Tuning of Higher-Order Topological Corner States in a Honeycomb Elastic Plate

Different from the traditional bulk-edge correspondence principle, the discovery of higher-order topological states has generated widespread interest. In a second-order, two-dimensional elastic wave topological insulator, the fluctuation information can be confined to the corners, with the state being topologically protected. To better apply the topological corner states, a cleverly designed mechanical structure is used to achieve the adjustment of the topological corner states on a two-dimensional elastic plate. The energy-band inversion can be achieved by switching the material of the inner and outer brackets of the unit cell of the honeycomb elastic plate. The topological corner states can be observed at the boundary of two different topological phase structures in the finite lattice using finite-element software. The robustness of the topological corner state is verified by setting up defective control groups at the corner. In addition, the topological corner states of this honeycomb elastic plate are discussed in terms of positional tunability. The honeycomb elastic plate can provide a reference for the design of devices for the local control of elastic waves and energy harvesting due to the possibility of modulation of topological corner states, which is beneficial for the application of topological corner states in practice.

Figure 1

(a) Schematic diagram of the unit-cell structure of the honeycomb elastic plate. The yellow honeycomb support is used as an elastic substrate. Steel or aluminum brackets are attached to the top and bottom of the substrate, where the purple sections represent the intracellular brackets and the blue sections represent the extracellular brackets. The green cylinders are magnetic cylinder oscillators. (b) The composite cell structure of the honeycomb elastic plate.

Figure 2

(a) The energy-band structure of the unit cell with the external brackets of steel and the internal brackets of aluminum and the eigenmodes of the $d$ and $p$ states at the top and bottom of the band gap; (b) the energy-band structure of the unit cell when both the inner and outer brackets are aluminum; (c) the energy band structure of the unit cell with the external brackets of aluminum and the internal brackets of steel and the eigenmodes of the $p$ and $d$ states at the top and bottom of the band gap.

Figure 3

(a) The supercell energy-band structure; (b) schematic diagram of the supercell structure and pseudospin boundary-state modes, with the horizontal color bar indicating the magnitude of the absolute value of the out-of-plane displacement; (c) out-of-plane displacement phase-field clouds and mechanical energy-flow distributions at the supercell interface in the pseudospin boundary states, the color bar indicates the magnitude of the displacement phase in the $z$ direction and the yellow arrows indicate the distributions of mechanical energy flows.

Figure 4

Characteristic spectrum analysis of the honeycomb elastic plate finite lattice. (a) Schematic diagram of the rhombic finite lattice; (b) the numerically calculated eigenfrequency distribution, where red rhombuses represent topological corner states, grey circles, blue regular triangles, and green inverted triangle represent bulk, edge, and trivial corner states, respectively, and the light gray area indicates the distribution of the bulk band gap; (c)–(f) the vibration modes of the edge state (3556.9 Hz), the bulk state (3679.2 Hz), the topological corner state (3488.5 Hz), and the trivial corner state (3473.4 Hz) are shown, respectively, and the color bar indicates the magnitude of the absolute value of the displacement in the $z$ direction.

Figure 5

Analysis of the strong robustness of the topological corner states of the honeycomb elastic plate. (a)–(c) The no-defect finite-lattice model, the disturbance finite-lattice model, and the vacancy finite-lattice model, respectively, where the purple represents the steel brackets, the blue represents the aluminum brackets, and the nontrivial phase structure is outside the corners and the trivial phase structure is inside the corners; (d) normalized energy spectrum curves for no defect, disturbance, and vacancy finite-lattice models. The gray bar represents the bulk band-gap frequency distribution of the supercell.

Figure 6

Tunability study of the topological corner states of the honeycomb elastic plate. (a),(b) Schematic diagrams of the inward and outward shifts of the topological corner states of the finite lattice ($13\times 13$ unit cells), respectively; (c),(d) the vibration modes of the inward and outward shifts of the topological corner states are shown, respectively.

Figure 7

Topological eigenmode characterization.

Figure 8

Topological corner modes at $\pi /3$ and $2\pi /3$ in the rhombic finite lattice. The blue and red represent the chiral charge symbols of $+$ and $-$, respectively.