Static and Dynamics of Twisted Skyrmion Tubes in Frustrated Magnets

TL;DR

The paper demonstrates the stabilization of twisted skyrmion tubes (TSkTs) in frustrated magnets with competing next-nearest-neighbor exchange, showing a helicity twist along $z$ and a Hopf topology with $\\mathcal{Q}_{H}= \\frac{Q_{v} \\kappa L_{z}}{2\\pi}$, which scales with thickness. An analytic ansatz with $\\Theta(\\rho,\\phi,z)= f(\\rho- R)$ and $\\Phi= Q_{v}\\phi+\\kappa z+\\eta$ yields a conical background with $\\kappa = (|A|/(2 C_{1}))^{1/2}/2$ and an energy functional $E(R) \\approx A + (C_{1}+C_{2})/R^{2} + B_{z} R^{2}$, implying a stable radius $R_{0} = ((C_{1}+C_{2})/B_{z})^{1/4}$. Micromagnetic simulations using MuMax3 validate the analytical predictions, map stability regions in $B_{z}$-$L_{z}$ space, and show that $\\mathcal{Q}_{Sk}$ approaches $-1$ while $\\mathcal{Q}_{H}$ approaches $1$ near the stability window, with a critical field $B_{c}$ and wavelength $\\lambda_{c}=2\\pi/\\kappa$. Under spin-orbit torque, the TSkT exhibits helicity rotation with frequency $\\Omega \\approx -\\sigma_{z}(\\tau_{DL}-\\alpha\\tau_{FL})$, radius oscillations around $\\langle R\\rangle$, and a spin-motive-force–induced dc voltage $\\bar{V}_{DC} = 2\\hbar \\Omega/q_{e}$, pointing to a nanoscale storage or nano-battery functionality, while the nonzero toroidal moment $\\boldsymbol{\\mathcal{T}}$ and emergent-field effects enable nonreciprocal transport and detection via TEM, STXM, MOKE, or MFM.

Abstract

Stable three-dimensional topological skyrmion structures in frustrated magnets are investigated. The texture exhibits a helicoid pattern along the vertical direction, described by a position-dependent helicity, which interpolates between Neel- and hedgehog-like two-dimensional skyrmions, characterized by the Hopf index, and is referred to as "twisted skyrmion tubes" (TSkTs). The stability and topology of TSkTs are achieved by competing next-nearest-neighbor exchange interactions, the thickness of the magnet, and the applied magnetic field. The dynamical behavior of a twisted structure in frustrated magnets is determined. Specifically, we derive that the helicity dynamics of the TSkT can be driven by an electric current resulting from spin-orbit torque interaction. Furthermore, we address the study of the electronic scattering problem using a spin-orbit-torque-driven TSKT, which offers promising applications for low-power storage nanodevices and nanobatteries with enhanced control.

Figures (5)

• Figure 1: Stabilized TSkT, obtained by numerical simulation, in a nano cylinder of radius $R=150$ nm, thickness $L_{z}=100$ nm, and external magnetic field $B_{z}=0.2$ T. (a) Magnetization field in the iso-surface $m_{z}=0$. (b) Pre-images of $\boldsymbol{m}=\hat{\boldsymbol{x}}$ (red) and $\boldsymbol{m}=-\hat{\boldsymbol{x}}$ (blue). (c) Horizontal midplane cross section in the $xz$-plane of the TSkT.
• Figure 2: TSkT with (a) anticlockwise chirality, where $\eta=\pi/2$ and Hopf index $\mathcal{Q}_{H}=-2$, and (b) clockwise chirality with $\eta=3\pi/2$ and Hopf index $\mathcal{Q}_{H}=2$. Preimages showing $\boldsymbol{m}=\hat{\boldsymbol{y}}$ (green) and $\boldsymbol{m}=-\hat{\boldsymbol{y}}$ (blue) components for a stabilized TSkT in a box of size $L_{x}=L_{y}=300$ nm and thickness film twice the period of the conical phase, $L_{z}=46 \ \mathrm{nm} \approx 2\lambda_{c}$. (c) Energy of a TSkT stabilized by micromagnetic simulations as a function of the helicity, with $\Delta E_{\mathrm{tot}}=E_{\mathrm{tot}}(\gamma)-E_{\mathrm{tot}}(0)$ for the cases: without DDI (blue line) and with DDI (red line).
• Figure 3: (a) Stability diagram of the TSkT as a function of the external magnetic field $B_{z}$ and the thickness of the film $L_{z}$. Enclosed region corresponds to the value of the skyrmion charge between $-1\leq \mathcal{Q}_{Sk}<-0.9$. (b) Isosurface, $m_{z}=0$, representing the magnetization $(m_{x},m_{y})$ of a stabilized twisted skyrmion tube in a box of size $L_{x}=L_{y}=300$ nm, and $L_{z}=25\ \mathrm{nm} \approx \lambda_{c}$ ($Q_{H}\approx 1$) equal to the wave length of the conical state. (c),(d) Skyrmion topological charge $\mathcal{Q}_{Sk}$ (red line) and hopf index $\mathcal{Q}_{H}$ (blue line) as a function of $B_{z}$ and $L_{z}$, respectively.
• Figure 4: Representation of the SOT-driven TSkT (with $\mathcal{Q}_{H}=1$). (a) Helicity dynamics of a TSkT driven by spin orbit torque with spin polarization $\boldsymbol{\sigma}=-\hat{\boldsymbol{z}}$, indicating a clockwise rotation. (b) Time evolution of $R(t)$, and the steady state oscillations. (c) rotation of the pre-images $\boldsymbol{m}=\hat{\boldsymbol{x}}$ (red curve) and $\boldsymbol{m}=-\hat{\boldsymbol{x}}$ (blue curve). (d) Frequency of a TSkT shown at (b) as a function of $\tau_{\mathrm{DL}}$ with $\tau_{\mathrm{FL}}=\chi \tau_{\mathrm{DL}}$ for $\chi=0$ (red line), $\chi=2$ (blue line). (e) Average radius of a TSkT as a function of the $\tau_{\mathrm{DL}}$ with $\tau_{\mathrm{FL}}=\chi \tau_{\mathrm{DL}}$ for $\chi=0$ (red line), $\chi=2$ (blue line).
• Figure 5: Streamlines of the emergent magnetic field $\boldsymbol{B}^{e}$ of the stabilized TSkT displayed in the Fig. \ref{['fig1']} . The curves are described by circular helices with constant pitch given by $\kappa$.