A closed-loop platform for the design and nanoscale imaging of GHz acoustic metamaterials

Abstract

Band structure engineering in surface acoustic wave (SAW) metamaterials could advance both classical telecommunications and quantum information processing. However, no imaging technique has demonstrated the necessary capability to resolve sub-$μ$m traveling SAWs across wide GHz bandwidths. Existing methods capture only fragments of the dispersion at discrete frequencies, preventing systematic characterization and control of SAW-based metamaterials. Here, we develop electrostatic force microscopy (EFM) to enable real-space imaging of traveling SAWs in honeycomb metamaterials on LiNbO$_3$. Our application leverages sub-200 nm spatial resolution, broad GHz bandwidth, and non-contact imaging to map complex band structures with continuous frequency resolution and expanded frequency range, while preserving sub-lattice detail. Using EFM, we map the full relevant frequency range around the Dirac point of a SAW graphene analog, including the acoustic Dirac cones, and the transition from ballistic to diffusive SAW transport regime. Furthermore, by breaking sublattice symmetry, we tune the opening of a band gap at the Dirac point, and image frequency-dependent wave localization on sublattice sites. Our EFM technique closes the loop between design and real-space validation, streamlining the engineering of arbitrary SAW landscapes for next-generation applications spanning telecommunications, microfluidics, and quantum acoustics.

Figures (11)

