🔬 mesoscale physics

Twisted magnon frequency combs in ferromagnetic nanorings

This paper reports the discovery of twisted magnon frequency combs in ferromagnetic nanorings, demonstrating how their formation, mode density, and tunable spacing can be precisely controlled through geometric hole size and external magnetic fields via nonlinear coupling and angular momentum conservation.

Original authors: Liu Xuejuan, Zheng Xingen, Li Zhengyi, Zhang Zhizhi, Li Xiaoguang, Sun Haipeng, Li Hui, Zhou Cangtao

Published 2026-08-21

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CC BY 4.0

Original authors: Liu Xuejuan, Zheng Xingen, Li Zhengyi, Zhang Zhizhi, Li Xiaoguang, Sun Haipeng, Li Hui, Zhou Cangtao

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the microscopic world of magnetic materials, tiny whirlpools of magnetism known as magnetic vortices exist. Imagine a flat, circular disk made of a magnetic metal. Inside this disk, the tiny magnetic arrows that make up the material do not all point in the same direction. Instead, they curl around a central point, much like water swirling down a drain, with the very center of the swirl pointing either up or down. These structures are stable and fundamental, acting as the building blocks for future magnetic memory and computing devices. When these vortices are disturbed, they do not just sit still; they wobble and spin. They also interact with waves of magnetism that travel through the material, similar to ripples moving across a pond. Scientists have long known that if you shake these vortices hard enough, they can generate a very specific pattern of sound-like waves called a frequency comb. This pattern is a series of equally spaced lines, like the teeth of a comb, which can be used to measure time and frequency with extreme precision. The question researchers have been asking is whether changing the shape of the magnetic material from a solid disk to a ring with a hole in the middle would change how these waves behave, and if so, how.

A team of researchers set out to answer this by creating a detailed computer simulation of these magnetic rings. They modeled a thin, flat ring made of a magnetic alloy called Permalloy, which is commonly used in such studies. By adjusting the size of the hole in the center of the ring and applying a rotating magnetic field, they watched how the magnetic vortex inside reacted. Their goal was to see if the unique geometry of the ring would create new types of wave interactions that are not possible in a solid disk. The simulations revealed that the ring shape does indeed preserve the ability to create these frequency combs, but it also introduces a powerful new way to control them. The researchers found that the size of the central hole acts like a tuning knob. When the hole is very small, the magnetic vortex behaves much like it does in a solid disk, producing a standard set of wave frequencies. However, as the hole is made larger, the physics changes dramatically.

When the researchers increased the hole size to a diameter of 50 nanometers, a new and unexpected phenomenon emerged. The larger hole forced the magnetic vortex to move into a different part of the ring, where it encountered a steeper magnetic landscape. This change caused the vortex to wobble at a much faster rate, increasing its frequency from 0.075 gigahertz to 0.45 gigahertz. More importantly, the larger hole acted as a structural defect that excited a completely new type of magnetic wave at 3.4 gigahertz. This new wave did not just sit alongside the others; it began to mix with the existing waves in a complex dance of energy exchange. This mixing process, known as four-wave mixing, generated a dense forest of new frequencies, effectively multiplying the number of lines in the frequency comb by an order of magnitude. The result was a much richer and more detailed spectrum of signals than what could be achieved with a solid disk or a ring with a tiny hole.

The team also discovered that they could tune these signals continuously by applying a steady magnetic field from the side. In a solid disk, the vortex moves smoothly and predictably as the field changes, creating a symmetric pattern. In the ring, however, the hole creates a sort of magnetic trap that pins the vortex in place. This pinning effect causes the system to behave differently depending on the direction of the magnetic field. If the field is applied in one direction, the vortex might be stable and produce a strong signal. If the field is reversed, the vortex might become unstable and disappear, causing the signal to vanish entirely. This creates a kind of magnetic switch that remembers its history, a property known as hysteresis. This asymmetry means that the frequency comb can be turned on or off simply by flipping the direction of the magnetic field, offering a new level of control for potential devices.

The researchers confirmed that these new patterns of waves carry a specific type of rotation, known as orbital angular momentum, which is a fundamental property of the waves themselves. The simulations showed that the waves maintain their shape and structure as they interact, obeying strict rules about how energy and rotation are conserved. The study demonstrates that by simply changing the geometry of the material—making a hole in the center—and applying a magnetic field, scientists can tailor the behavior of these magnetic waves with high precision. This work establishes the magnetic ring as a versatile platform for creating and controlling these frequency combs. The ability to generate such dense and tunable signals suggests that these structures could be used in future technologies for high-sensitivity sensing and ultra-precise measurement, turning a simple change in shape into a powerful tool for manipulating the magnetic world.

