Magnetic field-tuned size and dual annihilation pathways of chiral magnetic bobbers

TL;DR

The paper tackles how to control the size and annihilation of three-dimensional chiral bobbers (CBs) by developing a 3D analytical energy model that yields a closed-form radius $R(h_{ext})$ balancing exchange, DMI, anisotropy, demagnetization, and Zeeman terms. It validates the model with micromagnetic simulations in a bilayer geometry, revealing robust scaling relations and two distinct annihilation routes: a dynamic Weber-number–driven droplet-like collapse and a quasi-static Bloch-point depinning at interfaces, plus a novel fragmentation pathway $CB ightarrow ST ightarrow 4 imes HCB ightarrow FM$ unique to 3D topology. The radius scales as $R(h_{ext}) = π^2 l_D / [2(κ^2 + 4π^2 η^2 - 3 h_{ext} - 1)]$ with $η = l_{ex}/h$, $l_{ex} = \,sqrt{A/(μ_0 M_s^2)}$, $κ = \,sqrt{K_1/(μ_0 M_s^2)}$, and $l_D = D/(μ_0 M_s^2)$, and annihilation thresholds obey $δB_c ∝ 1/R^2$ and $B_{th} ∝ (K_2 - K_1)$ with a correction from $D$. These results provide quantitative design rules for stabilizing and manipulating CBs in 3D spintronic devices, with implications for high-density memory and neuromorphic architectures. $R(h_{ext})$ and $B_{th}$ relations offer predictive control of 3D topological textures under field tuning.

Abstract

Magnetic chiral bobbers (CBs) are three-dimensional (3D) topological spin textures that consist of a tapered skyrmion tube terminating in a Bloch point, promising applications in high-density spintronics. However, the mechanisms controlling their size and the dynamics of their annihilation are still not fully understood. In this study, we present an analytical model that predicts the radius $R$ of the CB as a function of the external magnetic field, the Dzyaloshinskii-Moriya interaction (DMI), the magnetic anisotropy, and the exchange interaction. The micromagnetic simulations validate this model across a broad range of parameters. We also identify two mechanisms of annihilation of CBs: (i) a droplet-like instability that occurs under rapid changes in the magnetic field, which we describe using a proposed magnetic Weber number $We$ and its critical field step scaling; and (ii) Bloch point depinning mechanism at interfaces, for which we determine the threshold magnetic field $B_{\text{th}}$ for annihilation. Importantly, we uncover a novel fragmentation pathway in which CBs transform into skyrmion tubes, then into half-CBs, and finally into ferromagnetic states. These findings lay the groundwork for understanding and manipulating 3D CBs as next-generation devices.

Figures (5)

• Figure 1: (a) Bilayer magnetic nanostructures and the generated single CB structure with the radius $R$ and the depth $h$. (b) Variation of radius $R$ of CBs with external magnetic field B$_{\text{ext}}$ for the different DMI $D$. (c) The radius $R$ of CBs vs. $D$ with different B$_{\text{ext}}$. (d) The radius $R$ of CBs vs. PMA $K_1$ with the different B$_{\text{ext}}$.
• Figure 2: (a) The magnetic Weber number $We$ as the function of the CB radius $R$. (b) The maximum magnetic field step $\delta B_c$ by the simulations as a function of $1/R^2$ of CBs.
• Figure 3: (a) The depth $h$ of the CB as a function of external fields B$_{\text{ext}}$. (b) Threshold magnetic field B$_{\text{th}}$ for the first magnetic bobber annihilation dependent on $K_2-K_1$.
• Figure 4: (a) The radius $R$ and the second annihilation of CBs with external magnetic field $B$ for different DMI $D$. (b) The second annihilation into CBs from a CB to an ST, an HCB, and an FM state. (c) Average magnetization $m_z$ as the magnetic field changes for different $K_2$. (d) The total magnetic energy $E_{\text{tot}}$, exchange energy $E_{\text{ex}}$, the PMA energy $E_{\text{ani}}$ and the demagnetization energy $E_{\text{deg}}$ as a function of external magnetic fields.
• Figure 5: The snapshot of the annihilation process of a CB from a CB to an ST, an HCB, and an FM state.