Second-order topological solitonic insulator in a breathing square lattice of magnetic vortices

Z.-X. Li, Yunshan Cao, X. R. Wang, and Peng Yan

Phys. Rev. B 101, 184404 – Published 4 May 2020

Abstract

We study the topological phase in a dipolar-coupled two-dimensional breathing square lattice of magnetic vortices. By evaluating the quantized Chern number and $Z_{4}$ Berry phase, we obtain the phase diagram and identify that the second-order topological corner states appear only when the ratio of alternating bond lengths goes beyond a critical value. Interestingly, we uncover three corner states at different frequencies ranging from subgigahertz to tens of gigahertz by solving the generalized Thiele's equation, which has no counterpart in natural materials. We show that the emerging corner states are topologically protected by a generalized chiral symmetry of the quadripartite lattice, leading to particular robustness against disorder and defects. Full micromagnetic simulations confirm theoretical predictions with great agreement. A vortex-based display device is designed as a demonstration of the real-world application of the second-order magnetic topological insulator. Our findings provide a route for realizing symmetry-protected multiband corner states that are promising to achieve spintronic higher-order topological devices.

Received 15 January 2020
Accepted 16 April 2020

DOI:https://doi.org/10.1103/PhysRevB.101.184404

Physics Subject Headings (PhySH)

Edge states Magnetic vortices Phase diagrams Topological insulators

Chiral symmetry

Micromagnetic modeling

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Z.-X. Li¹, Yunshan Cao¹, X. R. Wang^2,3, and Peng Yan ^1,*

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

^*Corresponding author: yan@uestc.edu.cn

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Issue

Vol. 101, Iss. 18 — 1 May 2020

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Images

Figure 1
(a) Illustration of the breathing square lattice of magnetic vortices, with $d_{1}$ and $d_{2}$ denoting the alternating lengths of intercellular and intracellular bonds, respectively. The dashed black rectangle is the unit cell for calculating the band structure and topological invariants, with $a_{1}$ and $a_{2}$ denoting the basis vectors. The inset shows the micromagnetic structure of the vortex, with thickness $w = 10$ nm and radius $r = 50$ nm. (b) The first Brillouin zone, with the high-symmetry points $Γ, K$ , and $M$ locating at $(k_{x}, k_{y}) = (0, 0), (\frac{π}{a}, \frac{π}{a})$ , and $(\frac{π}{a}, 0)$ , respectively. The (lowest-four) band structures along the path $Γ - K - M - Γ$ for different geometric parameters: $d_{1} = 3.6 r, d_{2} = 2.4 r$ (c), $d_{1} = d_{2} = 3.6 r$ (d), and $d_{1} = 2.08 r, d_{2} = 3.6 r$ (e). (f) Dependence of Chern number and $Z_{4}$ Berry phase on the ratio $d_{2} / d_{1}$ with $d_{1}$ being fixed to $3 r$ .
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Figure 2
(a) Eigenfrequencies of collective vortex gyrations under different ratios $d_{2} / d_{1}$ in a finite lattice of the size $(3 d_{1} + 4 d_{2}) \times (3 d_{1} + 4 d_{2})$ . We set $d_{1}$ to $2.08 r$ in the calculation. The schematic plot of the vortex lattice in (a) for two limit cases, $d_{2} \to \infty$ (b) and $d_{1} \to \infty$ (c).
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Figure 3
(a) Eigenfrequencies of the square-shaped vortex lattice with $d_{1} = 2.08 r$ and $d_{2} = 3.6 r$ . The spatial distribution of vortex gyrations for the corner (b), edge (c), and bulk (d) states with the frequency 0.939, 0.986, and 1.047 GHz, respectively.
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Figure 4
(a) Eigenfrequencies of the square-shaped vortex lattice in the absence of deformations (black balls) and in the presence of deformations (red balls). The blue circle indicates the topologically stable corner state with the inset showing the corresponding spatial distribution of vortex gyrations in pristine lattice. (b) Eigenfrequencies of the square-shaped vortex lattice under different disorder strengths.
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Figure 5
(a) The temporal Fourier spectra of the vortex oscillations at different positions. (b1)–(d3) The spatial distribution of oscillation amplitude under the exciting field with different frequencies indicated in (a). Since the oscillation amplitudes of the vortex core are too small, we have magnified them by different times labeled in each figure.
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Figure 6
(a) The schematic plot of the vortex lattice for “H” display. (b) Micromagnetic simulation of vortex gyrations with frequency $f = 0.939$ GHz. The oscillation amplitudes of the vortex core have been magnified by three times.
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