Topological analysis and experimental control of transformations of domain walls in magnetic cylindrical nanowires

TL;DR

This work investigates the topology of three-dimensional magnetization textures in cylindrical nanowires, focusing on the Bloch-Point-Wall (BPW) and Transverse-Vortex-Wall (TVW) configurations and their transformations under current-induced Oersted fields. By combining TXM-XMCD experiments with micromagnetic simulations, the authors reveal nanosecond-scale topology changes and track both bulk and surface topological signatures, introducing a bulk–surface invariant that connects $w_{vol}$ and $w_{surf}$ through the polarity of surface vortices/antivortices. The invariant, expressed as $DeltaOmega = w_vol - 1/2 sum(w_surf * p_n)$, remains conserved during dynamics, enabling a predictive framework for domain-wall transformations. Importantly, the final topology can be deterministically selected by tuning the current-pulse duration, offering a general pathway to control 3D magnetization textures with potential spintronic applications in 3D electronics.

Abstract

Topology is a powerful tool for categorizing magnetization textures by defining a topological index in both two-dimensional (2D) systems, such as thin films or curved surfaces, and in 3D bulk systems. In the emerging field of 3D nanomagnetism, both volume and surface topological numbers must be considered, requiring the identification of a proper global topological invariant to support categorization. Here we consider domain walls in cylindrical nanowires as an excellent playground for 3D nanomagnetic systems, excited by a charge current, that generates an OErsted field. We first provide experimental evidence of previously unreported domain-wall transformations of topology occurring at the nanosecond timescale. We investigate these transformations with micromagnetic simulations, tracking both bulk and surface topological signatures.We demonstrate a topological invariant combining both signatures, while the topological charge varies from bulk to surface during the dynamics. The experimental change of topology is reproduced when the pulse duration matches the timescale of the internal transformations of the wall, and the current is switched off before the transformation is complete. We expect that the topological categorization and dynamical exploitation apply to any 3D nanomagnetic system.

Figures (5)

• Figure 1: (a) Schematics of the geometry of the cylindrical nanowire considered, and the current-induced Œ rsted field (b) Sketch of head-to-head BPW texture with positive (C+) and negative (C-) circulation, defined with respect to $\hat{\boldsymbol{z}}$, and head-to-head TVW texture. Light arrows sketch the direction of magnetization around the surface vortex (resp. antivortex), here with positive V+ (resp. negative AV-) polarity, reflecting the sign of radial magnetization at the core of the object.
• Figure 2: Transmission X-ray microscopy (TXM) imaging of current-induced domain-wall transformations in a 120nm-diameter Permalloy nanowire. (a) Schematic of a TXM setup for imaging an electrically-contacted nanowire exhibiting the magnetic state of a BPW. (b) Electron microscopy image of an electrically contacted Permalloy nanowire. (c) and (f) TXM images at the Fe L$_{3}$ edge of different regions of the nanowire. The corresponding initial magnetic configurations are shown in the XMCD images in (d) and (g). The black arrows indicate the direction of magnetization. Subsequent XMCD images, after the application of the indicated current pulse, show (e) BPW to TVW transformation, (h) TVW to BPW transformation and (i) BPW circulation switching.
• Figure 3: Switching of circulation of a head-to-head BPW in a 90nm-diameter homogeneous Permalloy nanowire, for damping values $\alpha=1$ and $\alpha=0.1$. A pulse of current density of $-1.50e12A\per m\squared$ is applied in a step-like manner at $t=0$, resulting in an Œ rsted field anti-parallel to the initial BPW circulation, with a magnitude of 42.5mT at the surface of the wire. (a) Domain wall width $\Delta_{\tau}$ as a function of time for both damping parameters. (b) and (c): $z$-axis trace of the topological objects versus time for $\alpha =1$ and $\alpha = 0.1$, respectively. The colors indicate different topological features: black for Bloch points, red (blue) for vortices with positive (negative) polarity, orange (green) for anti-vortices with positive (negative) polarity. (d) and (e): Snapshots of the magnetic texture at selected times, for $\alpha = 1$ and $\alpha = 0.1$, respectively. The selected times are highlighted in (b) and (c), respectively. The top rows show 3-dimensional external surface views of the nanowire, while the bottom rows despict its unrolled 2-dimensional ($\hat{\boldsymbol{z}}$ - $\hat{\boldsymbol{\upvarphi}}$) cylinder surface, viewed from the outside. The color map represents the radial magnetization component, where $m_{\rho}=1$ is red and $m_{\rho}=-1$ is blue. Dashed lines highlight the trajectories of $\mathrm{V}_1$ and $\mathrm{AV}_1$.
• Figure 4: Identification of the type of the first vortex nucleated based on the sign of the Œ rsted field radial torque during BPW circulation switching from C+ to C-. (a) A simplified illustration of the Œ rsted field induced torque ($\partial_{t}\mathbf{m}$) along the $m_{\varphi}=0$ isolines (blue and red dashed lines) prior to the switching process. To the left of the BPW center (black dashed line), the magnetization aligns towards negative radial values ($m_{\rho}<0$) while it aligns towards positive radial values ($m_{\rho}>0$) on the right side. (b) Illustration of the vortex in-plane orientation (clockwise or counterclockwise) while switching.
• Figure 5: Final state dependence on pulse length. The unrolled 2-dimensional ($\hat{\boldsymbol{z}}$ - $\hat{\boldsymbol{\upvarphi}}$) cylinder surfaces are viewed from the outside. The color map represents the radial magnetization component, where $m_{\rho}=1$ is red and $m_{\rho}=-1$ is blue. (a) The final state after the current is switched off at event (iii) ($t= 2.80ns$) of \ref{['fig:BPW-switching']} shows V-AV stabilization, i.e., a TVW configuration. (b) The final state after the current is switched off at event (iv) ($t= 2.98ns$) of \ref{['fig:BPW-switching']} shows V-AV annihilation, i.e., a BPW configuration.