Technical Summary: Twisted Magnon Frequency Combs in Ferromagnetic Nanorings

Problem Statement
Magnetic vortices in ferromagnetic nanostructures exhibit rich dynamic properties, including low-frequency gyrotropic modes and quantized azimuthal spin-wave modes (twisted magnons). While the generation of twisted magnon frequency combs (tMFCs) via strong nonlinear coupling between vortex-core (VC) gyration and twisted spin waves has been established in nanodisks, it remains an open question whether these selection rules and phenomena persist in nanoring geometries. Specifically, it is unclear how the removal of the central singularity and the introduction of an inner boundary affect the tMFC formation, selection rules, and tunability. Furthermore, the potential for nanorings to offer new pathways for tailoring these selection rules through geometric parameters (such as hole size) and external fields requires systematic investigation.

Methodology
The authors employed micromagnetic simulations using the open-source package Mumax3 to numerically solve the Landau-Lifshitz-Gilbert (LLG) equation. The study utilized Permalloy (Py) material parameters with a cell size of $1 \times 1 \times 5$ nm $^3$ . Two primary geometries were investigated: a ferromagnetic nanodisk (outer diameter 300 nm, thickness 5 nm) and ferromagnetic nanorings with varying inner hole diameters ( $2r$ ).

The system was initialized in a vortex state. To characterize the excitation spectrum, an in-plane sinc-shaped magnetic field was applied. To generate and analyze tMFCs, the system was driven by a rotating in-plane magnetic field resonant with specific azimuthal spin-wave modes ( $l = +1$ ). The spectral power was obtained via Fast Fourier Transform (FFT) of the spatially averaged out-of-plane magnetization ( $\delta m_z$ ). The study systematically varied the inner hole diameter (comparing $2r = 5$ nm and $2r = 50$ nm) and applied static in-plane magnetic fields ( $H_y$ ) to observe their effects on the comb structure, mode frequencies, and selection rules.

Key Contributions and Results

Preservation of Selection Rules in Nanorings:
The study confirms that nanorings support tMFCs arising from the strong nonlinear coupling between the VC gyration and azimuthal spin-wave modes. The comb lines carry distinct orbital angular momentum (OAM) with quantum numbers spaced by unity. The formation of these combs obeys selection rules governed by the simultaneous conservation of energy and angular momentum. Specifically, three-magnon scattering processes conserve the radial index ( $n=0$ or $n=1$ ) while allowing the azimuthal index ( $l$ ) to shift by the order of the mixing process ( $m$ ), resulting in sum-frequency ( $f_0 + m f_g$ ) and difference-frequency ( $f_0 - m f_g$ ) modes.
Geometric Tuning via Hole Size:
The inner hole diameter serves as a critical tuning parameter.
- Small Hole ( $2r = 5$ nm): Under strong driving, the VC is displaced from the center, restoring a confining potential and enabling tMFC generation. In this configuration, the gyrotropic mode frequency is $f_g = 0.075$ GHz.
- Large Hole ( $2r = 50$ nm): Increasing the hole size forces the VC into the annular region, significantly increasing the restoring force and the gyrotropic frequency ( $f_g$ ) from 0.075 GHz to 0.45 GHz. Crucially, the larger hole acts as a structural defect that excites an additional resonant mode at $f_a = 3.4$ GHz. This new mode interacts with the drive and harmonics via nonlinear four-wave mixing, generating sum-frequency components that further couple with the gyrotropic mode. This process dramatically densifies the frequency comb, increasing sideband multiplicity by an order of magnitude compared to the small-hole ring or nanodisk.
Magnetic Field Tuning and Asymmetric Switching:
An external in-plane magnetic field ( $H_y$ ) enables continuous, reversible tuning of the comb spacing by displacing the VC and modifying its confinement potential.
- Nanodisk: Exhibits symmetric spectra with respect to field polarity, with comb spacing increasing monotonically with $|H_y|$ .
- Nanoring ( $2r = 5$ nm): The small hole introduces a geometric pinning potential that lifts the degeneracy of the VC confinement along opposite field directions. This leads to asymmetric nucleation and annihilation thresholds. In the positive field range ( $2 \text{ mT} \lesssim H_y \lesssim 4 \text{ mT}$ ), the VC annihilates, suppressing the tMFC, whereas in the corresponding negative range, the VC remains stable. This results in a field-polarity-dependent switching and hysteresis in the low-field regime.
- Nanoring ( $2r = 50$ nm): Displays a dense comb structure due to the additional mode, with the fundamental drive peak remaining prominent while sideband multiplicity is significantly enhanced.

Significance
The paper establishes the ferromagnetic nanoring as a versatile platform for nonlinear magnonics. The results demonstrate that the tMFC selection rules established in nanodisks persist in nanorings, but the additional boundary degrees of freedom allow for significant tailoring of the spectral properties. By adjusting the hole size, one can control the density of the frequency comb and the number of active nonlinear mixing channels. Furthermore, the geometric pinning in nanorings introduces unique hysteretic and asymmetric switching behaviors not present in disks. The authors posit that these findings provide a foundation for integrating tMFC functionality into complex magnonic circuits with geometric tunability, with potential applications in tunable frequency comb generation and precision metrology.

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