Enhanced Stability of Skyrmions in Two-Dimensional Chiral Magnets with Rashba Spin-Orbit Coupling

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Enhanced Stability of Skyrmions in Two-Dimensional Chiral Magnets with Rashba Spin-Orbit Coupling

Sumilan Banerjee, James Rowland, Onur Erten, and Mohit Randeria

Phys. Rev. X 4, 031045 – Published 9 September 2014

Abstract

Recent developments have led to an explosion of activity on skyrmions in three-dimensional (3D) chiral magnets. Experiments have directly probed these topological spin textures, revealed their nontrivial properties, and led to suggestions for novel applications. However, in 3D the skyrmion crystal phase is observed only in a narrow region of the temperature-field phase diagram. We show here, using a general analysis based on symmetry, that skyrmions are much more readily stabilized in two-dimensional (2D) systems with Rashba spin-orbit coupling. This enhanced stability arises from the competition between field and easy-plane magnetic anisotropy and results in a nontrivial structure in the topological charge density in the core of the skyrmions. We further show that, in a variety of microscopic models for magnetic exchange, the required easy-plane anisotropy naturally arises from the same spin-orbit coupling that is responsible for the chiral Dzyaloshinskii-Moriya interactions. Our results are of particular interest for 2D materials like thin films, surfaces, and oxide interfaces, where broken surface-inversion symmetry and Rashba spin-orbit coupling naturally lead to chiral exchange and easy-plane compass anisotropy. Our theory gives a clear direction for experimental studies of 2D magnetic materials to stabilize skyrmions over a large range of magnetic fields down to $T = 0$ .

Received 19 April 2014

DOI:https://doi.org/10.1103/PhysRevX.4.031045

This article is available under the terms of the Creative Commons Attribution 3.0 License. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.

Published by the American Physical Society

Authors & Affiliations

Sumilan Banerjee¹, James Rowland¹, Onur Erten^1,2, and Mohit Randeria¹

¹Department of Physics, The Ohio State University, Columbus, Ohio 43210, USA
²Department of Physics and Astronomy, Rutgers University, Piscataway, New Jersey 08854, USA

Popular Summary

New phases of matter with unexpected properties often arise from emergent mesoscale structures. Some of the most interesting phases include crystalline arrays of topological excitations with a periodicity that is orders of magnitude larger than that of the underlying atomic lattice. Recently, certain magnetic materials have been found to exhibit novel crystals made out of skyrmions, which are magnetic textures whose swirling pattern of spins has nontrivial topological properties. We show that skyrmion crystals have a much larger domain of stability in two-dimensional magnetic systems, such as thin films, surfaces or interfaces, compared with three-dimensional bulk materials. Our theoretical analysis shows that the spin-orbit coupling present in such systems naturally leads to magnetic interactions that stabilize skyrmions.

Skyrmions are of fundamental interest in diverse branches of physics, and their realization in magnetic materials has the promise of new functionalities and applications. Despite the explosion of research into skyrmion crystals, skyrmion phases have been found to occur only in a very narrow range of temperature-field phase parameters for three-dimensional bulk materials. We use numerical and analytical approaches to demonstrate that skyrmion crystals are much more stable in two-dimensional magnetic systems. Two-dimensional systems grown on a substrate necessarily break inversion symmetry, and our theoretical analysis shows that the spin-orbit coupling present in such systems naturally leads to magnetic interactions that stabilize skyrmions. Our results are not limited to monolayer materials and are also applicable to thin films.

Our theory gives a clear direction for experimental studies of two-dimensional magnetic materials to realize skyrmion crystals over a large range of magnetic fields down to zero temperature. The results demonstrated here have potential applications in spintronics and memory technologies.

