Acoustic Analogue of Graphene: Observation of Dirac Cones in Acoustic Surface Waves

We demonstrate the presence of Dirac cones in the dispersion relation of acoustic waves propagating on the surface of a plate of methyl methacrylate containing a honeycomb lattice of cylindrical boreholes. This structure represents the acoustic analogue of graphene, the cylindrical cavities playing the role of carbon atoms while acoustic surface waves are the equivalent of electronic waves in graphene. Analytical expressions for the Dirac frequency and Dirac velocity in acoustics are given as a function of the radius and depth of boreholes. These parameters have been experimentally determined for a constructed structure and the data are in fairly good agreement with the predicted values.

Figure 1

Schematic view of the acoustic graphene studied in this work. It consists of a honeycomb lattice of boreholes drilled in a rigid plate. $a_1$ and $a_2$ represent the primitive vectors of the hexagonal unit cell containing two acoustic cavities. The distance between nearest neighbors $d=|{\mathit{R}}_{\alpha \beta }|=a/\sqrt{3}$, where $a=|{\mathbf{a}}_1|=|{\mathbf{a}}_2|$.

Figure 2

Band structure of the acoustic surface waves propagating on a honeycomb lattice of boreholes drilled in a Plexiglas plate. The radius and depth of the boreholes are $R_0=0.26a$ and $L=0.5a$, respectively, where $a$ is the lattice constant of the underlaying hexagonal lattice (see Fig. 1). The upper right inset shows the Brillouin zone with the high symmetry points. The central inset shows a zoom of the dispersion relation around the Dirac point.

Figure 3

Dirac frequency (upper panel) and Dirac velocity (lower panel) as a function of boreholes depth $L$. Both graphs are obtained for a fixed boreholes radius of $R_0=0.26a$. The circles indicates the values at $L=0.5a$, which is the value used in the sample characterized.

Figure 4

Picture of the experimental setup. A loudspeaker at grazing incidence is employed to excite acoustic surface waves propagating on a Plexiglas plate with cylindrical boreholes distributed in a honeycomb lattice. Boreholes appear in the picture as white points due to refraction of light. The inset shows a schematic view of the boreholes geometry. Two microphones located at different points on the surface measure the phase delay between the propagating surface waves.

Figure 5

Dispersion relation of the acoustic surface waves guided in a Plexiglas plate with a honeycomb array of cylindrical perforations. The inset shows the phase delay between the two microphones employed in the experimental setup (see Fig. 4). Note that the origin of the horizontal axis actually corresponds to the $K$ point of the Brillouin zone.