Nonlinear Mode Coupling in Silicon Nitride Membrane Resonators

Abstract

Nonlinear interactions between vibrational modes play a crucial role in understanding the dynamical response of nanomechanical resonators. Here, we report the experimental observation and theoretical modeling of nonlinear mode coupling in a high-stress square silicon nitride membrane resonator. We quantify frequency shifts of the fundamental mode arising from tension-mediated geometric nonlinearity by increasing the amplitude of the fundamental mode and higher-order flexural modes. A quantitative theoretical framework based on Kirchhoff-Love plate theory is developed, which incorporates both intrinsic Duffing nonlinearity and nonlinear intermodal coupling and shows good agreement with experimental measurements for the (1,1)-(2,1) and (1,1)-(2,2) mode pairs. We further compute the nonlinear coupling matrix across mode families, revealing the role of mode symmetry and spatial overlap in governing intermodal interactions. These results establish nonlinear mode coupling as a controllable resource for multimode frequency tuning and mechanical transduction.

Figures (4)

• Figure 1: Measurement setup and vibrational modes of the device. (a) Schematic of the laser Doppler vibrometry (LDV) based measurement setup. A Polytec MSA-500 system is used to optically probe the out-of-plane motion of $\mathrm{Si_3N_4}$ membrane, while electrical actuation and signal detection are performed using a Zurich instruments HF2LI lock-in amplifier. (b) Optical micrograph of the device, showing a $500\,\mu\mathrm{m} \times 500\,\mu\mathrm{m}\ \mathrm{Si_3N_4}$ membrane suspended over a silicon substrate. (c) Measured displacement amplitude spectrum with drive frequency showing multiple mechanical resonances. Higher frequency modes are scaled by a factor of $\times 5$ and $\times 25$ for visibility. Inset shows the corresponding spatial mode shapes of the resonant modes.
• Figure 2: Intramodal coupling and Duffing constant. (a) Nonlinear frequency response of the $(1,1)$ mode, measured at increasing drive strengths. (b) Linear response of the $(1,1)$ mode at the lowest drive amplitude; solid lines indicate Lorentzian fits used to extract the resonance frequency and quality factor. (c) Amplitude-dependent frequency shift corresponding to the maximum displacement amplitudes extracted from (a). Solid line show fit using Eq. (\ref{['6']}) to extract the Duffing coefficient. (d) Comparison of the experimentally extracted Duffing coefficients with theoretical values calculated from Eq. (\ref{['5']}) for the $(1,1)$, $(2,1)$, and $(2,2)$ modes.
• Figure 3: Nonlinear intermodal coupling. (a,c) Resonance frequency-shifting phenomena of the fundamental $(1,1)$ mode induced by strong excitation of the $(2,2)$ and $(2,1)$ modes, respectively.(b,d) Extracted resonance frequency of the $(1,1)$ mode as a function of the displacement amplitude of the driven $(2,2)$ and $(2,1)$ modes, respectively. Solid lines represent fits to the data using Eq. (\ref{['9']}) to extract the nonlinear intermodal coupling constants. (e) Comparison of experimentally extracted coupling constants (stars) with theoretical values (circles) calculated from Eq. S40.
• Figure 4: Theoretical estimation of the nonlinear mode coupling matrix. (a–d) Calculated coupling constants as a function of the driven mode indices $(p,q)$ for $p,q = 1$–$10$, coupled to the probe modes $(1,1)$, $(2,2)$, $(3,3)$, and $(4,4)$, respectively. (e) Predicted frequency shift of the fundamental mode as a function of the coupled-mode amplitude, calculated using the theoretical coupling constants listed in Table \ref{['tab:Table 1']}.