Skyrmion Echo in a System of Interacting Skyrmions

X.-G. Wang, Guang-hua Guo, A. Dyrdał, J. Barnaś, V. K. Dugaev, S. S. P. Parkin, A. Ernst, and L. Chotorlishvili

Phys. Rev. Lett. 129, 126101 – Published 14 September 2022

Abstract

We consider helical rotation of skyrmions confined in the potentials formed by nanodisks. Based on numerical and analytical calculations we propose the skyrmion echo phenomenon. The physical mechanism of the skyrmion echo formation is also proposed. Because of the distortion of the lattice, impurities, or pinning effect, confined skyrmions experience slightly different local fields, which leads to dephasing of the initial signal. The interaction between skyrmions also can contribute to the dephasing process. However, switching the magnetization direction in the nanodiscs (e.g., by spin transfer torque) also switches the helical rotation of the skyrmions from clockwise to anticlockwise (or vice versa), and this restores the initial signal (which is the essence of skyrmion echo).

Received 8 October 2021
Revised 24 August 2022
Accepted 26 August 2022

DOI:https://doi.org/10.1103/PhysRevLett.129.126101

Physics Subject Headings (PhySH)

Magnetic interactions Magnetic texture Magnetization dynamics Micromagnetism Topological phases of matter

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

X.-G. Wang¹, Guang-hua Guo ¹, A. Dyrdał ², J. Barnaś ², V. K. Dugaev ³, S. S. P. Parkin ⁴, A. Ernst^4,5,6, and L. Chotorlishvili ³

¹School of Physics and Electronics, Central South University, Changsha 410083, China
²Faculty of Physics, Adam Mickiewicz University, ul. Uniwersytetu Poznańskiego 2, 61-614 Poznań, Poland
³Department of Physics and Medical Engineering, Rzeszów University of Technology, 35-959 Rzeszów, Poland
⁴Max Planck Institute of Microstructure Physics, Weinberg 2, D-06120 Halle, Germany
⁵Institute for Theoretical Physics, Johannes Kepler University, Altenberger Straße 69, 4040 Linz, Austria
⁶Donostia International Physics Center (DIPC), 20018 Donostia-San Sebastián, Basque Country, Spain

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Vol. 129, Iss. 12 — 16 September 2022

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Images

Figure 1
(a) Schematic of a chain of pinned skyrmions. Skyrmions are created in a thin magnetic film, on top of which there are magnetic nanodisks. The blue arrows show the direction of the field exerted on the skyrmions by the nanodisks. The skyrmions are confined in the regions below the nanodisks. Initially a particular skyrmion is embedded in the vicinity of the center of the corresponding nanodisk. (b) The skyrmion dynamics is described in terms of the position of the skyrmion center $(q_{x}, q_{y})$ with respect to the center of the corresponding nanodisk. For a negative field $H_{b} = - 15 mT$ , the skyrmion moves along the helix trajectory towards the center of the nanodisk (0,0). Upon the sudden quenching of the field $H_{b}$ from $- 15$ to 15 mT, at $t = 150 ns$ , the skyrmion moves along the helix trajectory away from the center of the nanodisk. (c) Frequencies of the clockwise ( $f_{-}$ , $H_{b} < 0$ ) and anticlockwise ( $f_{+}$ , $H_{b} > 0$ ) precessions as a function of the amplitude of the field exerted on the skyrmion by the nanodisk, $| H_{b} |$ . The anticlockwise precession frequency is larger than that of the clockwise one.
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Figure 2
(a) Evolution of the precession phases $φ_{i} = \arctan (q_{y}^{i} / q_{x}^{i})$ for 10 skyrmions. After the time interval 150 ns, the direction of magnetic field $H_{b}$ in all nanodisks is reversed periodically. (b) The time dependence of the amplitude of skyrmion echo, $⟨ q_{x y} ⟩ = (1 / n) [(\sum_{i = 1}^{n} q_{x}^{i})^{2} + (\sum_{i = 1}^{n} q_{y}^{i})^{2}]^{1 / 2}$ (here the average is over all $n = 10$ ) skyrmions. Blue dashed lines mark the position of the skyrmion echo. (c) The time dependence of the $x$ component of the spin-pumping current $I_{sp}^{x}$ into an additional heavy metal layer adjacent to the magnetic layer. (d)–(e) Skyrmion echo induced by a current pulse. Dynamics of the upper nanodisks is included via the spin transfer torque due to a current pulse in an additional magnetic layer adjacent on top of the layer of nanodisks (saturation magnetization $M_{s} = 1.4 \times 10^{6} A / m$ , exchange constant $A_{ex} = 30 pJ / m$ , and uniaxial anisotropy along $z$ with constant $K_{z} = 1 \times 10^{6} J / m^{3}$ ). After a time interval $= 150 ns$ , we periodically reverse the magnetization direction of the nanodisks via a strong spin transfer torque, and plot (d) the evolution of $x$ and $y$ coordinates $q_{x}$ and $q_{y}$ of the skyrmion position (single skyrmion is plotted while the induced reversal in skyrmion precession applies to all 10 skyrmions), and (e) the averaged amplitude $⟨ q_{x y} ⟩$ of all 10 skyrmions.
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Figure 3
(a) Averaged (over skyrmion area) magnetic free energy density $F_{M}$ of the skyrmion as a function of the skyrmion position $q_{x}$ with respect to the corresponding nanodisk center, $q_{y} = 0$ . The magnetic free energy is extracted from the simulation results, and the data are fitted to the function $F_{M} = A r^{2}$ , where $r^{2} = q_{x}^{2} + q_{y}^{2}$ . (b) and (c) Time evolution of the precession phases (b) and amplitude $⟨ q_{x y} ⟩ = (1 / n) [(\sum_{i = 1}^{n} q_{x}^{i})^{2} + (\sum_{i = 1}^{n} q_{y}^{i})^{2}]^{1 / 2}$ of 10 skyrmions (c), calculated from Thiele’s equation.
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Figure 4
(a) Dependence of the magnetic free energy density $F_{c}$ on the distance $d_{s}$ between neighboring skyrmions. The magnetic free energy is extracted from the simulation results. The energy $F_{c} (d_{s})$ fits to the exponential function $F_{c} = A_{0} \exp (- d_{s} / ξ)$ with the decaying length $ξ = 6.4 nm$ . (b) The coupling strength $c_{u}$ extracted from linear expansion of the energy profile in the vicinity of the skyrmion center. (c) The amplitude $⟨ q_{x y} ⟩$ of the skyrmion echo as a function of the distance $d_{s}$ between neighboring skyrmions. Amplitudes are fitted to the exponential function $⟨ q_{x y} ⟩ = I_{1} - I_{0} \exp (- d_{s} / ξ)$ . The bias field is $H_{z} = 100 mT$ .
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