Extended topological valley-locked surface acoustic waves

Stable and efficient guided waves are essential for information transmission and processing. Recently, topological valley-contrasting materials in condensed matter systems have been revealed as promising infrastructures for guiding classical waves, for they can provide broadband, non-dispersive and reflection-free electromagnetic/mechanical wave transport with a high degree of freedom. In this work, by designing and manufacturing miniaturized phononic crystals on a semi-infinite substrate, we experimentally realized a valley-locked edge transport for surface acoustic waves (SAWs). Critically, original one-dimensional edge transports could be extended to quasi-two-dimensional ones by doping SAW Dirac “semimetal” layers at the boundaries. We demonstrate that SAWs in the extended topological valley-locked edges are robust against bending and wavelength-scaled defects. Also, this mechanism is configurable and robust depending on the doping, offering various on-chip acoustic manipulation, e.g., SAW routing, focusing, splitting, and converging, all flexible and high-flow. This work may promote future hybrid phononic circuits for acoustic information processing, sensing, and manipulation. Here the authors provide experimental evidence of extended topological valley-locked states. By splicing together Dirac semimetals and topological insulators, they demonstrate reduced backscattering and enhanced matching of SAW with interdigital transducers proposing this system for topological acoustics devices.

S urface acoustic waves (SAWs) have gradually become an essential part of modern civilization since interdigital transducers (IDTs) were invented in 1965 1 . Using IDTs to generate, receive and control on-chip SAWs in wavelengths ranging from 10 −8 to 10 −4 m, the unique advantages of SAWs can be well applied to today's microwave electronics in wireless communicating, sensing, navigating [2][3][4][5] . Compared with electromagnetic waves, SAWs of the same frequency have a much shorter wavelength (up to 10 −5 ), which dramatically increases the possibility of device miniaturization. Also, as quasi-two-dimensional waves on solids, the transmission mode of SAWs is relatively simple, easy to distinguish, and low loss. Today, SAW technology is rapidly developing in the interdisciplinary fields of photonics, biomedical, and quantum science, giving birth to a significant number of interests in, e.g., acousto-optic modulations [6][7][8] , phonon-based quantum information [9][10][11][12][13] , spin-phonon interactions 14,15 , and acoustic microfluidics 16,17 .
In this article, we further use the topological advantage to propose and experimentally realize a monolithic SAW technology to simultaneously solve the aforementioned essential challenges. In principle, by designing the band structure of the SAWs in artificial periodic microstructure on a semi-infinite substrate, we have realized "Dirac semimetals" and "insulators" for the SAWs, respectively. Then, inspired by a band engineering methodology recently proposed in sonic 65 and electromagnetic 66 crystals, we spliced our SAW "semimetals" and "insulators" and successfully constructed a new kind of SAW guiding states with simultaneously topological protection and a wide equivalent width, called SAW extended topological valley-locked states (ETVSs). Through experiments, we have verified that these ETVSs have antireflection ability, high-flow, considerable working bandwidth for SAWs, and are configurable, demonstrating promising prospects for future large-scale phononic integrated circuits with versatile applications.

