Tunable Topological Phononic Crystals

Topological insulators first observed in electronic systems have inspired many analogues in photonic and phononic crystals in which remarkable one-way propagation edge states are supported by topologically nontrivial band gaps. Such band gaps can be achieved by breaking the time-reversal symmetry to lift the degeneracy associated with Dirac cones at the corners of the Brillouin zone. Here, we report on our construction of a phononic crystal exhibiting a Dirac-like cone in the Brillouin zone center. We demonstrate that simultaneously breaking the time-reversal symmetry and altering the geometric size of the unit cell result in a topological transition that we verify by the Chern number calculation and edge-mode analysis. We develop a complete model based on the tight binding to uncover the physical mechanisms of the topological transition. Both the model and numerical simulations show that the topology of the band gap is tunable by varying both the velocity field and the geometric size; such tunability may dramatically enrich the design and use of acoustic topological insulators.

Figure 1

(a) Schematics of a unit cell of our phononic crystal. The ring is connected by rectangular waveguides. The lower inset illustrates the reciprocal lattice. (b)–(d) Band structure of the phononic crystal whose rectangular waveguide has a width of $d_0=0.07336\mathrm{m}$, $d_1=0.04\mathrm{m}$, and $d_2=0.1\mathrm{m}$ without an airflow (shown in blue curves) and with a circulating airflow at $v=10\mathrm{m}/\mathrm{s}$ (shown in red curves). When airflow is introduced, the Chern numbers are marked on their corresponding bands under the band gap.

Figure 2

(a) Pressure-field distributions of three Bloch eigenstates at $\mathrm{\Gamma }$ point marked as ${\phi }_d$, ${\phi }_{px}$, and ${\phi }_{py}$. Dark red and dark blue indicate the positive and negative maxima. (b) The eigenfrequency of the eigenstates varies as functions of $d$ when no airflow is introduced. The black and red curves correspond to the eigenfrequencies of ${\phi }_d$ and ${\phi }_{px}$ (${\phi }_{py}$) eigenstates, respectively. Purple area indicates the system has a complete band gap. (c) The eigenfrequency of ${\phi }_d$, ${\phi }_{p_{+}}$, and ${\phi }_{p_{-}}$ versus the velocity field of the induced airflow. (d) The same as (b) but an airflow with $v=10\mathrm{m}/\mathrm{s}$ is introduced. The purple and blue areas indicate the regions with a trivial and nontrivial band gap, respectively.

Figure 3

Phase diagram of the topology property of the band gap with combined modulation of the width of the waveguide and the intensity of the airflow. Topological transition occurs across the solid line.

Figure 4

(a) Band structure of a $1\times 8$ supercell ($d=d_2$, $v=10\mathrm{m}/\mathrm{s}$). Dispersion relations of the edge states are shown in colored curves, and the bulk bands are plotted in gray curves. Red and blue curves represent different edge modes. (b) The pressure-field distribution of edge modes for $A$, $k=-0.1\pi /a$ and $B$, $k=0.1\pi /a$.

Figure 5

Demonstration of properties resulting from nontrivial band gaps by tuning the geometric size of the phononic crystal. The source indicated by a star is a point source at 139 Hz. (a) Topologically protected one-way propagation that is immune to defects (marked by a green circle) and without backscattering. Here, the width of the waveguide is $d=d_2$ and the velocity of the airflow is $v=10\mathrm{m}/\mathrm{s}$. (b) One-way interface state propagation on the interface between two different lattices. The width of the waveguide is $d=d_1$ in the upper half and $d=d_2$ in the lower half. The arrows in (a) and (b) indicate the directions of propagation of the edge or interface mode.

Figure 6

(a) The eigenfrequencies of the eigenstates vary as functions of $v$. Black and red curves correspond to the ${\phi }_d$ and ${\phi }_{p_{+}}$ modes, respectively. The purple and blue areas indicate the region of the band gap with a different topological invariant, where the intersection is the topological transition point. (b) The simulated pressure-field distribution excited by a point source in a phononic crystal with airflow ($v=5\mathrm{m}/\mathrm{s}$). (c) The same as (b), but the velocity field of the airflow is $v=15\mathrm{m}/\mathrm{s}$. In both cases, the rectangular waveguide in the phononic crystal has a width of $d=0.065\mathrm{m}$. The source has a frequency of $f=138.4\mathrm{Hz}$ and is marked by a star. The arrow indicates the direction of propagation of the edge mode.