Nonlinear interaction theory for parametrically-excited spin-wave modes in confined micromagnetic systems

TL;DR

The paper develops a nonlinear interaction theory for parametrically excited spin-wave modes in confined micromagnets under parallel pumping. By formulating a reduced-order Normal Modes Model (NMM) and its two-mode extension, it yields analytical expressions for instability thresholds, nonlinear frequency shifts, and steady-state amplitudes, revealing phenomena such as hysteresis, non-reciprocal coupling, and quasiperiodic modes. The theory is validated against full micromagnetic simulations on a nanoscale YIG disk, producing phase diagrams that map regimes of coupled and uncoupled oscillations under CW and PWM protocols. This framework provides a predictive tool for designing and interpreting spin-wave dynamics in magnonic devices and spintronic applications.

Abstract

We present a general theoretical approach for the quantitative description of parametric excitation of spin-wave modes in confined micromagnetic systems. This type of problem belongs to a broader class of nonlinear modal dynamics that arise across many areas of physics and engineering. The ferromagnetic sample is driven by parallel pumping with an external applied magnetic field having two tones at different frequencies, which are able to trigger parametric instability of two resonant modes. The two excited spin-wave modes interact in a strongly nonlinear fashion giving rise to quasiperiodicity, hysteresis and non-commutativity of steady-state oscillation regimes. To disentangle such a complex variety of dynamics, we develop a reduced-order model based on magnetization normal modes that is amenable of appropriate analytical treatment, leading to quantitative description of parametric instability thresholds, post-instability steady-state amplitude saturation and complete determination of phase diagrams for steady-state oscillation regimes. We have performed validation of the theory using numerical simulations. The phase diagrams allow to predict and explain all the features of the nonlinear interaction between the parametrically-excited spin-wave modes and can be directly compared with experimental results.

Figures (27)

• Figure 1: Graphical representation of conditions for parametric excitation of mode $h$ via parallel pumping. The shaded region depicts pairs of excitation parameters $(\omega_\mathrm{rf},\delta h)$ that trigger the amplitude growth of mode $h$ from nonzero initial value according to eq.\ref{['eq:detuning cone']}. The solid red (dashed blue) line refers to eq.\ref{['eq:critical field cone']} when the damping $\lambda_h>0$ (resp. $\lambda_h=0$). The minimum threshold $\delta h_\mathrm{thres,h}$ (see eq.\ref{['eq:threshold field parametric resonance']}) is also reported.
• Figure 2: Steady-state amplitude $|a_h|^2$ of mode $h$ as function of ac field frequency $\omega_\mathrm{rf}$ after parametric excitation via parallel pumping with given ac field amplitude above threshold $\delta h>\delta h_\mathrm{thres,h}$ starting from tiny nonzero mode amplitude (PWM excitation). Here it is assumed that the NFS $N_h>0$. The solid red (resp. blue) line refers to $|a_h|^2_+$ (resp. $|a_h|^2_-$) in eq.\ref{['eq:saturation amplitude mode h']}. When $N_h<0$ red and blue curves undergo a mirror-reflection around the abscissa axis.
• Figure 3: Steady-state amplitude $|a_h|$ of mode $h$ as function of ac field amplitude $\delta h$ after parametric excitation via parallel pumping with given ac field frequency starting from tiny nonzero mode amplitude (PWM excitation). Here it is assumed that the detuning is positive $\omega_\mathrm{rf}-2\omega_h>0$ and NFS $N_h>0$. The solid red (resp. blue) line refers to $|a_h|^2_+$ (resp. $|a_h|^2_-$) in eq.\ref{['eq:saturation amplitude mode h pm']}. When $N_h<0$ red and blue curves undergo a mirror-reflection around the abscissa axis.
• Figure 4: Steady-state amplitude of mode $h$ as function of ac field frequency detuning $\epsilon_h$ under parametric excitation via parallel pumping with given ac field amplitude above threshold $\delta h>\delta h_\mathrm{thres,h}$. Solid (dashed) lines refer to stable (unstable) P-modes. Red (blue) lines refer $N_h>0$ ($N_h<0$), green lines refer to P-mode $|a_h|^2=0$. Arrows elucidate the hysteresis in the case $N_h>0$ and the irreversible downward jump that may occur for large positive detuning (notice the axis breaks in the lower panel) not described by the theory.
• Figure 5: Graphical representation of conditions for parametric excitation of modes $h$ and $n$ via parallel pumping with two-tones signal. The shaded regions depict pairs of excitation parameters $(\omega_\mathrm{rf1},\delta h_1)$ and $(\omega_\mathrm{rf2},\delta h_2)$ that trigger the amplitude growth of mode $h$ and $n$ from nonzero initial values according to eq.\ref{['eq:detuning cone']}. The solid red (dashed blue) and solid purple (dashed green) lines refer to eq.\ref{['eq:critical field cone']} when the damping $\lambda_h, \lambda_n>0$ (resp. $\lambda_h=\lambda_n=0$). The minimum thresholds $\delta h_\mathrm{thres,h}$ and $\delta h_\mathrm{thres,n}$ (see eq.\ref{['eq:threshold field parametric resonance']}) are also reported.