Nanoscale imaging of super-high-frequency microelectromechanical resonators with femtometer sensitivity

The study addresses the need for direct, nanoscale visualization of acoustic mode profiles in super-high-frequency (SHF, 3–30 GHz) MEMS resonators to understand key device characteristics such as energy dissipation channels, spurious modes, and fabrication-induced imperfections. Conventional transducer-based electrical readouts overlook such spatial features, creating a bottleneck for designing efficient acoustic and cross-domain microsystems. Direct imaging at SHF is challenging due to sub-micrometer wavelengths requiring <100 nm spatial resolution, reduced vibration amplitudes at high frequencies demanding sub-picometer sensitivity, and the predominance of in-plane oscillations in many piezoelectrics that are not accessible to out-of-plane sensing techniques. The purpose of this work is to develop and demonstrate a microscopy method capable of imaging laterally polarized SHF acoustic waves with high spatial resolution and femtometer-level displacement sensitivity, thereby enabling improved design and performance of resonators for telecommunications, sensing, and quantum information applications.

The paper reviews existing GHz acoustic imaging techniques and their limitations. SEM-based acoustic imaging is limited to sub-GHz due to charging effects. AFM approaches rely on nonlinear detection of surface displacements, which is unfavorable for low signal levels. Stroboscopic X-ray imaging requires highly coherent sources, restricting usability and sensitivity. Optical techniques, including homodyne and heterodyne interferometry and pump-probe methods, are diffraction-limited to ~0.5 µm spatial resolution and detect only out-of-plane displacement, which is inadequate for laterally polarized modes in transparent piezoelectric membranes common in SHF resonators. These constraints motivate a new method that can image in-plane, laterally polarized SHF acoustic waves with nanoscale resolution and high sensitivity.

Devices: Freestanding lateral overtone bulk acoustic resonators (LOBARs) were fabricated on transferred 510 nm thick 128° Y-cut LiNbO3 membranes released from a silicon wafer. Aluminum (50 nm) interdigitated transducers (IDTs) were patterned on the membrane to excite the first-order antisymmetric (A1) Lamb mode via the large transverse piezoelectric coefficient d15. Vector network analyzer (VNA) admittance measurements revealed >20 resonances between 4.4–5.6 GHz corresponding to lateral overtones; frequency-domain finite-element analysis (FEA) reproduced the admittance, validating the model. Imaging technique: Transmission-mode microwave impedance microscopy (TMIM) integrated into an AFM was used to map the piezoelectric surface potential associated with in-plane A1 displacement fields. The TMIM tip acts as an electrically floating nanoscale (~100 nm) metal sphere, minimally perturbing the local potential. The received GHz signal is amplified and demodulated with an in-phase/quadrature (I/Q) mixer referenced to the same source, yielding two channels Vch1 ∝ cos(kx+φ) and Vch2 ∝ sin(kx+φ), providing phase-sensitive complex signal Vch1 + iVch2 ∝ e^{-ikx}. Fast Fourier transform (FFT) of images provides reciprocal-space (k-space) information on wave propagation and standing-wave formation. Imaging conditions: Typical scans covered 35 µm × 30 µm areas (slightly smaller than the 38 µm × 32 µm suspended membrane) at 3.33 s per line with AFM contact force <1 nN. Broadband TMIM imaging across multiple overtones was performed; for standing waves, the mixer phase φ was adjusted to place most signal in a single channel. Spatial resolution in unsuspended regions was ~100 nm. Analysis: Real-space TMIM maps and k-space FFTs were used to identify standing waves inside the cavity and leakage through anchors. The square modulus of the complex TMIM signal (Mod² = Ch1² + Ch2²) was integrated over the scanned area to estimate stored acoustic power, compared against Y² (square of admittance) from VNA. FEA provided simulated piezoelectric potential distributions and in-plane displacement fields at the same IDT drive conditions. Sensitivity calibration and noise analysis: An impedance-matching section was added to route the tip impedance to 50 Ω and enhance sensitivity near 5 GHz. Line profiles across the cavity were recorded at varying input powers Pin to the resonator; the maximum TMIM peak near the IDT was plotted versus Pin to identify the instrument noise floor. The IDT voltage was computed from Pin, Z0=50 Ω, and reflection coefficient Γ measured by VNA. FEA at the corresponding IDT voltage yielded surface potential and in-plane displacement amplitudes, enabling determination of a conversion factor (~−0.01 mV/fm). Noise power spectral density (PSD) of line scans was computed; taking the square root and applying the conversion factor produced amplitude spectral density (ASD) curves, from which line-noise peaks and broadband white noise levels were assessed.