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Vol. 4, Iss. 3 — July - September 2014

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Figure 1
Skyrmion and $T = 0$ phase diagram. (a) A skyrmion configuration. (b) The anisotropy-field phase diagram with ferromagnetic (FM), spiral, and skyrmion crystal (SkX) phases for $D / J = 0.01$ and $A_{c} J / D^{2} = 1 / 2$ , with $A = A_{c} + A_{s}$ . Double lines denote first-order transitions, while the single line is an unusual first-order transition with a divergent length scale; see text. The dashed line $H = 2 A$ separates the out-of-plane FM from the tilted FM. For the FM, $m^{z}$ is shown in the color bar. Results are obtained from a circular-cell variational calculation.
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Figure 2
Internal structure of skyrmions. Skyrmion core structure from circular-cell calculation with $D / J = 0.01$ and $A_{c} J / D^{2} = 1 / 2$ . Here, $L_{D} = (J / D) a$ , where $a$ is the microscopic lattice spacing. (a, b) False color plots of $m^{z}$ (shown in the color bar). (c, d) Angle-averaged topological charge density $| 2 π r χ (r) |$ and $m^{z} (r)$ (right axes). Panels (a) and (c) correspond to easy-axis anisotropy $A J / D^{2} = - 0.5$ and $H J / D^{2} = 0.28$ . The skyrmion core is conventional, with a single peak in the topological charge density. Panels (b) and (d) are for easy-plane anisotropy $A J / D^{2} = 1.35$ and $H J / D^{2} = 1.96$ . Here, the core has a large “transition” region (yellow-orange) from down (center) to up (boundary) in $m$ , leading to an unusual two-peak structure for $| 2 π r χ |$ .
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Figure 3
$T = 0$ anisotropy-field phase diagram from a linear ansatz and 2D square-cell calculation. The phase diagram shown here is obtained as a result of a full 2D variational calculation, distinct from the effectively 1D variational calculation shown in Fig. 1. The symbols and parameters used are exactly the same as described in the caption for Fig. 1. Note that the 2D square-cell calculation and the 1D variational calculation, although quite different in their computational complexity, nevertheless lead to essentially identical results for the overall phase diagram. The dotted boundaries shown here are obtained from the simplest “linear” variational ansatz for SkX described in text.
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Figure 4
Internal structure of the skyrmion from 2D square-cell calculation. We show the skyrmion core structure with $D / J = 0.01$ and $A_{c} J / D^{2} = 1 / 2$ obtained from a full 2D variational calculation, which should be compared with circular-cell results shown in Figs. 2 and 2. False color plots of $m^{z}$ are shown in the color bar. (c, d) Angle-averaged topological charge density $| 2 π r χ (r) |$ and $m^{z} (r)$ (right axes). Panels (a) and (c) correspond to easy-axis anisotropy $A J / D^{2} = - 0.5$ and $H J / D^{2} = 0.3$ . Panels (b) and (d) are for easy-plane anisotropy $A J / D^{2} = 1.2$ and $H J / D^{2} = 1.1$ . Note that the parameters used here are slightly different from those used in Fig. 2; however, the nontrivial structure of the skyrmion core in the easy-axis case is qualitatively similar to that in the circular-cell calculations.
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Figure 5
Skyrmion length scales. We show plots of the $H$ dependence of the optimal skyrmion cell radius $R^{*}$ and the core radii defined by the location of the maxima of $| d m^{z} / d r |$ . For the ansatz of Eq. (5) in the text, $d m^{z} / d r = | 2 π r χ (r) |$ . (a) In the easy-axis region, both $R^{*}$ and the core radius $R_{c}$ are finite at the spiral-to-SkX phase boundary, but $R^{*}$ diverges while $R_{c}$ remains finite at the SkX-to-FM transition. The vertical dashed lines indicate phase transitions from the SkX to the spiral state (at small $H$ ) and to the FM (at large $H$ ). (b) In the easy-plane region, there are two core radii corresponding to the two maxima in $| 2 π r χ (r) |$ . These inner and outer core radii $R_{c 1}$ and $R_{c 2}$ , and the cell radius $R^{*}$ all remain finite at the two phase transitions out of the SkX phase. Here, the vertical dashed lines indicate SkX-FM transitions.
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