Topological Phononic Crystals with One-Way Elastic Edge Waves

We report a new type of phononic crystals with topologically nontrivial band gaps for both longitudinal and transverse polarizations, resulting in protected one-way elastic edge waves. In our design, gyroscopic inertial effects are used to break the time-reversal symmetry and realize the phononic analogue of the electronic quantum (anomalous) Hall effect. We investigate the response of both hexagonal and square gyroscopic lattices and observe bulk Chern numbers of 1 and 2, indicating that these structures support single and multimode edge elastic waves immune to backscattering. These robust one-way phononic waveguides could potentially lead to the design of a novel class of surface wave devices that are widely used in electronics, telecommunication, and acoustic imaging.

Figure 1

Ordinary and gyroscopic phononic crystals: (a) Schematic of the hexagonal lattice. The blue and grey spheres represent concentrated masses $m_1$ and $m_2=m_1$, respectively. The red and black straight rods represent massless linear springs with stiffness $k_1$ and $k_2=k_1/20$, respectively. The dashed cell is the primitive cell of the lattice. (b) Unit cell for the ordinary (nongyroscopic) phononic crystal. (c) Band structure of the ordinary (nongyroscopic) phononic crystal. The inset is the Brillouin zone. (d) Schematic of a gyroscope with the top tip pinned to a mass in the lattice. (e) Unit cell for the gyroscopic phononic crystal. (f) Band structure of the gyroscopic phononic crystal (${\alpha }_1={\alpha }_2=0.3m_1$) with the Chern numbers labeled on the bulk bands. The frequencies are normalized by ${\omega }_0=\sqrt{k_1/m_1}$.

Figure 2

Edge modes in gyroscopic phononic crystal: (a) 1D band structure showing bulk bands (black dots) and edge bands (colored lines). Red solid lines represent edge modes bound to the top boundary, while blue dashed lines represent edge modes bound to the bottom boundary. (b) Modal displacement fields of top edge states with negative group velocities. (c) Modal displacement fields of bottom edge states with positive group velocities.

Figure 3

Transient response of a gyroscopic phononic crystal consisting of $20\times 20$ unit cells with a line defect on the right boundary: Snapshots of the displacement field at (a) $t=2T_0$, (b) $t=12T_0$, (c) $t=22T_0$ and (d) $t=32T_0$, where $T_0=\sqrt{m_1/k_1}$ is the characteristic time scale of the system. Starting from $t=0$, a time-harmonic excitation force $\mathbf{F}(t)=[F_x(t),F_y(t)]=[1,1]F_0{e}^{-i\omega t}$ is prescribed at the site indicated by the red arrow.

Figure 4

Square lattice results: (a) Schematic of the square lattice. The blue and grey spheres represent concentrated masses $m_1$ and $m_2=m_1$, respectively. The black and red straight rods represent massless linear springs with stiffness $k_1$ and $k_2=2k_1$, respectively. (b) Band structure of the ordinary (nongyroscopic) phononic crystal. The insets are the Brillouin zone and the unit cell. (c) Band structure of the gyroscopic phononic crystal (${\alpha }_1={\alpha }_2=0.3m_1$) with the Chern numbers labeled on the bulk bands. Frequencies are normalized by ${\omega }_0=\sqrt{k_1/m_1}$. The inset is the unit cell. (d) 1D band structure showing bulk bands (black dots) and two topological edge bands (red solid lines) bound to the top boundary. Note that the edge bands that are bound to the bottom boundary are not shown here. (e) Modal displacement fields of the edge states shown in (d).