Synchronous In Situ Harmonic Calibration of Vibration Amplitude and Gap in 2D Nanomechanical Resonators

Reliable calibration that converts transduced signals into physical displacement is essential for quantitative studies and applications of nanoelectromechanical resonators, including nonlinear dynamics, precision sensing, and optomechanical and electromechanical coupling. Existing harmonic calibration based on nonlinear optical transduction is generally restricted to systems with optically thin suspended layers and highly reflective substrates. Their weak higher-order harmonic signals are also susceptible to noise, drift, and inconsistencies between separately acquired frequency sweeps. Here, we generalize this approach to hexagonal boron nitride/graphene (h-BN/Gra) heterostructure resonators without local metallic reflectors. Multilayer thin-film interference calculations show that a branch-local phase correction enables the effective two-beam inversion to recover vibration amplitude and local static gap within acceptable error bounds. Experimentally, we use multi-demodulator lock-in detection to acquire the ω, 2ω, and 3ω optical responses simultaneously at each frequency point. Ratios among these harmonics then yield frequency-resolved vibration amplitude and local static gap. Repeated frequency sweeps at a constant gate bias simultaneously track the resonance characteristics and local static gap, revealing a time-dependent relaxation of approximately 24 nm in the local device configuration. This work provides a practical in situ route for simultaneously resolving resonance characteristics and configurational evolution across a broader range of nanomechanical resonator architectures.