Results
SAW valley vortices and valley-locked states on a piezoelectric LiNbO 3 half-space. The Quantum Valley Hall Effect (QVHE) was first found in two-dimensional (2D) hexagonal crystals, e.g., graphene 71 , double-layer graphene 72 , and transition metal dichalcogenides 73 . It originates as a result of the broken space inversion-symmetry. In those quantum states of matters, Diracfermions that correspond to different valleys move to opposite transverse edges in the presence of an in-plane electric field. In recent years, analogous to the Fermi electronic system, bosonic QVHE for photons [41][42][43][44][45][46][47] and phonons [54][55][56][57][58][59][60][65][66][67][68][69] have also been demonstrated. In this work, our first step is to realize the valley vortices 66,73 and the QVHE for SAWs. SAW phononic crystals are artificial mechanical microstructures based on semi-infinite substrates developed in this century, which can realize the dispersion modulation for SAWs, deriving, e.g., SAW bandgaps 74-79 , guiding 18,19,80,81 , localization 82 . By utilizing a pillar-type SAW phononic crystal 19,75,79 , we have successfully mimicked the phenomenon of electrons in those 2D hexagonal crystals. Specifically, honeycomb-patterned mechanical micro-resonator pillars, coupled with each other through a half-infinite substrate, constitute our phononic crystal. To ensure electrical pumping/ transducing for the SAWs, a piezoelectric crystal [in particular, a y-cut lithium niobate (LiNbO 3 )] is chosen as our substrate. A schematic diagram of our phononic crystals is shown in Fig. 1a-c, there are two inequivalent phononic 'atoms' (micro-resonator pillars) in each unit cell, marked as pillar A and pillar B, respectively. These two sets of pillars have the same height (h = 8 μm) and side angle (θ = 6°) but may differ in their radius. . d and f Two SAW valley "insulators" with inequivalent A/B pillars in their radius, both having omnidirectional bandgaps at their K and K' points of the 1st Brillouin zones, i.e., the valleys. We call the "insulator" in d (r A = 5.9 μm r B = 3.9 μm) as valley insulator A (VA) and the "insulator" in f (r A = 3.9 μm r B = 5.9 μm) as valley insulator B (VB). Insets show the SAW flux vortices observed at their valleys.
We define a relative mass Δm for all these A and B pillars and Δm¼ r A Àr B r A þr B for their contrast. When Δm = 0 (e.g., r A = r B = 5.8 μm in Fig. 1e), a "semimetal" for SAWs could be formed 80 , the symmetry of the honeycomb crystal preserves a pair of SAW Dirac degeneracy around the K and K' points of the 1st Brillouin zone (considering the influence of the anisotropic LiNbO 3 substrate, K/ K' are used here to distinguish them from the high symmetry points K/K' of the traditional honeycomb lattice. See details in SI Note 7). When Δm ≠ 0 (e.g., r A = 5.9 μm r B = 3.9 μm in Fig. 1d; and r A = 3.9 μm r B = 5.9 μm in Fig. 1f), the spatial inversion symmetry (along the y-axis) of the crystal is then broken, thus leading to "insulators" for SAWs with omnidirectional bandgaps.
In these SAW insulators, flux vortices of the SAWs could be observed at their frequency extrema K ( K') points [insets of  73 . By constructing zigzag boundary between different SAW insulators in Fig. 1d and f, we realized the analogous QVHE for SAWs, or say SAW topological valleylocked states (TVSs). Simulation and experimental results are shown in Fig. S10 of SI. Note that the symmetry of the y-cut LiNbO 3 substrate slightly affects our phononic crystal's symmetry. However, this effect is faint and does not affect the presence of TVSs (see SI Note 7).
Extended topological valley-locked states for SAWs. Waveguides built from topological edge states may provide a disruptive advantage over conventionally designed ones, e.g., backscattering suppression, single-mode operation, and linear dispersion at their working bandwidth. It is especially the case for SAWs. Due to the relatively low impedance mismatch (generally <10 1 ) in different solid-state materials, and the extensively high sensitivity of acoustic waves in solid-state media, conventional designed solidstate acoustic waveguiding are pretty susceptible to intersections, sharp bends, and wavelength-scale defects.
Meanwhile, for (future) electro-acoustic SAW devices with SAW waveguides functioned between dual-(or multiple-) port IDTs, there is a crucial trade-off between the signal fidelity and the working bandwidth 17 . Broadband operating frequency is one of the core requirements of advanced signal processing. Generally, a broadband IDT requires a relatively large "aperture/wavelength" ratio, which is bound to reduce the proportion of the SAWs injected into the waveguide to the total SAWs that the IDT pumped. For the device as a whole, this requirement would increase the insertion loss of the waveguide signal and reduce its fidelity (see more in SI, Note 10). Consequently, to the practical use of SAW waveguides in concreting, e.g., phononic integrated circuits, a topological SAW waveguide with a wide aperture that can match broadband IDTs, is highly promising.
In this work, our second step (and the main achievement) is to realize such wide-aperture topological waveguides for SAWs by bringing about the ETVSs. Specifically, as shown in the upper panel of Fig. 2, based on the valley-type phononic crystal present in Section I, we doped our SAW semimetal (Δm = 0, r A = r B = 5.8 μm) in the interface between two SAW insulators with opposite valleys [i.e., one Δm = 0.2 (r A = 5.9 μm r B = 3.9 μm) and one Δm = −0.2 (r A = 3.9 μm r B = 5.9 μm)], thus directly making a heterostructured interface. This doping could extend the onedimensional (1D) TVSs to quasi-two-dimensional (quasi-2D) ETVSs, with an additional degree of freedom (DOF) in the width of the doping area. Topological characteristics carried by the bulk on both sides of the heterostructured interface will tunnel through the doping area with Dirac dispersion 83,84 , thus forming the ETVSs.
We numerically modeled several of those heterostructured interfaces with increasing doping areas. Projected band structures of these heterostructured interfaces and their SAW field/phase distributions are shown in the middle and bottom panels of Fig. 2. All dispersions of these ETVSs share the same characteristics as the conventional TVSs. Differently, for the TVSs, their energy is highly localized around the 1D interfaces. For the ETVSs, their energy fills the entire 2D doping area, thus offering a topological waveguide with a much wider aperture. As the doping area increases, the waveguide aperture may expand several times or even dozens of times to the original (none doped) one, at the price of a gradual decrease in working bandwidth. (see SI, Note 8).
Experimentally, we fabricated three different heterostructured samples with 3, 5, and 7 molecular doping layers. All the samples are in the same configuration, i.e., phononic crystal containing the heterostructured interface is prepared in the middle; two broadband IDTs, acting as SAW emitter and SAW receiver, are prepared on both sides, respectively. In the experiments, the emitter IDTs pump planar SAWs into the heterostructured interfaces (i.e., the waveguides), exciting the ETVSs. A laser vibrometer imaged the out-of-plane displacement of the excited ETVSs, confirming the SAWs do strongly localized in the whole doping area, together with uniform phases. After the SAWs passed through the heterostructured interfaces, they were captured by the receiver IDTs and read out by a network analyzer, demonstrating considerable transmittance (i.e., S 21 , Fig. S18d, f, h of SI) at their operating frequencies. For the 3, 5, and 7-layer doping ETVSs, their maximum working bandwidths are expected to reach 7.8%, 6.7%, and 5.3%, respectively, without additional challenge for sample processing (see SI, Note 9).
ETVS based splitter with high SAW throughput. A three-port SAW splitter sample was fabricated to confirm that the ETVSs hold the same topological properties as TVSs. As shown in Fig. 3, this sample consists of five different parts, thus forming four ports labeled in the figure. Ports #1 and #3 are located on the straight SAW transmission route but are different in their valley pseudospins. Ports #2 and #4 are located on two "zigzag" SAW transmission routes with 120°sharp bends but hold the same valley pseudospin as Port#1. Planer SAWs wave generated by IDTs and are injected into the multi-port device through Port#1.
In conventional SAW waveguide devices, the input SAWs will mainly directly exit the device via Port #4 but partially go through the "zigzag" routes to reach Ports #2 and #3. However, in this topological valley-locked waveguide, the input SAWs from Port #1 could only exit the system via the ports that hold the same valley-pseudospin. Hence, these SAWs should pass through those "zigzag" routes and eventually be captured at Ports #2 and #3. Experimental imaged SAW distributions around the input and output ports confirm that the SAWs experience this "Port#1 → Ports# 2/3" transport. These results agree well with theoretical expectations, proving that in our SAW system, the ETVSs do hold the same function as conventional TVSs. Our SAW splitter is a demo for the valley-pseudospin locking and a promising primary component for future SAW integrated circuits with high throughput and energy capacity.
Defects immunity and phase reconstruction for SAWs. In the QHEs and QSHEs 35,85 , the nonzero (spin) Chern number originates from integrating the Berry curvature over the whole Brillouin zone. For the QVHEs 58,66,69 , although the integration is zero, the Berry curvature is highly localized around the K and K' points (i.e., the valleys), giving birth to nonzero valley Chern number with opposite signs at paired valleys.
When a wavelength-scale defect is introduced into a conventional QVHE system, the Berry curvature will become less localized, and the intervalley mixing will become stronger, thus destroying the TVSs 61 . Such destruction can be observed in our SAW system, as shown in the left panel of Fig. 4. We experimentally prepared a conventional (non-doping) valleylocked edge with two wavelength-scale vacuum defects deliberately embedded. When time-harmonic SAWs are pumped into this edge, relatively few SAWs can be detected from the exit. Meanwhile, the phase of the transmitted SAWs is wholly disturbed, confirming the destruction of the edge states.
Unlike non-doped TVSs in conventional QVHE systems, ETVSs have an excellent tolerance for the same wavelength scale defects. The additional width (DOF) of ETVSs endows them with stronger robustness in their transmission routes since the same defect should impose much less disturbance on quasi-2D ETVSs than 1D TVSs. This robustness was observed in a comparative experiment, as shown in the right panel of Fig. 4. When timeharmonic SAWs are pumped into a doped edge, much more SAWs can now be detected from the exit. Along the transmission route, the phase of the SAWs is disturbed around the defect. However, the phase uniformity would quickly emerge as the SAWs propagate away from the defect for several wavelengths. This intriguing phenomenon would bring us some novel SAW applications with precise wavefront control, e.g., SAW cloaking, focusing, collimating, and beam expanding.
ETVS's stability and configurability. The existence of the topological edge states is originated and preserved by the nontrivial topology of the bulk bands. Although the characteristics and functionalities of those edge states, e.g., defects immunity and insensitivity of the external field perturbation, have been widely demonstrated in electronics, photonics, and phononics, doping (and manipulating) on these topological edges are currently underdeveloped and reveals a promising research area to explore complex nontrivial physics and applications 86 in, e.g., signal processing and information communications.
For our present SAW ETVSs, we will demonstrate their stability and configurability to the different doping. Our experiments are shown in Fig. 5. The doping width is adjustable by adopting different doping layers, i.e., layers of the SAW semimetal embedded in the doping area. As the doping width increase, the frequency range in which the ETVSs exist would continue to narrow and eventually disappear. Except that, in a fixed doping width, the doping "height" is also adjustable. This "height" can be defined as the Dirac frequency of the SAW semimetal. By changing the radius of the micropillars, the doping "height" changes from 77.90 MHz, 75.99 MHz, 74.94 MHz (Fig. 5a, b, and c, respectively). Calculated projected band structures of these different heterostructured interfaces (see Fig. S20 of SI) show that the ETVSs crossing the whole bulk band stably exist in all these cases, except for slight frequency shift. Experimentally, these configurable ETVSs are imaged (see Fig. 5e-g), all exhibit phase uniformity with slightly changed patterns. When the doping "height" is relatively high, the interaction between adjacent micropillars seems to be relatively weak; SAWs in the doping area (i.e., the ETVSs) are more similar to plane waves. As the doping "height" decreases, the inter-pillar coupling becomes stronger; SAWs in the doping area are more localized around the micropillars, exhibiting a "fish scales" style. Notably, the transmittances of the three ETVS waveguides are unity since they are all valley pseudospin locked and robust to wavelength-scale defects and interactions. Practically, it provides a new strategy to manipulate SAWs in the ETVS waveguide, e.g., operation frequencies and distributions 84 .

