# Subwavelength Chiral Spiral Acoustic Metamaterials for a Robust Topological Acoustic Insulator
## 3 Results  
### Band Structure  
Employing the finite element software COMSOL Multiphysics to calculate the band structure of spiral metamaterials. Note that the whole metamaterial is a periodic structure, and the Floquet Bloch periodic conditions are imposed on the boundaries of the unit cell. Along the boundary of the first Brillouin zone, M-T-K-M scanning wave vector is adopted to obtain the energy band of the whole structure. The band structure is shown in Figure 2a, where \(f_{c}=c_{0}/a\). There exist three band gaps during the normalized frequency range of 0-0.6: the first band gap (0.117-0.141), the second band gap (0.354-0.371), and the third band gap (0.498-0.528). Among them, the first and the second band gaps are the local resonant band gap thanks to the lengthy spiral design, whose normalized frequency is less than 0.5. The third band gap is the Bragg scattering band gap owing to the symmetry-broken, which forms the band gap at the high symmetry point K. To better understand these three band gaps, the eigenmodes of the four energy bands corresponding to the K point in the Brillouin region are analyzed. These four eigenmodes are labeled as 1, 2, 3, and 4, respectively, and the corresponding normalized frequencies are 0.117 (994 Hz), 0.35 (2975 Hz), 0.496 (4218 Hz), and 0.535 (4545 Hz), respectively. Figure 2b shows the eigenmodes of each energy band. It suggests that the first and the second eigenmodes are excited by the local resonance effect since the resonance part is mainly distributed in the spiral structure. The third and the fourth eigenmodes are excited by the Bragg scattering effect since the resonance part is mainly distributed outside the spiral structure. The eigenmodes of the left-handed and right-handed structures corresponding to each band are mirror symmetric. It can be concluded that the spiral structure can effectively lower the whole system frequency, which possesses the subwavelength characteristics. Next, we investigated the effect of the geometric parameters on the band gaps. First, we rotated the spiral structure around the center of the unit cell, which can be characterized by \(\theta\). Figure 3a shows the variation of the first, second and third band gap widths with the rotation angle \(\theta\), marked by grey, blue, and orange areas, respectively. It can be derived from the figure that the rotation angle \(\theta\) hardly affects the first and second band gaps, but has a certain effect on the third band gap. This is because the first and second band gaps are formed by the local resonance effect of the spiral structure which is bound to the length of the spiral, and the change of the rotation angle hardly affects their eigenfrequency. The third band gap is formed by the Bragg scattering effect. Therefore, the variation of rotation angle will change the unit cell symmetry, and the eigenfrequency will be influenced by the rotation angle. It can be seen that when the rotation angle is 70\({}^{\circ}\), the third band gap is the narrowest. Figure 1: (a) Schematic of an Archimedean spiral with its geometric parameters. (b) The left-handed spiral distributed in honeycomb unit cell. To explore the effect of width of spiral \(w\) on the band gaps, the rotation angle is set to \(\theta=0^{\circ}\). Figure 3b shows the variation of the first, second and third band gap widths with the width of spiral \(w\) ranging from 0.05 cm to 0.5 cm, marked by grey, blue, and orange areas, respectively. From the diagram, with the increase of the width of the spiral, the frequency of the first, second and third band gap decreases gradually, indicating that the local resonance effect is enhanced. Furthermore, the width of the first and second band gaps decreases with the increase of the spiral width, and the width of the third band gap is less affected by the spiral width. Then, the number of turns of the spiral \(n\) is changed to explore its effect on the energy band.