Topological Magnonics: A Paradigm for Spin-Wave Manipulation and Device Design

X. S. Wang, H. W. Zhang, and X. R. Wang

Phys. Rev. Applied 9, 024029 – Published 27 February 2018

Abstract

Conventional magnonic devices use magnetostatic waves whose properties are sensitive to device geometry and the details of magnetization structure, so the design and the scalability of the device or circuitry are difficult. We propose topological magnonics, in which topological exchange spin waves are used as information carriers, that do not suffer from conventional problems of magnonic devices with additional nice features of nanoscale wavelength and high frequency. We show that a perpendicularly magnetized ferromagnet on a honeycomb lattice is generically a topological magnetic material in the sense that topologically protected chiral edge spin waves exist in the band gap as long as a spin-orbit-induced nearest-neighbor pseudodipolar interaction (and/or a next-nearest-neighbor Dzyaloshinskii-Moriya interaction) is present. The edge spin waves propagate unidirectionally along sample edges and domain walls regardless of the system geometry and defects. As a proof of concept, spin-wave diodes, spin-wave beam splitters, and spin-wave interferometers are designed by using sample edges and domain walls to manipulate the propagation of topologically protected chiral spin waves. Since magnetic domain walls can be controlled by magnetic fields or electric current or fields, one can essentially draw, erase, and redraw different spin-wave devices and circuitry on the same magnetic plate so that the proposed devices are reconfigurable and tunable. The topological magnonics opens up an alternative direction towards a robust, reconfigurable and scalable spin-wave circuitry.

Received 7 June 2017
Revised 29 June 2017

DOI:https://doi.org/10.1103/PhysRevApplied.9.024029

Physics Subject Headings (PhySH)

Domain walls Edge states Magnons Spin waves Spintronics Topological phases of matter

Devices

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

X. S. Wang^1,2, H. W. Zhang¹, and X. R. Wang^2,3,*

¹School of Electronic Science and Engineering and State Key Laboratory of Electronic Thin Film and Integrated Devices, University of Electronic Science and Technology of China, Chengdu, Sichuan 610054, China
²Department of Physics, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon 999077, Hong Kong
³HKUST Shenzhen Research Institute, Shenzhen 518057, China

^*Corresponding author. phxwan@ust.hk

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Vol. 9, Iss. 2 — February 2018

