Field-driven triggering of self-induced Floquet magnons in a magnetic vortex

This paper demonstrates the experimental control and hysteretic switching between regular and self-induced Floquet magnons in magnetic vortex state tunnel junctions by manipulating the vortex core orbit with an external magnetic field, revealing a mechanism where Floquet-mediated feedback creates multiple stable gyration radii.

The Gist

The Big Picture: A Magnetic Dance Floor

Imagine a tiny, circular dance floor made of magnetic material (a nanodisk). On this floor, there is a special "dance partner" called a magnetic vortex.

• The Vortex: Think of the magnetism in the disk like a swirling whirlpool. The water spins around in circles, but right in the very center, there is a tiny, distinct point (the "core") that points straight up or down.
• The Goal: The scientists wanted to see what happens when they shake this dance floor with microwaves (like a DJ playing a beat) and whether they could control the dance moves in a special, predictable way.

The Discovery: Two Different Ways to Dance

When the scientists played their microwave "beat," they discovered the vortex didn't just spin in one way. It could do two very different things, depending on how they started the dance:

1. The "Sleepy" Dance (Regular Magnons): If the vortex starts right in the dead center of the disk, it stays relatively calm. It wiggles a little, but it doesn't create a complex pattern. It's like a dancer just tapping their foot to the beat.
2. The "Wild" Dance (Floquet Magnons): If the vortex is nudged slightly away from the center before the music starts, it goes wild. It starts spinning in a large circle. As it spins, it creates a complex "frequency comb"—a pattern of many different frequencies that look like the teeth of a comb. This is the Floquet state.

The Magic Trick: The most surprising part is that they could use the exact same microwave power and frequency to make the vortex do either the calm dance or the wild dance. The only thing that changed was where the vortex was sitting when the music started.

The Analogy: The Swing Set

To understand why this happens, imagine a child on a swing set.

• The Setup: The swing is the magnetic vortex. The microwave field is a parent pushing the swing.
• The Calm State: If the child is sitting perfectly still in the middle of the swing, and you give a gentle push, they just rock back and forth a little. They stay in the middle.
• The Wild State: Now, imagine you pull the child back a few feet (using a magnetic field) and then let go before you start pushing. When you start pushing, the child swings much higher and faster.
• The Hysteresis (The "Memory" Effect): Here is the tricky part. If the child is already swinging high, you can stop pushing for a moment, and they will keep swinging high for a while. But if they are sitting still, you have to push much harder to get them to start swinging high.
  • The system has a "memory." It remembers if it was already moving or if it was still.
  • This creates a hysteresis loop: The path to get the swing going is different from the path to get it to stop.

What Did They Actually Do?

1. The Experiment: They used a special device called a Magnetic Tunnel Junction (MTJ). Think of this as a super-sensitive microphone that listens to the magnetic dance floor. It turns the magnetic movements into electrical signals so they can see the "music" the vortex is making.
2. The Control Knob: They used a small magnetic field to physically move the vortex core away from the center (like pulling the swing back).
3. The Result:
   • Centered Vortex: They turned on the microwaves. The vortex stayed calm. No complex "frequency comb" appeared.
   • Displaced Vortex: They moved the vortex, then turned on the exact same microwaves. Suddenly, the vortex started spinning wildly, creating the complex "frequency comb" pattern.

Why Does This Matter?

This isn't just about watching magnets spin. It's about control.

• New Technology: This discovery shows that we can use the "history" of a magnetic state (where it was before) to switch between two different modes of operation without changing the power or the frequency.
• The "Floquet" Concept: In physics, "Floquet" refers to systems that are driven by a repeating force (like the microwaves). The scientists found that these systems can have multiple stable states at the same time.
• The Takeaway: By simply "priming" the system (moving the vortex first), they can force it to switch into a high-performance mode that generates complex signals. This could be a blueprint for future ultra-fast, low-power computer memory or communication devices that can switch states instantly based on their initial conditions.

Summary in One Sentence

The scientists discovered that by simply moving a magnetic "whirlpool" to a new starting spot before turning on the power, they could force it to switch from a calm wobble to a wild, complex spin, proving that where you start determines how you dance, even if the music stays the same.

Technical Summary

1. Problem Statement

Magnetic vortices in nanodisks exhibit complex nonlinear dynamics where the vortex core couples with high-frequency spin-wave modes (magnons). Under strong microwave excitation, this coupling can lead to the formation of Floquet magnons, characterized by frequency combs (sidebands) generated through time-periodic modulation of the spin waves by the gyrating vortex core.

While theoretical models predict that Floquet-mediated feedback can create multiple stable vortex gyration radii under identical drive conditions (multistability), experimental verification has been challenging. The central open question addressed in this work is: Can these multistable Floquet states be experimentally accessed and controlled? Specifically, can the system be switched between a "regular" state (no Floquet sidebands) and a "Floquet" state (frequency combs) without changing the external microwave drive parameters, but rather by manipulating the initial magnetic state?

