This repository contains the MuMax3 simulation code and post-processing scripts used to generate and drive magnetic Hopfions in a chiral ferromagnetic nanodisk, as reported in our study of nonlinear magnon dynamics and frequency multiplication.
Note: The associated manuscript is currently under review at Physical Review Applied. This repository will be updated with the full citation and DOI once the paper is published.
Hopfions are three-dimensional topological solitons characterized by the Hopf index, formed from a closed-loop, twisted skyrmion-string structure. This project uses systematic micromagnetic simulations to study how a magnetic Hopfion responds nonlinearly to microwave driving fields, with a focus on:
- Stabilizing a Hopfion in a chiral ferromagnetic nanodisk sandwiched between two perpendicular-magnetic-anisotropy (PMA) layers.
- Characterizing the localized magnon spectrum (breathing and gyrotropic modes) under in-plane and out-of-plane excitation.
- Demonstrating second- and third-harmonic generation (frequency multiplication) driven by the noncollinear spin texture.
- Exciting localized magnon modes by driving at fractional multiples of their eigenfrequency (e.g., f/2, f/3, f/4).
- Assessing robustness against thermal fluctuations and quenched DMI disorder, and comparing isolated versus bulk Hopfion response.
- MuMax3 (GPU-accelerated micromagnetic simulator)
- A CUDA-capable NVIDIA GPU
- Python 3.x for post-processing (numpy, scipy, matplotlib) to compute FFT spectra and generate figures
The default system is a circular chiral ferromagnetic disk (diameter 200 nm, height 90 nm) with two 2 nm PMA capping layers required to stabilize the Hopfion.
Key material parameters (unless otherwise stated):
| Parameter | Symbol | Value |
|---|---|---|
| Saturation magnetization | Ms | 384 kA/m |
| Exchange constant | A | 2.19 pJ/m |
| DMI strength | D | 340–395 µJ/m² |
| Uniaxial anisotropy (PMA layers) | Ku | 8 × 10⁵ J/m³ |
| Gilbert damping | α | 0.01 |
| Cell size | — | 2 × 2 × 2 nm³ |
The Hopfion is initialized from an analytic ansatz, relaxed, and then evolved to a stable state before microwave excitation is applied.
- Hopfion generation — Relax the system from the initial ansatz to obtain a stable Hopfion ground state.
- Spectrum characterization — Apply a sinc-pulse excitation to extract the spin-wave resonance spectrum and identify localized magnon modes.
- Nonlinear driving — Apply single-frequency sinusoidal microwave fields (in-plane or out-of-plane) to study harmonic generation.
- Fractional-resonance excitation — Drive at f/2, f/3, f/4 of a target mode to selectively excite localized states.
- Robustness studies — Run stochastic-LLG (finite-temperature) and quenched-DMI-disorder simulations, and compare isolated versus bulk configurations.
- Post-processing — Compute FFT power spectra of the averaged magnetization to visualize harmonics.
- Scripts for Hopfion generation and relaxation
- Microwave excitation drivers (in-plane and out-of-plane)
- Post-processing scripts for FFT / spectral analysis
Please refer to the individual scripts for detailed, per-simulation parameters.
The codes for Hopfion generation and microwave excitation are available in this repository: https://github.com/waleedwaseer/Hopfion
A citation will be added here upon publication. The manuscript is under review at Physical Review Applied.
This project was supported by the National Key R&D Program of China (Grant No. 2022YFA1402802), the National Natural Science Foundation of China (Grant Nos. 12374103 and 12434003), and the Sichuan Science and Technology Program (No. 2025NSFJQ0045).
For questions regarding the code or the study, please open an issue in this repository or contact the corresponding author.