Plasmonic Skyrmion Lattice Based on the Magnetoelectric Effect

Open Access

Plasmonic Skyrmion Lattice Based on the Magnetoelectric Effect

X.-G. Wang, L. Chotorlishvili, N. Arnold, V. K. Dugaev, I. Maznichenko, J. Barnaś, P. A. Buczek, S. S. P. Parkin, and A. Ernst

Phys. Rev. Lett. 125, 227201 – Published 23 November 2020

Abstract

The physical mechanism of the plasmonic skyrmion lattice formation in a magnetic layer deposited on a metallic substrate is studied theoretically. The optical lattice is the essence of the standing interference pattern of the surface plasmon polaritons created through coherent or incoherent laser sources. The nodal points of the interference pattern play the role of lattice sites where skyrmions are confined. The confinement appears as a result of the magnetoelectric effect and the electric field associated with the plasmon waves. The proposed model is applicable to yttrium iron garnet and single-phase multiferroics and combines plasmonics and skyrmionics.

Received 29 May 2020
Revised 4 August 2020
Accepted 1 October 2020

DOI:https://doi.org/10.1103/PhysRevLett.125.227201

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Open access publication funded by the Max Planck Society.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Magnetic interactions Magnetization dynamics Methods in magnetism Micromagnetism Spintronics

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

X.-G. Wang¹, L. Chotorlishvili ², N. Arnold ³, V. K. Dugaev⁴, I. Maznichenko², J. Barnaś^5,6, P. A. Buczek⁷, S. S. P. Parkin⁸, and A. Ernst ^8,9

¹School of Physics and Electronics, Central South University, Changsha 410083, China
²Institut für Physik, Martin-Luther Universität Halle-Wittenberg, D-06120 Halle/Saale, Germany
³Soft Materials Lab, Linz Institute of Technology LIT, Johannes Kepler University Linz, Altenberger Straße 69, 4040 Linz, Austria
⁴Department of Physics and Medical Engineering, Rzeszów University of Technology, 35-959 Rzeszów, Poland
⁵Faculty of Physics, Adam Mickiewicz University, ul. Uniwersytetu Poznańskiego 2, 61-614 Poznań, Poland
⁶Institute of Molecular Physics, Polish Academy of Sciences, ul. M. Smoluchowskiego 17, 60-179 Poznańn, Poland
⁷Department of Engineering and Computer Sciences, Hamburg University of Applied Sciences, Berliner Tor 7, 20099 Hamburg, Germany
⁸Max Planck Institute of Microstructure Physics, Weinberg 2, D-06120 Halle, Germany
⁹Institute for Theoretical Physics, Johannes Kepler University, Altenberger Straße 69, 4040 Linz, Austria

Article Text

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Supplemental Material

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References

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Issue

Vol. 125, Iss. 22 — 27 November 2020

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Images

Figure 1
The plasmonic confinement lattice. The plot shows the distribution of the plasmonic wave intensity $I = 2 E_{l s}^{2} [\cos (k_{x} x + φ_{x}) + \cos (k_{y} y + φ_{y})]^{2}$ normalized to its maximal value $8 E_{l s}^{2}$ in the center of a given lattice site. The radius of the lattice sites is about 70–100 nm, and the distance between them 250–300 nm. The distance between lattice sites exceeds the characteristic skyrmion interaction length (see below). Therefore skyrmions confined in the lattice sites can be treated as independent objects. Nevertheless, to explore the robustness of the confinement, we also consider the case when the distance between lattice sites is comparable to the skyrmion interaction length.
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Figure 2
(a) The center of the plasmonic beam $(x_{0}, y_{0})$ is steered with the velocity $v_{y} = 1 m / s$ and $E_{0} = 0.2 MV / cm$ . The spatial profile of $m_{z}$ at time $t = 0$ and $t = 100 ns$ (from micromagnetic simulations). The color specifies the value of the $m_{z}$ component of magnetization. (b) The temperature profile $T (x, y, t)$ induced by plasmonic surface waves. The maximum of the temperature coincides with interference peaks of plasmonic surface waves (plasmonic lattice sites). The drift of the interference peaks with the velocity $v_{x} = 8 m / s$ forms the temperature pattern. The maximal temperature for the electric field $E_{0} = 0.8 MV / cm$ and $l_{g} = 37.5 nm$ is $T_{m} = 150 K$ . (c) Centers of the plasmonic beam $(x_{0}, y_{0})$ are steered (blue dashed lines) along the $y$ axis. The curly trajectories of two skyrmion centers (from micromagnetic simulations) are tracked for different velocities of the plasmonic beam: $v_{y} = 2.5 m / s$ (black solid lines) and $2.54 m / s$ (red dashed lines). Blue lines represent the trajectory of the two laser peak centers. Here, we use $E_{0} = 0.2 MV / cm$ and $l_{g} = 37.5 nm$ . (d) By solving Thiele equations (see Supplemental Material [9]), the trajectories of two skyrmion centers are tracked for different velocities of the plasmonic beam: $v_{y} = 10 m / s$ (black solid lines) and $24 m / s$ (red dashed lines) for $E_{0} = 0.8 MV / cm$ and $l_{g} = 37.5 nm$ .
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Figure 3
(a) Centers of plasmonic beams $(x_{0}, y_{0})$ are steered with the velocity $v_{y} = 1 m / s$ and $E_{0} = 0.2 MV / cm$ . Centers of skyrmions $(q_{x, 1}, q_{y, 1})$ and $(q_{x, 2}, q_{y, 2})$ trapped by the plasmonic field follow two peaks of the plasmonic beam. (b) The critical velocity $v_{y}^{c}$ as a function of the plasmonic electric field $E_{0}$ . The distance between peaks $l_{g} = 37.5 nm$ . (c) Deviation along $x$ axis of the positions of skyrmions centers $q_{x, 1 (2)}^{0}$ , from the centers of the plasmonic beam. For elevated electric field and stronger confinement $E_{0}$ , the deviation is smaller. Here, $l_{g} = 37.5 nm$ (blue dashed line). All results in this figure are from micromagnetic simulations.
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Figure 4
(a) The critical velocity $v_{y}^{c}$ as a function of the plasmonic lattice constant $l_{g}$ . The electric field $E_{0} = 0.8 MV / cm$ . (b) Positions of the skyrmion centers $q_{x, 1 (2)}^{0}$ with increasing plasmonic lattice constant $l_{g}$ . The blue dotted lines show the positions of the two plasmonic peaks. The amplitude of the plasmonic field $E_{0} = 0.8 MV / cm$ . (c) The total driving force $τ_{s, x}$ in the Thiele equation as a function of skyrmion position $q_{s}$ . (d) Dependence of the magnetic free energy density $F_{M}$ on the distance $2 q$ between two skyrmions, $2 q = q_{x, 1}^{0} - q_{x, 2}^{0}$ . The magnetic free energy is extracted from the simulation results. The interaction part of the free energy $F_{M} (q)$ precisely fits to the exponential function $F_{M} = A_{0} \exp (- q / ξ)$ with $A_{0} = 110 J / m^{3}$ and $ξ = 11 nm$ . The distance between skyrmion is controlled by varying distance between the plasmonic beams $l_{g}$ for (a) and (b).
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