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Symmetry-Protected Zero Modes in Metamaterials Based on Topological Spin Texture

Z.-X. Li, Yunshan Cao, X. R. Wang, and Peng Yan
Phys. Rev. Applied 13, 064058 – Published 24 June 2020
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  • INTRODUCTION
  • MODEL, METHOD, AND BULK PROPERTIES
  • CORNER STATES
  • DISCUSSION AND CONCLUSION
  • ACKNOWLEDGMENTS
  • APPENDICES
    • References

    Abstract

    We predict the emergence of second-order topological insulators in the dynamics of spin-texture-based metamaterials. We propose that the quantized Chern number and the Z6 Berry phase can completely characterize the nontrivial topology. By studying the collective gyration of magnetic vortices in a breathing honeycomb lattice, we derive the full phase diagram and show that the topological “zero-energy” corner mode is protected by a generalized chiral symmetry in the sexpartite lattice, leading to particular robustness against disorder and defects. Interestingly, we observe corner states at either obtuse-angled or acute-angled corners, depending on whether the lattice boundary has an armchair or a zigzag shape. Full micromagnetic simulations confirm the theoretical predictions with good agreement. Our findings open up a promising route to realizing higher-order symmetry-protected corner states in magnetic systems and to finally achieving topological spintronic memories, imaging, and computing.

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    • Received 9 October 2019
    • Revised 8 December 2019
    • Accepted 20 May 2020

    DOI:https://doi.org/10.1103/PhysRevApplied.13.064058

    © 2020 American Physical Society

    Physics Subject Headings (PhySH)

    1. Research Areas
    Magnetic textureMagnetic vorticesSpintronicsTopological phases of matter
    1. Physical Systems
    Magnonic crystalsMetamaterialsNanodisks
    1. Techniques
    Landau-Lifschitz-Gilbert equation
    Condensed Matter, Materials & Applied Physics

    Authors & Affiliations

    Z.-X. Li1, Yunshan Cao1, X. R. Wang2,3, and Peng Yan1,*

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

    • *yan@uestc.edu.cn

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    Vol. 13, Iss. 6 — June 2020

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    Images

    • Figure 1
      Figure 1

      Illustration (top view) of a breathing honeycomb lattice of magnetic vortices, with d1 and d2 representing the alternating lengths of intercellular and intracellular bonds, respectively. The radius of each nanodisk is r=50nm, and the thickness is w=10nm. The dashed black rectangle is the unit cell used for calculating the band structure, with a1 and a2 denoting the basis vectors.

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    • Figure 2
      Figure 2

      (a) First Brillouin zone of the breathing honeycomb lattice, with high-symmetry points Γ, G, M, and K located at (kx,ky)=(0,0), (2π/3a,23π/3a), (π/a,3π/3a), and (4π/3a,0), respectively. (b)–(d) Band structures along the loop Γ-M-K-Γ for different lattice parameters: d1=3.6r,d2=2.08r (b); d1=d2=3.6r (c); and d1=2.08r,d2=3.6r (d).

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    • Figure 3
      Figure 3

      (a) Dependence of the topological invariants of the Chern number and Z6 Berry phase on the ratio d2/d1 when d1 is fixed at 2.5r. (b) Schematic illustration of a parallelogram-shaped vortex lattice with armchair edges. (c) Eigenfrequencies of collective vortex gyration for different ratios d2/d1, with the red segment denoting the corner-state phase. (d) Phase diagram of the system, with five-pointed stars of different colors representing three typical pairs of parameters d1 and d2 for the different phases considered in the subsequent calculations and analyses.

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    • Figure 4
      Figure 4

      (a) Nanoribbon with armchair edges (closed boundaries in the x direction and open boundaries in the y direction). The dashed black rectangle is the unit cell. (b)–(d) Band dispersions for different geometric parameters as shown in Fig. 3: d1=3.6r,d2=2.08r (b); d1=d2=3.6r (c); and d1=2.08r,d2=3.6r (d). The dashed blue frame in (d) indicates the band of nonchiral edge states. D is the width of the nanoribbon.

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    • Figure 5
      Figure 5

      (a) Eigenfrequencies of a finite system with parameters d1=2.08r and d2=3.6r for a parallelogram-shaped structure [see Fig. 3]. (b)–(d) Spatial distribution of vortex gyrations for the bulk (b), corner [(c),(d), and (f)], and edge (e) states at five representative frequencies.

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    • Figure 6
      Figure 6

      (a) Temporal Fourier spectra of vortex oscillations at different positions, marked by dashed black rectangles in Fig. 3. (b)–(e) Gyration paths of all vortices under excitation fields at different frequencies: 872 (b), 934 (c), 944 (d), and 948 MHz (e). Since the oscillation amplitudes of the vortex centers are too small, we magnify them by 2, 5, or 10 times, as labeled in each part of the figure.

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    • Figure 7
      Figure 7

      (a) Eigenfrequencies of a parallelogram-shaped vortex lattice with armchair edges in the absence (black balls) and presence (red balls) of defects. The blue circle indicates the topologically stable corner state 3 (type III), with the inset plotting the spatial distribution of the vortex gyrations. (b) Eigenfrequencies of a parallelogram-shaped vortex lattice for different disorder strengths. In the calculations, we set the alternating bond lengths to d1=2.08r and d2=3.6r.

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    • Figure 8
      Figure 8

      Configurations of breathing honeycomb lattices of magnetic vortices with (a) armchair and (d) zigzag edges. (b) and (e) [(c) and (f)] show the configurations corresponding to (a) and (d), respectively, in the zero-correlation-length limit d2→∞ (d1→∞), which consist of isolated dimers (hexamers). Green and black balls indicate eigenvalues +1 and −1, respectively, of the chiral-symmetry operator. The shaded areas represent the unit cell at different positions.

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    • Figure 9
      Figure 9

      (a) Schematic illustration of a parallelogram-shaped vortex lattice with zigzag edges. (b) Eigenfrequencies for the parameters d1=2.08r and d2=3.6r. (c)–(f) Spatial distributions of vortex gyrations for the bulk (c) and corner (d)–(f) states at four representative frequencies.

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    • Figure 10
      Figure 10

      (a) Eigenfrequencies of a parallelogram-shaped vortex lattice with zigzag edges in the absence (black balls) and presence (red balls) of defects. The blue circle indicates corner state 6 (type III), with the inset plotting the corresponding spatial distribution of vortex gyrations. (b) Eigenfrequencies of the system for different disorder strengths. In the calculations, we use the same parameters as in Fig. 7.

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