Valley, as a new degree of freedom, raises the valleytronics in fundamental and applied science. The elastic analogs of valley states have been proposed by mimicking the symmetrical structure of either two-dimensional materials or photonic valley crystals. However, the asymmetrical valley construction remains unfulfilled. Here, we present the valley anisotropy by introducing asymmetrical design into elastic metamaterials. The elastic valley metamaterials are composed of bio-inspired hard spirals and soft materials. The anisotropic topological nature of valley is verified by asymmetrical distribution of the Berry curvature. We show the high tunability of the Berry curvature both in magnitude and sign enabled by our anisotropic valley metamaterials. Finally, we demonstrate the creation of valley topological insulators and show topologically protected propagation of transverse elastic waves relying on operating frequency. The proposed topological properties of elastic valley metamaterials pave the way to better understanding the valley topology and to creating a new type of topological insulators enabled by an additional valley degree of freedom.
IntroductionElastic waves, possessing plenty of degree of freedoms (DOFs) including frequency, phase and polarization, have demonstrated tremendous promise in a variety of applications including target detection, information processing and biomedical imaging 1-4 . Recently, topology has been proposed as a new DOF in manipulating waves in both photonic and phononic systems, exhibiting remarkable impact not only on fundamental science such as condensed matter physics, but also on engineering applications such as low loss devices and waveguides 5-9 . In photonics, the photonic spin Hall effect has been achieved by taking advantage of spin DOF, which opens up an avenue of spin-dependent light transport and one-way spin transport 10-13 . In phononics, mechanical patterns and deformation have been employed as a new DOF to enable the elastic topological states 8,9,[14][15][16][17][18][19] .Recently, valley, the degenerate yet inequivalent energy extrema in momentum space, has emerged as a new dimension in manipulating waves in electronics, photonics and phononics 5,6,[20][21][22][23][24]
. In graphene and transition metal dichalcogenides, valley-selective circular dichroism and valleyHall effect due to the long lifetime of valley polarization and non-zero Berry curvature have been studied for the promising applications in information carrier and storage 20,21,23,24 . As the concept of valley is introduced into the classic system, the photonic and phononic valley crystals are proposed, showing potential applications such as information processing via valley-dependent transportation 5,6,22 . However, existing designs of valley metamaterials are limited to the inherent spatial inversion symmetry of the physical system, where the typical Berry curvature distribution in the Brillouin zone follows Ω −# = Ω # . The valley metamaterials without spatial inversion symmetry have not yet bee...