Discussions
We proposed and experimentally verified a SAW analog of quantum valley Hall system, accompanied by the ETVEs for SAWs, i.e., an ideal candidate for SAW integrated circuits. The SAWs ETVEs are anti-reflection (making the waveguide have a high degree of freedom and low loss), high-flow and configurable. Based on them, a series of application-driven SAW prototype devices, e.g., waveguides and beam splitters with flexible path and defects immunity, are fabricated and experimentally demonstrated. Although the operating frequency of the SAW prototypes in this article is in the tens of megahertz, future devices based on the same principle can exceed several gigahertz or even higher through advanced acoustic MEMS manufacturing technology. Similarly, the material systems used to build this type of ETVSs can also be varied, such as traditional piezoelectric materials for SAWs (e.g., LiTaO 3 , AlN, GaN), mainstream silicon-based materials (e.g., Si, SiO 2 , SiN, SiC), and mechanical or even piezoelectric 2D materials (e.g., monolayer transition metal dichalcogenides, group III-V binary compounds and group IV monochalcogenides). Future designs can comprehensively consider using these materials based on their piezoelectric characteristics, electromechanical coupling characteristics, velocity, temperature drift, loss, and convenience of micro-nano processing. This universal design principle may flourish and promote future topological-related applications, such as (topological) photonic/phononic integrated circuits for classical and quantum photonics/phononics. One challenge is how to increase the working bandwidth of these ETVSs further. If the bandwidth can Sim. Exp.

