and subsidiary absorption (3 and 9 GHz) from room temperature down to 1.6K. The observed fine structure consists of several series of closely spaced resonances, indicating a critical wave vectork≠0 even for half the pumping frequency lower than the 45° spin-wave branch, which is in contrast to the prediction of Suhl's theory. The spacing of these resonances was analyzed in terms of spherical modes. We were able to determine the indices and the effective wave numbers (≈105 cm−1) of the critical modes. The very different dynamic behaviour observed at room temperature and at 1.6K can be attributed to a competition of different damping mechanisms—mainly surface pit scattering and three-magnon confluence processes — and their dependences on field, frequency, and temperature.

Multistability and chaos at parametrically excited spherical modes in an yttrium iron garnet sphere

The different behaviors observed in different regimes of field of recent high-power ferromagnetic resonance experiments on yttrium iron garnet spheres within the coincidence regime of the first-order Suhl instability, which showed a complex multistable behavior beneath auto-oscillations, are examined on the basis of a detailed model which utilizes the true eigenmodes of the sphere instead of plane spin waves.

Theory for the first-order spin-wave instability threshold in ferromagnetic insulating thin films

A theory for the first-order Suhl and the parallel pumping instability in thin films is presented. Significant differences for the critical microwave field and wave vector to former calculations occur, which discuss the problem in terms of bulk spin-waves neglecting boundary conditions. A coupling matrixCkk′ is introduced, which describes the couplings between the modes and the driving microwave field. For bulk standing spin-wavesCkk′ is always diagonal. For the true discrete standing modes of a thin filmCkk′ changes only in case of 1. Suhl instability and if the wavevector has a non vanishing component perpendicular to the film plane. Here the diagonal bulk couplings have to be replaced in part by off diagonal terms, describing couplings between modes, which perpendicular wave vector componentk⊥ differs byπ/d (d=film thickness). The decisive quantity, which decides if the finite thickness of the film is of importance or if the film can be treated as a bulk system, is the frequency difference δωk of the coupled modes. For δωk much smaller than the spin-wave damping ηk the bulk approximation is correct. For\(\delta \omega _k > > \eta _k \) two experimental situations for 1. Suhl instability are discussed: For a perpendicular to the film plane magnetized film the critical microwave field is by π/2 bigger than in the bulk case. In an in-plane magnetized film the critical spin-waves propagate always in the film plane, as only hereCkk′ remains identical to the bulk case.

Different types of intermittency observed in transverse-pumped spin-wave instabilities

The dynamic behaviour of magnetization in YIG spheres was studied by FMR within the coincidence regime of the first-order Suhl instability. For certain ranges of the control parameters (magnetic field, microwave frequency and amplitude, etc.) the time-resolved absorption signal shows auto-oscillations changing over to chaos by intermettency. We clearly observed each of the types I–III and rather often crises.

Period-doubling cascade and chaos in YIG at the first-order Suhl instability

Transverse pumping at the first-order Suhl instability in YIG has been performed at a pumping frequency of 3.3 GHz at 4.2 K. Above the spin-wave instability threshold, period-doubling cascades and chaotic auto-oscillations are observed. The largest Lyapunov exponent as a function of driving microwave power shows the characteristics of the onset of chaos. The fractal structure is elucidated by a correlation dimension and an f(α) spectrum.

Fractal Structure of Strange Attractor at the Subsidiary Resonance in Yttrium Iron Garnet

Transverse pumping at the first-order Suhl instability in YIG has been performed at a pumping frequency of 3.3 GHz at 4.2 K. Above the spin-wave instability threshold, period-doubling cascades and chaotic auto-oscillations are observed. From time-series data, return maps, fractal dimensions, phase space portraits of strange attractors, two-dimensional Poincare sections, and singularity spectrum f (α) are obtained. The Poincare sections of the phase space trajectories exhibit the stretching and folding which are characteristics of strange attractors. At the onset of chaos, the f (α) spectrum is consistent with the universal one.

Transition to chaotic state in spin-wave turbulence

Parametric excitation of spin-waves in an yttrium iron garnet is studied within the coincidence regime of the first-order Suhl instability. Above the spin-wave instability threshold, a period-doubling cascade is observed. At this onset of chaos, the singularity spectrum f(α) is obtained.

Spin-wave instabilities, auto-oscillations, and chaos in yttrium-iron-garnet

A review is presented of recent experimental results in yttrium-iron-garnet subject to three different spin-wave pumping mechanisms: parallel pumping, subsidiary resonance (first-order Suhl process), and premature saturation of the main resonance (second-order Suhl process). A theoretical model derived from first principles and leading to coupled nonlinear spin-wave equations, is used to interpret the observed spin-wave instabilities, auto-oscillations, and chaotic dynamics. It is shown that a two-mode model explains the main features of the experimental observations, both in parallel and perpendicular pumping. Improvements needed in the model are indicated. >

Multistability and chaos by parametric excitation of longwave modes in a YIG sphere

Spin-wave instabilities in spheres of yttrium iron garnet have been studied by ferromagnetic resonance within the coincidence regime of the first-order Suhl instability (1.8–3.4 GHz), i.e. both the uniform mode and pairs of spin-waves were simultaneously excited on resonance. From the characteristic behaviour above the threshold, three different regimes of resonance fields (and frequencies) could be distinguished: For low fields (640–680 Oe) and for high fields (950–1210 Oe) the amplitude of magnetization remains time independent and only at high input power becomes suddenly chaotic. In the intermediate regime (680–950 Oe) a very complex multistable behaviour occurs and a variety of oscillations and sequences of bifurcations are observed. We discuss our experimental results in terms of a multi-mode model which, beyond spin-waves, considers the excitation of longwave modes with wavelengths in the order of sample dimension. Magnetostatic modes and spin-waves can be described within a unified picture by introducing “spherical spin-waves” which are classified by symmetry. A new mechanism based on the indirect excitation of the magnetostatic (4,3,0) mode is introduced. Numerical simulations show that this mechanism can explain both the multistability and the complex dynamic behaviour of the system.