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Images

Figure 1
(a) Schematic illustration for the TESW states (left panels) and the domain-wall SWBS (right panels). The red and cyan regions denote domains in which spins point to the $+ z$ and $- z$ directions, respectively. The yellow arrows denote the spin-wave propagation direction. (b) Various phases in the $K / J − F / J$ plane when $D = Δ = 0$ . The spin arrangements in these phases are shown in the insets. (c) Topological phase diagram in the $D / J − Δ / J$ plane (for $K = J$ and $F = 0.01 J$ ). The cyan (pink) region is a topologically nontrivial phase with the Chern number $C_{c} = - 1$ ( $C_{c} = + 1$ ) for the conduction band, and the TESWs propagate counterclockwise (clockwise) with respect to the magnetization direction, as illustrated in the insets. The white regions represent a topologically trivial phase with $C_{c} = 0$ .
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Figure 2
(a) The density plots of the spectral function at the strip edge or domain wall (indicated by the dashed boxes) for (upper panel) a zigzag strip and (lower panel) an armchair strip. An abrupt domain wall, which separates an upper domain of $m_{z} = 1$ from a lower domain of $m_{z} = - 1$ , sits in the middle of each strip. (b) (Middle panel) The distribution of wave-function amplitude on sublattice $A$ for the edge states of $ω = 12$ along the strip-width direction (the $y$ direction). The symbols are numerical results for a 100-wide strip, and the solid lines are the analytical results. The left (right) panel is the corresponding spatial distribution of the TESW eigenstate along the armchair (zigzag) edge. In the left and right panels, the symbol shape traces the spin-precession trajectories, and the size of the symbols denotes the amplitude of the TESW at each site. The azimuthal angles of spins on the lattice at $t = 0$ are encoded by the symbol colors with the color ring shown in the inset.
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Figure 3
(a) Snapshots of the spin waves injected at the site marked by the black arrows at $t = 5$ . The upper (lower) panel is for $D = 0$ ( $D = 0.4$ ). Only the parts between the fifth site (①) and the tenth site (②) at the top edge are shown. (b) Snapshots of a spin wave at $t = 15$ ( $t = 15.5$ ) after leaving a 12-long (upper panel) [15-long (lower panel)] domain wall. The spin wave is injected at the site marked by the black arrow in each sample. Only the portions near the domain wall are shown. In (a) [(b)], the pink (cyan) regions represent the $m_{z} = + 1$ ( $m_{z} = - 1$ ) domains. The radius of each circle is proportional to $\sqrt{m_{x}^{2} + m_{y}^{2}}$ , and the color encodes the azimuthal angles of the spins. (c) The domain-wall length dependence of spin-wave power division ratio. The symbols are simulation results, and the red dashed line is $η = 0$ (for a $1 ∶ 1$ splitting). (Inset) $η$ approaches 0 for a large $L_{y}$ value. (d) Spatial distribution of two topologically protected edge spin waves of $ω = 12$ inside a domain wall parallel to the armchair edges with wave number $k_{1} = - 0.0209$ (left panel) and $k_{2} = 0.821$ (right panel). (Insets) Spin precession is mirror symmetric (antisymmetric) as $m_{∥} \to m_{∥}$ and $m_{⊥} \to - m_{⊥}$ ( $m_{∥} \to - m_{∥}$ and $m_{⊥} \to m_{⊥}$ ) with respect to the domain wall central line for a $k_{1} = 0.0209$ ( $k_{2} = 0.821$ ) state. Here, $m_{∥}$ and $m_{⊥}$ are the magnetization components parallel and perpendicular to the domain wall. The meaning of the circle symbols is the same as that in Fig. 2. (e) The spatial period (the vertical axis) of the power division ratio from the LLG simulations verses the period of beat (the horizontal axis). The green line is $y = x$ . The model parameters for squares, circles, and triangles are, respectively, $F = 5 J$ and $K = 10 J$ , $F = 6 J$ and $K = 10 J$ , and $F = 5 J$ and $K = 9 J$ . $α$ is set to $10^{- 4}$ , and several different frequencies inside the gap are calculated for each set of parameters.
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Figure 4
A snapshot of a spin wave at $t = 65$ under a continuous microwave excitation of $ω = 12$ at the site marked by the inward arrow on the lower edge. The outward arrows denote the output signals. The device geometry, with a zigzag edge along the $x$ direction and armchair edges along the $y$ direction, is $L_{x} = 40 \sqrt{3}$ , $L_{y} = 63$ , $ℓ_{y} = 25.5$ , and $ℓ_{x} = 12 \sqrt{3}$ . $α = 1$ in the gray areas.
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Figure 5
(a) The spin-wave beam of $ω = 12$ , generated at the site marked by the inward arrow by a microwave field, is split into beams $I$ and $II$ by the domain wall $A B$ (the first SWBS). The two beams recombine in the domain wall $C D$ (the second SWBS). The spin-wave pattern at $t = 70$ is represented by the size and color of the symbols, which have the same meaning as in Fig. 3. The device geometry, with armchair edges along the $x$ direction and zigzag edges along the $y$ direction, is $L_{x} = 90$ and $L_{y} = 40 \sqrt{3}$ . The first domain wall, which is $12 \sqrt{3}$ long, is placed at $x = L_{x} / 2$ and splits the incoming spin wave evenly. An area of $30 \times 6 \sqrt{3}$ is removed from the center of the device so that the two split spin-wave beams can propagate along the internal boundary. The two beams recombine at the second domain wall, which is $12 \sqrt{3}$ long, at $X$ . The gray parts are absorbing areas with a large damping constant of $α = 1$ . (Inset) $X$ dependence of the power division ratio $η$ . (b) Schematic diagram of the interferometer in (a). (Inset) In the optical Mach-Zehnder interferometer, a light beam enters optical beam splitter ① and splits into two beams, $I$ and $II$ . The two beams recombine at the second optical beam splitter ②. The outputs ③ and ④ depend on the interference of two beams at beam splitter ②.
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