Observation of zone folding induced acoustic topological insulators and the role of spin-mixing defects

This article reports on the experimental realization of a flow-free, pseudospin-based acoustic topological insulator designed using the strategy of zone folding. Robust sound one-way propagation is demonstrated with the presence of non-spin-mixing defects. On the other hand, it is shown that spin-mixing defects, which break the geometric symmetry and therefore the pseudo-time-reversal symmetry, can open up nontrivial band gaps within the edge state frequency band, and their width can be tailored by the extent of the defect. This provides a possible route for realizing tunable acoustic topological insulators.

Figure 1

(a) Structure of the phononic crystal. The rods are periodically arranged with a lattice constant $a_0$. The conventional unit cell of the triangular lattice is marked by the blue line whereas the expanded unit cell of the triangular lattice is marked by the orange line. (b) Top view of the phononic crystal. $r$ is the radius of the center “atom” whereas $r+\mathrm{\Delta }r$ is the modified radius for breaking the translational symmetry. (c) The conventional Brillouin zone and new Brillouin zone correspond to the two different unit cells. The “zone folding” mechanism shows how high symmetry points $K$ and $K^{\prime }$ are mapped to $\mathrm{\Gamma }$ point in the new Brillouin zone.

Figure 2

Band structures for various unit cells. Unit cells are depicted at the bottom left corner in each subfigure. The rods are assumed to be steel and the background medium is air. (a) Original unit cell with a single atom. The single Dirac cone at $K$ is marked by the red dot. (b) Expanded unit cell with double Dirac cones at $\mathrm{\Gamma }$. (c), (d) Symmetry-broken unit cells with $\mathrm{\Delta }r=1$ mm and $\mathrm{\Delta }r=-1$ mm, respectively. Topological band gaps are marked with pink color. $p/d$ eigenmodes are shown with their locations separated by the band gap which indicates the occurrence of band inversion.

Figure 3

Band structures of supercells. (a) Band structure of the supercell with $\mathrm{\Delta }r=1.0\mathrm{mm}$. (b) Band structure of the supercell with $\mathrm{\Delta }r=-1.0\mathrm{mm}$. (c) Band structures when the two supercells are joined together. One pair of edge states at $f_0=17800$ Hz (arbitrarily chosen within the edge state frequency band) are shown on the right. The inset figure shows a zoom-in of the minigap.

Figure 4

(a), (b) Simulated transmission spectra with two classes of defects in topological acoustic channels and an ordinary waveguide, respectively. Black curves represent the no defect case, red curves represent the cavity case, and blue curves represent the bending case. (c), (d) Experimental results of the transmission spectra corresponding to (a) and (b), respectively. The topologically protected edge state frequency range is indicated by the light region.

Figure 5

(a), (b) Acoustic pressure (left) and intensity (right) distributions in the topological insulator with a cavity and a V shape bending. (c), (d) Acoustic pressure and intensity distributions in the ordinary waveguide with the same defects. Considerably lower sound transmissions are observed in the ordinary phononic crystal waveguides. Results are shown at 18 400 Hz.

Figure 6

(a) Simulated band structure of the combined supercell with the cut $d=4.0$ mm. (b) Simulated band structure of the combined supercell with the cut $d=3.5$ mm. (c) Simulated band structure of the combined supercell with the cut $d=3.0$ mm. (d) Supercell figure with the cut $d=3.0$ mm.

Figure 7

(a) Simulated transmission spectra of the straight waveguides with different cuts. The black curve represents the case without a cut. The green, red and blue curves represent the cases where $d$ equals 3.0, 3.5, and $4.0\mathrm{mm}$, respectively. The cut is shown in the inset figure with $d=3.0\mathrm{mm}$. (b) The experimental results for the waveguide without a cut (black) and with $d=3.0\mathrm{mm}$ (green).

Figure 8

(a) Transmission spectra of the topological waveguide with a cavity and cut. The black curve represents the case without the cut. Blue, red, and green curves represent the cases with $d=4.0$, 3.5, and 3.0 mm, respectively. (b) Acoustic pressure and intensity distributions of the topologically protected waveguide with a cavity and no cut. (c) Acoustic pressure and intensity distributions of the case with the same cavity and a cut of $d=4.0$ mm. Results are simulated at the frequency of 18 400 Hz, chosen to be within the band-gap frequency created by the spin-mixing defect.

Figure 9

(a)–(d) Band structures of unit cells with filling ratio 0.25, 0.3, 0.35, and 0.4. Top views of the unit cells are located at the bottom left of each subfigure.

Figure 10

(a)–(c) Band structures of the expanded unit cells with $\mathrm{\Delta }r$ from 0.5 to 1.5 mm. (d)–(f) Band structures of the expanded unit cells with $\mathrm{\Delta }r$ from $-0.5$ to $-1.5$ mm. The band gaps are marked with pink color.

Figure 11

(a), (b) Acoustic topological insulators structure with defects of cavity and bending. (c), (d) Ordinary waveguide structure with defects of cavity and bending. (e) The experimental system for testing the sound transmission.