Topological Transition in Spiral Elastic Valley Metamaterials

Elastic valley metamaterials offer an excellent platform to manipulate elastic waves and have potential applications in energy harvesting and elastography. Here we introduce a series of strategies to realize a topological transition in spiral elastic valley metamaterials by parameter modulations. We show the evolution of the Berry curvature and valley Chern number as a function of inherent parameters of a spiral, which further results in a general scheme to achieve topological valley edge states. Our strategy leverages multiple degrees of freedom in spiral elastic valley metamaterials to provide enhanced opportunities for desired topological states.

Figure 1

Schematics of spiral elastic valley metamaterials. The scatterer (Archimedean-spiral structure) is shown in purple and the soft matrix is shown in beige. Modulations of chirality, rotation angle, number of turns, and thickness of the scatterer are illustrated on the right, at the top, on the left, and at the bottom, respectively.

Figure 2

(a) Band evolution as a function of rotation angle. The blue and red lines represent the maximum frequency of the second band and the minimum frequency of the third band, respectively, as the spiral rotates. The green line indicates the valley Chern number as a function of rotation angle. (b) Band structure of (right-handed, 22°, 2, 2) elastic metamaterial, which is marked by yellow diamonds in (a). (c) The corresponding Berry curvature. Black crosses indicate the corners of the Brillouin zone (K point). (d) Band structure of (right-handed, 38°, 2, 2) elastic metamaterial, which is marked by cyan diamonds in (a). (e) The corresponding Berry curvature. Black crosses indicate the corners of the Brillouin zone (K point). The area for the calculation of the Berry curvature is shown in (b),(d) in black lines, which spans k_x from $8\pi /15c$ to $4\pi /5c$ and k_y from $\pi /(2\sqrt{3}c)$ to $5\pi /(6\sqrt{3}c)$.

Figure 3

(a),(b) Evolution of the valley Chern number for elastic metamaterials for different chiralities. (a) Valley Chern numbers for (right-handed, $\theta$, 1.5, 2) and (left-handed, $\theta$, 1.5, 2) for rotation angle $\theta$ varying from 0° to 360° shown as green filled circles and green unfilled circles, respectively. (b) Valley Chern numbers for (right-handed, $\theta$, 2, 2) and (left-handed, $\theta$, 2, 2) shown as magenta filled triangles and magenta unfilled triangles, respectively. (c) The left panel shows the Berry curvature of the third band and the second band of (left-handed, 60°, 1.5, 2) metamaterial, while the right panel shows the Berry curvature of (right-handed, 60°, 1.5, 2) metamaterial.

Figure 4

Evolution of valley Chern number for elastic metamaterials for different thicknesses. Valley Chern numbers for (right-handed, 60°, 1.5, d), (left-handed, 60°, 1.5, d), (right-handed, 60°, 2, d), and (left-handed, 60°, 2, d) metamaterials for thickness d varying from 1 to 4 mm, shown as the solid red line, the dashed red line, the solid blue line, and the dashed blue line, respectively. The corresponding Berry curvatures of the third band are shown beside the lines. The corresponding configurations are also shown beside the lines.

Figure 5

(a) The projected band structure of the (right-handed, 60°, 1.5, 2 | left-handed, 0°, 2, 1) metamaterial. Topological interface states are shown as red and blue lines. Bulk modes are shown in gray. Four markers are placed at k = 0.6 and k = 0.7, respectively. (b) The displacement fields corresponding to markers in (a). The arrows indicate the position of the interface. Enlarged displacement fields near the interfaces are shown beside the arrows. (c) Propagation of elastic waves along the interface. The directions of topological edge states projected by different valleys are marked by the yellow and cyan arrows. At a frequency of 143 Hz, which is shown by dashed green line in (a), elastic waves propagate along the interface and through the bend.

Figure 6

(a) Evolution of the second band at the K point as a function of rotation angle (blue line) and the evolution of the third band at the K point as a function of rotation angle (red line). The corresponding dotted lines show the pseudospin angular momentum in the unit cell as a function of the rotation angle. The dashed green line represents zero pseudospin angular momentum. (b),(c) Eigen-displacement-fields and pseudospin-angular-momentum-density distribution of the third bands of (right-handed, 22°, 2, 2) and (right-handed, 38°, 2, 2), respectively. (d),(e) Eigen-displacement-fields and pseudospin-angular-momentum-density distribution of the second bands of (right-handed, 22°, 2, 2) and (right-handed, 38°, 2, 2). The curved arrows indicate pseudospin up and pseudospin down.

Figure 7

Absolute values of valley Chern numbers for (right-handed, $\theta$, 1.5, 2) and (right-handed, $\theta$, 2, 2) shown as green filled circles and green unfilled circles, respectively (left panel), and absolute values of valley Chern numbers for (left-handed, $\theta$, 1.5, 2) and (left-handed, $\theta$, 2, 2) shown as magenta filled triangles and magenta unfilled triangles, respectively (right panel).

Figure 8

Variation of band-gap range as a function thickness d for (left-handed, 0°, 1.5, d) and (left-handed, 0°, 2, d).

Figure 9

Projected band structures calculated for the supercell (right-handed, 0°, 2, 1 | left-handed, 0°, 2, 1 | right-handed, 0°, 2, 1) (left panel) and for the supercell (right-handed, 60°, 2, 1 | left-handed, 0°, 2, 1 | right-handed, 60°, 2, 1) (right panel).