Sim.
Exp. exceed 15% or even 25% (i.e., the largest bandwidth in current 5G mobile networks), it will significantly facilitate their demonstrating functionalities in practical scenarios. Some valuable results have recently appeared in broadband topology materials, e.g., those using topology optimization 87 and inverse design 88 . From the perspective of condensed matter physics, due to the easy-to-process characteristics of artificial structures and SAWs' intuitive and high-fidelity characteristics, this research on doped topological edge states may provide an ideal platform for exploring topological properties and transport behavior. It may inspire other related studies on bosons and fermions.

Methods
Sample fabrications. We used the LIGA-like technique to fabricate our phononic crystals on the LiNbO 3 semi-half-space. It contains four following steps: a 10 nm Cr followed by a 50 nm Cu layer, working as a seed layer, is deposited on a 500 μm thickness y-cut LiNbO 3 substrate; (2) patterning of phononic crystal structures using UV lithography with 10 μm positive resist AZ9600; (3) Electrochemical plating of nickel (Ni) micro-resonator pillars on the exposed Cu-Cr seed layer; (4) removal of the resist. Note that these values are specific to our growth technique.
Reporting summary. Further information on research design is available in the Nature Research Reporting Summary linked to this article.

Data availability
All simulation data in the current study are performed using commercial finite element software COMSOL Multiphysics. All experimental measured SAW displacement/energy data are characterized by a commercial scanning vibrometer (Polytec UHF-120) with software PSV Acquisition. They are under private user license which cannot be made public. All data are available from the corresponding authors upon reasonable request.

Code availability
Numerical simulations in this work are all performed using the commercial finite element software COMSOL Multiphysics. All related codes can be built with the instructions in the "Methods" section and available from the corresponding authors upon reasonable request.