Model for spin-wave chaos in the coincidence regime of nonlinear ferromagnetic resonance

1. Model for spin-wave chaos in the coincidence regime of nonlinear ferromagnetic resonance 1 1,2 A. Krawiecki , A. Sukiennicki 1 Faculty of Physics, Warsaw University of Technology, Koszykowa 75, 00-662 Warsaw, Poland 2 Department of Solid State Physics, University of Łódź, Pomorska 149/153, 90-283 Łódź, Poland

2. Nonlinear ferromagnetic resonance • Ferromagnetic sample is placed in perpendicular dc and rf (with frequencies in the GHz range) magnetic fields. • The uniform precession of magnetization (uniform mode) is excited in the sample by the rf field. In the coincidence regime the rf field frequency wpis close to the uniform mode frequency wo. • If the rf field amplitude hTexceeds a certain threshold hthr, the uniform mode decays into spin-wave pairs. • The measured quantity is usually absorption in the sample, which is proportional to the uniform mode amplitude. • As the rf field amplitude is increased, periodic (with frequencies in the range of kHz) and then chaotic oscillations of absorption appear.

3. The 1st-order Suhl instability (coincidence regime) Decay of the uniform mode (pumped in resonance) into spin-wave pairs with half the pumping frequency and opposite wave vectors

4. Theoretical description The magnetic Hamiltonian The Hamiltonian contains the Zeeman energy, the energy of magnetic dipolar interactions, and the exchange energy; all magnetizations and fields are normalized to the saturation magnetization.

5. The Holstein-Primakoff canonical transformation The Fourier expansion The Bogolyubov transformation

6. The Hamiltonian in the canonical form  The above Hamiltonian contains non-resonant three-mode interaction terms, e.g., . Such terms should be removed by another canonical transformation, which, however, should leave the resonant terms (e.g., ) intact.  The removal of all non-resonant terms from H3 influences the higher-order terms in the Hamiltonian.  However, since the basic nonlinear process in the case under study is the 1st-order Suhl instability (the resonant three-mode process), the higher-order terms in the Hamiltonian can be subsequently neglected.

7. The second quasi-canonical transformation Let us assume that only the uniform mode (denoted by zero) is directly excited by the rf field and has frequencyclose to wp, and the spin-wave pairs have frequencies close to wp/2. Then the following transformation removes the non-resonant terms from H3: [ A similar transformation is well known in the case of parallel pumping: V.S. Zakharov et al., Usp. Fiz. Nauk114, 609 (1974)]

8. Equations of motion for the spin-wave amplitudes The Hamiltonian and canonical equations (with damping) I0 - interaction coefficient between the uniform mode and the rf field, a0, ak- complex amplitudes of the uniform mode and spin waves, h0, hk- phenomenological damping of the uniform mode and spin waves, V0,k- coefficients of nonlinear interactions between the uniform mode and spin-wave pairs.

9. Separation of the fast time dependence The 1st-order Suhl instability threshold Just above the threshold only one (critical) spin-wave pair is excited; if hT exceeds much the threshold, other pairs with frequency close to w p/2 can be excited. However, experimental results (low correlation dimension of chaotic attractors, etc.) suggest that even deeply in the chaotic regime the oscillations of absorption appear due to interactions of a small number of spin-wave pairs with the uniform mode.

10. Model with two spin-wave pairs Equations of motion in a dimensionless form • The model with one spin-wave pair ( with a2=0) shows transition to chaos via period-doubling, • Inclusion of a second spin-wave pair, with higher Suhl instability threshold, can lead to quasiperiodicity, Pomeau-Maneville type-III intermittency, etc. • The chaotic behavior of the models with one or two spin-wave pairs is in qualitative agreement with experiments on spin-wave chaos in the coincidence regime.

11. Example: route to chaos via period-doubling Model with one spin-wave pair, left column: time series of absorption, right column: chaotic attractor.

12. Example: route to chaos via quasiperiodicity Model with two spin-wave pairs, left column: time series of absorption, right column: power spectrum of absorption.

13. Example: type-III Pomeau-Maneville intermittency Model with two spin-wave pairs, (a) time series of absorption, (b) mean duration of laminar phases vs. the control parameter, (c) probability distribution of durations of laminar phases.

14. Conclusions • Systematic derivation of the equations of motion for spin-wave amplitudes in the coincidence regime of nonlinear ferromagnetic resonance above the 1-st order Suhl instability threshold was presented, • The non-resonant three-mode interaction terms can be removed by means of the quasi-canonical transformation, which leaves only resonant three-mode terms in the Hamiltonian, and the higher-order terms can be neglected, • The model equations with one or two parametric spin-wave pairs show transition to chaos via, e.g., period doubling, quasi-periodicity, Pomeau-Maneville intermittency, etc., and the results of simulations are in qualitative agreement with experimental results.

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