• Figure 1: Metamaterial design.a, Optical image of a single SAW metamaterial device. A lattice of gold pillars is deposited between two interdigital transducers (IDTs) used for SAW generation/detection in the 880-1150 MHz frequency range. b, SEM image of the metamaterial (rotated by $90^{\circ}$). SAWs propagate along $\hat{x}$, corresponding to the $\Gamma-\text{K}$ direction of the honeycomb lattice. c, SEM image of the pillars, where the white rhombus indicates the unit cell, which consists of two gold nanopillars with $a_0 = 1.06\ \mu$m, $r_0 = 300$ nm, $h = 400$ nm. d, Tight-binding band structure of graphene showing the Dirac cone. e, Simulated acoustic band structure of the graphene metamaterial. The color scale represents the ratio of surface-to-bulk acoustic energy density, with darker markers corresponding to surface acoustic waves (SAWs) and lighter regions to bulk acoustic waves (BAWs). The flat bands around 950 MHz come from the radial expansion modes of the pillars which are decoupled from the substrate and do not appear in the experiment.
• Figure 1: COMSOL Multiphysics acoustic mode simulations.a, Simulated acoustic band structure of the graphene metamaterial. The color scale represents the ratio of surface-to-bulk acoustic energy density, with darker markers corresponding to surface acoustic waves (SAWs) and lighter markers corresponding to bulk acoustic waves (BAWs). The purple dispersive band corresponds to the mode in which the pillars move out of plane in phase, which we refer to as the symmetric or "bonding" mode. This motion induces equal pressure and therefore equal piezoelectric response in the substrate beneath the pillars, producing no node between pillars in the EFM signal (see Fig. \ref{['fig:graphene']}c inset). The green band corresponds to the mode in which the pillars move out of plane with a $180^{\circ}$ phase difference. This creates high pressure under one pillar and low pressure under the other, generating opposite piezoelectric responses and a distinct node in the EFM signal (see Fig. \ref{['fig:graphene']}a inset). We refer to this as the antisymmetric or "antibonding" mode. b--c, Evolution of the mode shape across a phase cycle for the antisymmetric (b) and symmetric (c) modes. Simulations are shown at $k \approx \mathrm{K}$. The color intensity represents the displacement magnitude relative to the rest position. d, Simulated acoustic band structure of the hBN ($\delta r = 7.5\%$) metamaterial. The modes at the lower edge of the upper band (red) and the upper edge of the lower band (blue) show clear sublattice localization. e--f, Evolution of the mode shape across a phase cycle for the upper-band-edge mode (e) and lower-band-edge mode (f), shown at $k = \mathrm{K}$. In the upper-band-edge mode the participation localizes on the smaller pillar A, whereas in the lower-band-edge mode, the participation is dominated by the larger pillar B. This behavior is consistent with our EFM observations (see Fig. \ref{['fig:hbn']}k).
• Figure 2: Phase-resolved electrostatic force microscopy (EFM) of SAW metamaterials.a, Schematic of the experimental setup, including the metamaterial device and the EFM used for detecting traveling SAWs. The EFM is a modified optical-detection AFM system in which the RF voltage used to drive the IDT is also applied to a conductive cantilever tip, with the tip signal chopped at the cantilever's mechanical resonance frequency $f_{\mathrm{r}}$. b, Schematic of the tip–sample interaction at two different spatial locations, $x_1$ and $x_2$. c,d, Time evolution of the tip–sample interaction at $x = x_1$ where $V_{\mathrm{tip}}$ and $V_{\mathrm{SAW}}(x)$ are in-phase (c) and $x = x_2$ where $V_{\mathrm{tip}}$ and $V_{\mathrm{SAW}}(x)$ are $180^{\circ}$ out-of-phase (d). Both $V_{\mathrm{tip}}$ and $V_{\mathrm{IDT}}$ are driven by the same $V_{\mathrm{RF}}$; therefore each spatial location maintains its own constant tip-sample phase difference, conceptually akin to a path-difference interferometer. To understand the full tip-sample force during SAW imaging, we first consider the tip-sample force with no SAW (light blue), when $V_{\mathrm{IDT}}=0$, and the electrostatic interaction is driven entirely by $V_{\mathrm{tip}}$. In this case, the tip-sample force is always attractive, and proportional to $|V_{\mathrm{tip}}|$. When we turn on $V_{\mathrm{IDT}}$, the propagating SAW induces a weaker surface voltage $V_{\mathrm{SAW}}(x) \ll V_{\mathrm{tip}}$, producing a small additional tip-sample force contribution $\delta F(x)$ (dark orange). At locations such as $x=x_1$ (c), where $V_{\mathrm{SAW}}(x)$ and $V_{\mathrm{tip}}$ are in-phase, $\delta F(x)$ is maximally repulsive, reducing the total force $F_{\mathrm{tot}}(x)$ felt by the cantilever (purple). The cantilever responds primarily at its own mechanical resonance frequency (low-pass purple trace), resulting in minimum resonant oscillation amplitude $|\Delta z|_{f_\mathrm{r}}$. Conversely, at locations such as $x=x_2$ (d), where $V_{\mathrm{tip}}$ and $V_{\mathrm{SAW}}(x)$ are $180^{\circ}$ out-of-phase, $\delta F(x)$ is maximally attractive, increasing the total force $F_{\mathrm{tot}}(x)$, and inducing maximal $|\Delta z|_{f_\mathrm{r}}$.
• Figure 2: EFM characterization of the graphene metamaterial. Same data series as in Fig. \ref{['fig:graphene']}, but showing additional frequencies and analysis steps. a, EFM amplitude shift $\Delta z$ measured at room temperature and ambient pressure for the graphene metamaterial. A quadratic background was subtracted from each of the 128 scan lines. b, Raw Fourier transform of the data in (a). c, Six-fold symmetrized data derived from (b). d, Overlays of 32 distinct linecuts along the $\hat{x}$ direction from the real-space data in (a), equally spaced in $\hat{y}$.
• Figure 3: EFM observation of Dirac cones and transport regimes in a graphene-like SAW metamaterial.a--c, EFM amplitude shift $\Delta z$, measured at room temperature in air for frequencies above the Dirac frequency $f_{\mathrm{D}}$ (a), at $f_{\mathrm{D}}$ (b), and below $f_{\mathrm{D}}$ (c). A quadratic background was subtracted from each of the 128 scan lines for easier visualization of the spatial modulations. Below $f_{\mathrm{D}}$ (c), the linear wavefronts are clearly visible, while at $f_{\mathrm{D}}$ (b) the waves form a triangular interference pattern. Above $f_{\mathrm{D}}$ (a), the amplitude drops because the mode symmetry suppresses direct excitation by the $\hat{x}$-propagating SAW, allowing only "deaf" modes driven by secondary scattering. Insets in a and c show zoomed views of the unit cell, revealing antibonding (a) and bonding (c) modes. d--f, Six-fold symmetrized Fourier transforms of the real-space images in a--c (for raw FT data, see Extended Data Figs. \ref{['fig:graph_ext']}-\ref{['fig:gra_symm2']}). Below $f_{\mathrm{D}}$ (f), a faint ring appears around $\Gamma_1$, with its intensity peaking along the $\Gamma$--K direction, corresponding to the unscattered plane waves. Both the ring and its peaks expand outward as the frequency approaches $f_{\mathrm{D}}$. At $f_{\mathrm{D}}$ (e), sharp peaks localize at the K and K$^{\prime}$ corners of the BZ. The modes above $f_{\mathrm{D}}$ are visible in (d) as circular bands closing around $\Gamma_2$, consistent with a graphene-like band structure. g, Symmetrized band dispersion obtained from the $\Gamma_1$--K--M--K$^{\prime}$--$\Gamma_2$ normalized linecut versus frequency, showing Dirac cones at the K and K$^{\prime}$ points. The dashed green line is a tight-binding fit. h--j, Overlays of 32 distinct linecuts along the $\hat{x}$ direction from the real-space data in a--c. For $f < f_{\mathrm{D}}$ (j), transport is ballistic, with linear amplitude decay and coherent wavefronts. At $f_{\mathrm{D}}$ (i), we observe a crossover to a diffusive transport regime, where the amplitude decays exponentially with distance due to scattering; such diffusive behavior persists for all frequencies $f > f_{\mathrm{D}}$ (h).