Dynamic annihilation pathways of magnetic skyrmions

TL;DR

The paper tackles how numerical modeling choices affect the predicted dynamics and annihilation of 2D magnetic skyrmions as textures approach atomic scales. It introduces a pseudospectral Landau-Lifshitz framework with a dispersion-based nonlocal kernel that embeds exchange and interfacial DMI across scales, enabling seamless atomistic-to-continuum behavior and two exchange kernels for comparison. The study shows that the magnon kernel reproduces the full first Brillouin zone and yields immediate annihilation under a ramped field, while a micromagnetic kernel can overestimate exchange energy and induce breathing prior to annihilation; results depend strongly on discretization, ramp rate, and damping. These findings highlight the need for accurate dispersion treatment in multiscale simulations of skyrmions and other 2D/3D textures, with PS-LL offering a robust pathway for reliable material discovery and modeling.

Abstract

The investigation of magnetic solitons often relies on numerical modeling to determine key features such as stability, annihilation, nucleation, and motion. However, as soliton sizes approach atomic length scales, the accuracy of these predictions become increasingly sensitive to the details of the numerical model. Here, we study the annihilation of two-dimensional magnetic skyrmions using a pseudospectral approach and compare its performance to that of conventional micromagnetic simulations. A central distinction between the models lies in their treatment of the exchange interaction, which governs the magnon dispersion relation and plays a crucial role to balance the uniaxial anisotropy to stabilise skyrmions. We demonstrate that both the choice of model and spatial discretisation significantly influence the skyrmion dynamics and the magnetic field required for annihilation. The pseudospectral model provides a consistent description across length scales and captures complex behaviours such as skyrmion breathing on its path to annihilation. Our results have direct implications in the state-of-the-art modeling of skyrmions and other two-dimensional textures and will impact the modeling of three-dimensional textures such as hopfions. More broadly, our approach will contribute to the development seamless multiscale model and optimization machine learning approaches for material discovery.

Figures (6)

• Figure 1: Magnon dispersion relations extracted from dynamic simulations using three different models: the PS-LL model with the full magnon kernel, (a) and (b), the PS-LL model with the micromagnetic kernel, (c) and (d), and MuMax3 (e) and (f). The top row shows results at an atomic-scale discretization of 0.4 nm, while the bottom row uses a micromagnetic-scale discretization of 5.0 nm. (a) The magnon kernel at an atomic resolution reproduces the cosine-shaped dispersion relation for the first Brillouin zone. (b) The magnon kernel at 5.0 nm cell size produces a dispersion curve in a confined region [12.5 times smaller than panel (a)] of the first Brillouin zone, revealing a parabolic form consistent with the low-$k$ limit. (c) The micromagnetic kernel an atomic resolution produces a parabolic dispersion which approximates this dispersion well near the origin but fails to capture short-wavelength behavior. (d) The micromagnetic kernel at 5.0 nm produces a dispersion curve that shows excellent agreement with panel (b), confirming the expected parabolic behavior at this scale. (e) MuMax3 at an atomic resolution closely matches the magnon kernel at atomic resolution due to its nearest-neighbor finite-difference scheme. (f) MuMax3 at 5.0 nm produces a cosine-shaped dispersion due to the use of a finite-difference scheme and therefore deviates from the other two models at this scale. The color scheme is found in Ref. colorscheme.
• Figure 2: (a) Out-of-plane magnetization component $m_\mathrm{z}$ rendered as a color map for a $320~\mathrm{nm} \times 320~\mathrm{nm}$ thin film. A skyrmion is positioned in the center of the film according to the ansatz of Eq. \ref{['eq:skyrmion_profile']}. The inlay indicates the in-plane magnetization Néel-type configuration for the skyrmion. (b) One-dimensional cross-section of $m_\mathrm{z}$ taken through the skyrmion center and parallel to the $x$-axis.
• Figure 3: (a) Skyrmion radius versus external field for a $0.4$ nm cell size. The red curve uses the PS-LL model with the magnon kernel and MuMax3. The black curve is the PS-LL model with the micromagnetic kernel with the same ramp rate and alpha as the red curve. The blue curve is also the micromagnetic kernel but with a larger ramp rate. (b) Skyrmion radius versus external field for a $5.0$ nm cell size. Both PS-LL models behave nearly identically at a $5.0$ nm cell size. The orange and green curves individually represent both PS-LL models but for different values for the Gilbert damping constant $\alpha$. The purple curve is for MuMax3 with the same parameters as the orange curve.
• Figure 4: Color-plots indicating the field at which the skyrmion annihilates based on a change of the topological charge from 1 to 0. The top row pertains to simulations using a discretization of 0.4 nm for (a) magnon kernel, (c) micromagnetic kernel, and (e) MuMax3. The white boundary indicates the transition between skyrmion annihilation and breathing. To the left of the dividing line, the skyrmion annihilates at a relatively low field. To the right of the dividing line, the skyrmion's annihilation field increases significantly with increasing ramp rate and decreasing damping. The bottom row has a discretization of 5.0 nm for (b) magnon kernel, (d) micromagnetic kernel, and (f) MuMax3. In this case, we observe annihilation only after few breathing periods but the annihilation field shows the same trend with respect to ramp rate and damping.
• Figure 5: Color-plots indicated the number of breathing periods the skyrmion undergoes prior to annihilation. The top row pertains to simulations using a discretization of 0.4 nm for (a) magnon kernel, (c) micromagnetic kernel, and (e) MuMax3. The dark blue region to the left of the white line corresponds to zero breathing periods, signifying immediate annihilation following the initial contraction. The region to the right of the dividing line shows how the number of breathing periods increases with increasing ramp rate and decreasing damping. The bottom row has a discretization of 5.0 nm for (b) magnon kernel, (d) micromagnetic kernel, and (f) MuMax3. In this case, breathing is observed across the entire parameter space, with the number of periods generally increasing with higher ramp rates and lower damping.