Topological Surface Acoustic Waves

Topological materials for classical waves, e.g., electromagnetic and acoustic waves, have attracted growing interest, mainly due to the robustness, low loss, and new artificial degree of freedom conferred by their boundaries. Surface acoustic waves (SAWs), as widely used information carriers of microdevice relevance, are ubiquitous in today’s wireless communication and sensing networks. Herein, we report the implementation of a SAW topological insulator based on a monolithically integrated platform. By using a miniature acoustic resonator array working tens of megahertz on a piezoelectric half-space, we successfully endow electrically pumped Rayleigh-type SAWs with the “spin-momentum locking” feature, enabling solid-state acoustic waves on the “one-dimensional interface of the two-dimensional surface on the three-dimensional volume” detour arbitrarily and pass through defects and intersections with much smaller losses than those incurred with any other solutions. These revolutionary topological SAWs may open an avenue for monolithic electronic-(photonic)-phononic circuits with ultrahigh performance and advanced functionalities in, e.g., future mobile communicating, sensing, and quantum-information processing.

Figure 1

(a) Schematics of the $\mathrm{Pn}\mathrm{C}$ for SAWs used in this study, i.e., a honeycomb lattice composed of identical miniaturized resonator pillars on a ${\mathrm{Li}\mathrm{Nb}\mathrm{O}}_3$ half-space. (b) Enlarged view of each micropillar made of copper ($\mathrm{Cu}$) with an inverted truncated conical shape due to our fabrication process. (c) Top view of the unit cells of the SAW-$\mathrm{Pn}\mathrm{C}$: (I) original diamond cell with two micropillars and (II) threefold-larger hexagonal cell with six micropillars for the zone-folding implementation. Originally, the honeycomb lattice has a pillar-center distance R = a₀. (d), (e) SEM images of two elaborately fabricated SAW $\mathrm{Pn}\mathrm{Cs}$, one with a shrunken lattice (R = 0.91a₀) and another with an expanded lattice (R = 1.07 a₀), respectively. Insets are tilted views of the two samples, where the scale bars represent 10 µm. (f), (g) Calculated band structures of the two $\mathrm{Pn}\mathrm{Cs}$. The blue lines and dots represent Rayleigh-type SAW modes. The pink lines and dots represent ineffectual bulk acoustic wave modes brought by the calculation; see Notes S2 and S3 within the Supplemental Material. The two $\mathrm{Pn}\mathrm{Cs}$ are SAW insulators with an overlapping band gap from approximately 72 to 74 MHz. Essentially, there is a band inversion for their p_x/p_y and d_xy/d_x2−y2 modes (like p/d orbitals of electrons), corresponding to a topological transition from an SAW ordinary insulator [OI, zero spin Chern number (C_s)] to a SAW topological insulator (TI, none-zero C_s). (h), (i) Simulated field profiles of the vertical displacement at the top surface of the six micropillar unit cells for the p/d eigenmodes, corresponding to the Rayleigh-type SAW OI and TI, respectively. The black arrows indicate the directions of the surface mechanical vibrations.

Figure 2

(a) Adjacent SAW TI-OI sample used in our experiments. On the surface of a yz-cut ${\mathrm{Li}\mathrm{Nb}\mathrm{O}}_3$ substrate, planar SAWs of different frequencies are evoked by broadband IDTs and incident on a straight TI-OI interface (yellow dashed line). (b) Experimentally measured (heat map) and numerically simulated (solid lines) projected band diagrams of the interface, showing two gapless helical edge states of SAWs. (c), (d) Measured and simulated out-of-plane displacement fields and energy distributions in the vicinity of the TI-OI interface at 72 MHz, respectively. No TI-OI edge mode, but conducting SAWs are observed inside the TI and OI everywhere. (e), (f) Similar displacement fields and energy distributions at 75 MHz, i.e., in the TI-OI band gap, respectively. SAWs are observed only localized and propagating along with the TI-OI interface. (g), (h) Experimentally recorded temporal evolution of the out-of-plane displacement fields when the SAWs are selectively excited forward and backward along with the TI-OI interface, respectively, confirming the SAW pseudospins ±½ represented in the S/A basis: time-dependent anticlockwise SAW spin+½ (S+iA) and clockwise SAW spin−½ (S−iA), respectively.

Figure 3

Left: schematics of different SAW TI-OI interfaces to demonstrate (a) defect robustness. The orange dot indicates a random vacancy defect, (b) geometrical flexibility, and (c) splitting capability. The green and blue regions correspond to SAW OI and TI, respectively, the yellow dotted lines indicate the TI-OI interfaces, and the scale bars represent 400 µm. Right: heat maps are calculated elastic energy-density distributions for SAWs working at frequency within the bulk TI-OI band gap (74.5 MHz); cyan bars are experimental results measured along two straight lines cut through the TI-OI interfaces in front of and behind the defect (a), bends (b), and splitter (c).

Figure 4

(a) Schematic of our experiments in a SAW transmission line. IDTs are used to excite and receive SAWs. The black frame surrounds a periodic SAW-$\mathrm{Pn}\mathrm{C}$ structure. (b), (c) Top-view SEM images of the fabricated samples, including straight and “Z”-shaped topological interfaces, where the scale bars represent 200 µm. The red and blue regions correspond to SAW-OI and SAW-TI, respectively. (d), (e) Measured and simulated transmission spectra for the straight and “Z”-shaped TI-OI interfaces, respectively. The red- and green-shaded regions represent the band gaps of the OI and TI, respectively. (f)–(h), (i)–(k) Elastic stored energy densities and electric energy densities for the two distinctly shaped TI-OI interfaces at 75 MHz.

Figure 5

(a) Two choices of unit cells of the same phononic crystal in the honeycomb lattice. Cell I contains two pillars, triple-sized cell II has six pillars. (b) Two types of BZs according to cell I and cell 2, respectively. (c) SAW dispersions in the extended BZs, along the direction from M_II’*, $K_{\mathrm{I}}^{\mathrm{\prime }}$, $M_{\mathrm{II}}^{\mathrm{\prime }}$, Γ, M_II, K_I to $M_{\mathrm{II}}^{\ast }$. Red dotted arrows indicate the nonphysical band folding, i.e., G + k₀ → k₀, from extended Brillouin zones to the reduced (first) Brillouin zone. The red circle indicates the same mode. This mode “appears” in the reduced (first) Brillouin zone and inside the (green dotted) sound cone, but it locates at the higher Brillouin zones far away from the Γ point. From the perspective of the extended Brillouin zone, the real position of this mode is below the sound line. The fourfold degeneracy appears at the Γ point of the reduced (first) Brillouin zone is folded from two twofold degeneracies at the K_I and $K_{\mathrm{I}}^{\mathrm{\prime }}$ points of the extended Brillouin zones, which are physically below the sound line.

Figure 6

(a), (b) 3D model of unit cell I and II. (c), (d) Calculated band structures for the two cases. SAW degeneracy appears at the K_I (also $K_{\mathrm{I}}^{\mathrm{\prime }}$) point in (c), and appears at the Γ point in (d). (e), (f) Vertical displacement fields of two degenerate modes (① and ②) shown in (c) and (d), respectively. (g), (h) Total displacement as a function of depth of mode ① shown in (c) and (d), respectively, by using the same analysis of reference [55]. It confirmed that all these modes are true Rayleigh-type surface acoustic waves (RSAWs).

Figure 7

Projected SAW band structures for different TI-OI interfaces. (a), (c), (e) Schematic zigzag, armchair, and deformed armchair interfaces, respectively, between SAW OI and SAW TI. (b), (d), (f), SAW band structures of the three TI-OI interfaces.