2. Methodology

The authors employed a combination of experimental microwave spectroscopy and nonlinear theoretical modeling:

• Experimental Setup:

  • Device: Magnetic Tunnel Junctions (MTJs) with a 400 nm diameter free layer composed of a CoFeBSi alloy, designed to stabilize a magnetic vortex ground state. The reference layer was fixed, allowing electrical detection of vortex dynamics via magnetoresistance.
  • Excitation: A 3 µm-wide inductive antenna positioned above the MTJ generated a microwave magnetic field to excite both the vortex gyrotropic mode (sub-GHz) and higher-order spin-wave modes (GHz).
  • Detection: Voltage fluctuations across the biased MTJ were measured using a spectrum analyzer to resolve the power spectral density (PSD), capturing the gyration frequency, harmonics, and Floquet sidebands.
  • Control: An in-plane DC magnetic field was used to controllably displace the vortex core from the disk center prior to microwave excitation, serving as an initialization parameter.

• Theoretical Modeling:

  • The authors utilized a nonlinear vortex-magnon model based on the Thiele equation.
  • They calculated the nonlinear interaction contribution ($f(R)$), which represents the feedback between Floquet modes and the gyration radius.
  • Simulations were performed to map the stability of gyration radii ($R$) as a function of drive frequency and amplitude, identifying stable and unstable fixed points.

3. Key Contributions

• Experimental Demonstration of Hysteresis: The study provides the first experimental evidence of hysteresis in the formation of Floquet magnon spectra. The system exhibits bistability where two distinct steady states (centered vortex vs. large-orbit vortex with Floquet combs) coexist under identical microwave drive conditions.
• Field-Driven Switching: The authors demonstrate that the vortex core's initial position acts as a control knob. By displacing the core with a magnetic field before turning on the microwave drive, they can trigger the Floquet state at drive powers significantly below the spontaneous instability threshold.
• Validation of Multistability Theory: The experimental results directly validate recent theoretical predictions that Floquet-mediated feedback creates multiple stable gyration radii, linking the system's history to its spectral output.

4. Key Results

A. Emergence of Floquet Frequency Combs

• At specific drive frequencies (near the $(n=0, m=-1)$ spin-wave mode at ~4.9 GHz) and sufficient power, the spectrum transitions from simple harmonics to a frequency comb.
• The comb consists of sidebands spaced by the gyration frequency ($f_g$), following the relation $f = f_{in} + k f_g$.
• The appearance of these combs correlates with a sharp increase in the integrated power of the gyrotropic mode, indicating a transition from thermally driven Brownian motion to a driven, large-amplitude steady-state orbit.

B. Power Sweep Hysteresis

• Up-sweep: When microwave power is increased from zero, Floquet sidebands appear only above a threshold of ~7.8 dBm.
• Down-sweep: When power is decreased from a high-power state, the Floquet spectrum persists down to ~5.8 dBm.
• Bistability Window: In the range of 5.8–7.8 dBm, the system supports two stable states. If the vortex starts centered, it remains centered (no combs). If it is already in a large orbit, it sustains the orbit (combs present).

C. Magnetic Field Initialization (History Dependence)

• Centered Initialization: Starting with the vortex at the disk center requires a higher power threshold (~9 dBm) to trigger Floquet sidebands.
• Displaced Initialization: If an in-plane magnetic field is applied to displace the core (and then removed before microwave excitation), the system enters a large-amplitude orbit state.
• Result: In this displaced state, Floquet sidebands emerge at a significantly lower threshold (~5.5 dBm).
• This confirms that the initial position determines which stable branch (regular vs. Floquet) the system occupies, effectively allowing "switching" between states at identical drive conditions.

D. Theoretical Confirmation

• The nonlinear model predicts that the feedback function $f(R)$ intersects the relaxation function $\Gamma_g R$ at multiple points for specific drive amplitudes.
• At low amplitudes, only the $R=0$ solution is stable. At high amplitudes, only a finite $R$ is stable. In the intermediate range, bistability exists (stable $R=0$ and stable $R>0$).
• The system's final state depends on whether the initial condition lies within the basin of attraction of the $R=0$ or $R>0$ fixed point.

5. Significance

• Floquet Engineering in Magnonics: This work establishes magnetic state initialization as a robust method for Floquet engineering. It allows researchers to tune distinct frequency-comb spectra without altering the external drive frequency or power, simply by controlling the internal magnetic history.
• Nonlinear Dynamics Control: It highlights the critical role of nonlinear feedback in magnetic nanostructures, showing how vortex-magnon coupling can create complex multistable behaviors.
• Applications: The ability to switch between regular and Floquet states hysteretically suggests potential applications in:
  • Microwave Signal Processing: Creating tunable frequency combs for communication or sensing.
  • Neuromorphic Computing: Utilizing the bistable nature of the system for memory elements or logic gates where the state depends on the history of inputs.
  • Low-Power Switching: Triggering complex dynamical states with lower power thresholds by pre-conditioning the magnetic state.

In summary, the paper successfully bridges theory and experiment to demonstrate that self-induced Floquet magnons in magnetic vortices are not just a function of drive power, but are hysteretic and controllable via the initial magnetic configuration of the vortex core.