Slow Surface Acoustic Waves via Lattice Optimization of a Phononic Crystal on a Chip

Strategically reducing the speed of waves, which greatly improves both the energy density and information capacity of carrier signals in space, is a key enabling factor for signal-processing devices. Among these devices, especially in the prosperous wireless communication industry, surface acoustic wave (SAW) devices based on interdigital transducers (IDTs) currently hold an essential status. However, velocity reduction in traditional IDT-based SAW devices can be achieved only by using specific substrate materials that are generally of lower hardness, which inevitably leads to an increase in device size and less-optimal electromechanical coupling coefficients. Here, we demonstrate a technological means of realizing slow on-chip SAWs that is relevant for practical rf signal processing, gyrometers, sensing, and transduction. This method takes advantage of the gradual flattening of a Rayleigh-type dispersion band due to the spatial lattice evolution of a surface phononic crystal. In our experiment, the speed of an ultraslow SAW is measured to be approximately 200 m/s, which is even slower than the speed of sound in air and equivalent to 1/17.4 of the speed of the original Rayleigh waves in ${\mathrm{Li}\mathrm{Nb}\mathrm{O}}_3$. Such ultraslow SAWs may have promising applications in time-dependent SAW modulation, high-sensitivity SAW sensors, and SAW nonlinear even quantum-dynamic systems. Additionally, our technique can be similarly applied to a broad range of other two-dimensional or quasi-two-dimensional wave structures, e.g., in electronic, optical, acoustic, and thermal systems.

Figure 1

SAW dispersion in four basic SPnC configurations and symmetries. This research is based on the platform shown in (a), namely, a chip-scale SPnC consisting of solid pillars on an elastic substrate, through which planar SAWs pass. Calculated band structures of the four considered cases of SAW incidence (along directions of high symmetry with respect to the SPnC) are shown: (b),(c) correspond to cases in which the SAWs are incident on a square-lattice SPnC in the Γ-X and Γ-M directions, respectively; (d),(e) correspond to cases in which the SAWs are incident on a triangular-lattice SPnC in the Γ-M and Γ-K directions, respectively. All surface modes lie below the sound line (black), which represents the slowest bulk acoustic modes. The flat dispersion curves (gray) near the frequency of 1670 Hz m are the localized resonant (LR) modes. S₀ dispersion bands are manifestations of the Rayleigh modes in the SPnCs.

Figure 2

Distributions of elastic fields in the modes and dispersion bands of the four different configurations. (a),(b) correspond to square-lattice SPnCs with SAWs incident along the Γ-X and Γ-M directions, respectively; (c),(d) correspond to triangular-lattice SPnCs with SAWs incident along the Γ-M and Γ-K directions, respectively.

Figure 3

Theoretical evolution of the SAW dispersion under lattice deformation. Deformation of the SPnC lattice is schematically illustrated in (a): in a triangular lattice of thickness L (left), the distance from each unit cell to its next-nearest neighbor in the x direction remains constant, while the distance in the y direction is evenly spatially compressed such that $b_0\to b=r{b}_0$ (right). Corresponding band structures before and after the lattice deformation are illustrated in (b),(c), respectively. During the lattice deformation process, the initial $S_0-$ dispersion curve (with a negative slope) is gradually integrated into $S_0+$ (with a positive slope) and eventually vanishes when r reaches a certain value (near 0.75). In this exact configuration, the now-unified dispersion band $S_0$ possesses the smallest but positive mean group velocity over the broadest frequency range, from the lower frequencies to the band edge.

Figure 4

Chip-scale experiment and device based on a ${\mathrm{Li}\mathrm{Nb}\mathrm{O}}_3$ substrate. For the materials used in our practical experiment, i.e., electroplated nickel ($\mathrm{Ni}$) pillars on a ${\mathrm{Li}\mathrm{Nb}\mathrm{O}}_3$ substrate, the calculated diagram of the transformation of the $S_0$ dispersion during the theoretical lattice deformation process is illustrated in (a). Red hexagons indicate the boundary between $S_0+$ and $S_0-$. With compression of the lattice, dispersion subbranch $S_0-$ is gradually integrated into $S_0+$ and eventually vanishes at $r_t=0.80$, the phase-change point. Structure of a practical SPnC-based transmission line is schematically illustrated in (b). On y-z cut lithium niobate substrate, a broadband SAW ranging from 66 to 86 MHz is generated from one electroacoustic transducer; transmitted through the SPnC, in which the SAWs are slowed; and eventually received in the crystallographic z direction. Real device, prepared with practical SPnCs composed of electroplated nickel pillars attached to the substrate, is shown in (c); these pillars all have the same geometry, which is shown in (d). Typical sidewall angle $\theta$ for these pillars is approximately 84° (limited by the sample-processing technology). Top radius $r_{\mathrm{top}}$ and height H of the pillars are measured to be 6.0 and 7.4 µm, respectively.

Figure 5

Optimized slow SAWs with a broad operation bandwidth. Comparative experiments are conducted using one reference sample with the original triangular lattice $(b=b_0)$ and one lattice-optimized sample $(b=0.78b_0)$, slightly over the point of total transition $(r_t{b}_0=0.80b_0)$. The corresponding results are shown in (a),(c),(e),(g),(h) and (b),(d),(f),(h),(j), respectively. (a),(b) present SEMs of the SPnCs. (c),(d) present the calculated band structures along $k_x$ (Γ-K direction). Transmittance spectra in (e),(f) confirm that SAW-forbidden bands (gray) start at 73.6 and 78.8 MHz, respectively. In (g),(h), time-delay spectra of the whole transmission line are plotted. Group-index spectra are plotted in (i),(j), indicating the extent of SAW slowdown. In the lattice-optimized sample, SAWs are slowed down to a greater extent and over a much broader range, from frequencies of higher than 77.6 MHz to the forbidden band, as indicated by the yellow highlighted region. Ultraslow SAW is measured at 78.4 MHz, near the forbidden band but still with an appreciable transmittance (>−10 dB); its velocity reaches 200 m/s, even slower than the speed of sound in air.

Figure 6

Dynamic transduction or delay of signals in wave packets. SAW Gaussian pulses with identical bandwidths (1 MHz) and different center frequencies, f_c, are injected into the lattice-optimized sample depicted in Fig. 5, and corresponding time-dependent transmission spectra are shown. (a),(b) correspond to the case of f_c = 68 MHz; (c),(d) correspond to the case of f_c = 77.6 MHz; (e),(f) correspond to the case of f_c = 78.4 MHz. Spectra presented in (a),(c),(e) serve as references, representing the scenario without any SPnCs placed in the transmission and time-delay line; here, amplitudes of the transmitted electrical signals reach approximately 115 mV. Net time delays in these three cases are measured to be 0.13, 0.48, and 0.89 µs, respectively. All cases exhibit near-perfect properties of signal restoration and still-appreciable transmittance, with measured amplitudes of approximately 100, 50, and 30 mV, respectively.

Figure 7

Tight-binding model of lattice deformation. ${\overrightarrow{a}}_1$ and ${\overrightarrow{a}}_2$ are the basis vectors in real space. a and b are the distances between nearest-neighbor unit cells in the x and y directions, respectively. $s_0$ and s are the coupling strengths between pillars in the x direction and diagonal direction, respectively.

Figure 8

Results of the tight-binding analysis. (a) Relationship between the coupling strength and spatial compression ratio. (b),(c) Band structures according to 3D FEM calculations and two-dimensional tight-binding theory, respectively.