- TMIM directly visualized A1 Lamb-mode standing waves in a LiNbO3 LOBAR around 5 GHz with ~100 nm spatial resolution, revealing individual overtone mode profiles, higher-order transverse spurious modes, and anchor leakage. - FFT analysis showed equal-magnitude counterpropagating waves inside the cavity (standing-wave behavior) and outward-propagating leakage near anchors. TMIM signal at anchors was ~100× smaller than inside the resonator, implying ~10⁻³ acoustic power loss; with Q ≈ 1000 for that mode, anchor loss is not the dominant dissipation. - Broadband imaging across overtones confirmed decreasing fringe spacing with increasing mode index. Measured 1/λ and phase velocity vph trends versus frequency matched quasi-static approximations; vph reached ~12 km/s at ~5 GHz for the A1 mode, easing microfabrication compared to SH0 modes (~4 km/s). - Higher-order transverse modes near the 32nd overtone were imaged (fundamental, 2nd, 3rd transverse orders) and matched by FEA. These spurious modes, at slightly higher frequencies than the main tone, are expected to degrade Q and thus should be mitigated in design. - The integrated TMIM-Mod² over the cavity tracked Y² from VNA, both proportional to stored acoustic power. A Q-factor ~700 was extracted near the 32nd overtone main tone from both metrics. - Sensitivity: The instrument noise floor was reached at Pin = −68 dBm. FEA indicated an in-plane oscillation amplitude ~80 fm at this drive. A conversion factor of ~−0.01 mV/fm was obtained by correlating peak TMIM signals with simulated displacements. - Noise analysis: ASD revealed a power-line-noise-limited floor of ~10 fm/√Hz at ~55 Hz and harmonics, surpassing state-of-the-art optical methods by >5×. The broadband white-noise level was ~3–5 fm/√Hz, setting the ultimate room-temperature sensitivity, implying acoustic signals become unobservable for Pin < −74 to −77 dBm even after filtering line noise. - Finite-element modeling and measurements establish an equivalent in-plane displacement sensitivity of ~10 fm/√Hz at room temperature, with potential improvement to <1 fm/√Hz at 4 K and <0.1 fm/√Hz below 100 mK.

The results demonstrate that TMIM provides the required nanoscale, phase-sensitive, and in-plane displacement imaging capability for SHF MEMS resonators that conventional electrical and optical methods cannot offer. By mapping real- and reciprocal-space mode profiles, the method directly addresses the need to identify and quantify dissipation channels, anchor leakage, and spurious transverse modes. The agreement between integrated TMIM-Mod² and admittance-derived stored energy validates TMIM as a quantitative tool for resonator characterization. The observed small anchor leakage indicates that other mechanisms dominate loss in the tested LOBAR, guiding attention to alternative damping channels. Imaging of higher-order transverse modes near principal tones highlights design targets to suppress these spurious resonances and preserve Q. The measured displacement sensitivity at the femtometer/√Hz level, exceeding optical techniques, underscores TMIM’s significance for low-signal environments, including quantum acoustic systems. Collectively, these findings provide actionable insights for optimizing SHF resonator performance in telecommunications, sensing, and quantum information applications.

The study introduces and validates transmission-mode microwave impedance microscopy as a powerful nanoscale imaging method for SHF MEMS resonators, achieving ~100 nm spatial resolution and femtometer-level displacement sensitivity. TMIM visualizes individual overtone mode profiles, quantifies anchor leakage, and reveals higher-order transverse spurious modes. Integrated TMIM signals correlate with stored acoustic energy, enabling quantitative assessment of Q and energy distribution. The demonstrated equivalent in-plane displacement sensitivity is ~10 fm/√Hz at room temperature, with a path to sub-femtometer/√Hz under cryogenic operation. Future work should implement TMIM in cryogenic environments to suppress Johnson-Nyquist noise, further enhance sensitivity, and apply the technique to optimize resonator designs (e.g., suppressing higher-order transverse modes, engineering anchors) and to investigate quantum acoustic devices with low phonon numbers.

Current sensitivity is limited by environmental power-line noise (~10 fm/√Hz) and broadband Johnson-Nyquist white noise (~3–5 fm/√Hz) at room temperature; achieving sub-femtometer/√Hz sensitivity requires cryogenic operation. The quantitative calibration of displacement relies on FEA and device parameter estimates, leading to slight frequency discrepancies between simulation and experiment. Spatial scans are constrained to slightly smaller than the suspended area to avoid damage, and the technique requires impedance matching and careful mixer phase adjustment for optimal sensitivity.