Nonlinear Coupling of Phononic Resonators Induced by Surface Acoustic Waves

The rising need for hybrid physical platforms has triggered a renewed interest in the development of agile radio-frequency phononic circuits with complex functionalities. The combination of traveling waves with resonant mechanical elements appears as an appealing means of harnessing elastic vibration. In this work, we demonstrate that this combination can be further enriched by the occurrence of traveling surface acoustic waves (SAWs), induced by elastic nonlinearities, interacting with a pair of otherwise linear micron-scale mechanical resonators. Reduction of the resonator-gap distance and an increase in the SAW amplitude results in a frequency softening of the resonator-pair response that lies outside the usual picture of geometrical Duffing nonlinearities. The dynamics of the SAW excitation scheme allows further control of the resonator motion, notably leading to circular-polarization states. These results may pave the way toward versatile high-frequency phononic microelectromechanical-systems–nanoelectromechanical-systems circuits fitting both classical and quantum technologies.

Figure 1

The experimental frequency responses of SAW-driven phononic microresonators. (a) The response of a single pillar with a diameter of $4.4\mu \mathrm{m}$ and a height of about $4\mu \mathrm{m}$ excited by a SAW for different input powers applied to the IDT (10 dBm, 15 dBm, and 20 dBm). The light dotted lines correspond to data obtained by laser-scanning interferometry and acquired every 10 kHz. The solid lines are obtained by applying a Savitsky-Golay smoothing filter to the raw experimental data. The inset shows an angular plot of the distribution of the orientation of the flexural-mode vibration for frequencies between 65 and 75 MHz. (b) The frequency response for two pillars within a pair with a 6-$\mu \mathrm{m}$ gap distance excited by a longitudinal SAW wave vector under different drive powers. The solid line corresponds to the resonator located closest to the SAW source and the dashed line to the resonator located farthest from the source.

Figure 2

The experimental frequency and phase responses of a pillar pair excited by a transverse SAW. The gap distance is $1.5\mu \mathrm{m}$. (a) The response of resonator $T1$. (b) The response of resonator $T2$. (c) Experimental phase maps at 12 dBm for the lower-frequency mode (i) at 68.6 MHz (1) and for the higher-frequency mode (ii) at 71.8 MHz (2); and at 20 dBm for the lower-frequency mode (i) at 68.3 MHz (3) and for the higher-frequency mode (ii) at 71.5 MHz (4).

Figure 3

Finite-element method simulations of the von Mises stress distribution. The two 1.5-$\mu \mathrm{m}$-spaced resonators are excited by a linear elastic line source following a transverse excitation scheme. (a) Symmetric mode (i), here found at 70.73 MHz. (b) Antisymmetric mode (ii) at 73.34 MHz.

Figure 4

The coupled-oscillator model of a pillar pair excited by a transverse SAW. (a) The solid lines correspond to the experimental data for a pillar pair with a 1.5-$\mu \mathrm{m}$ gap distance, excited using a rf input power of 20 dBm. The dashed lines correspond to the theoretical frequency response obtained from the proposed coupled-oscillator model for an applied force amplitude $F_0=10\mu \mathrm{N}$. (b) The sum of the measured amplitudes of the two resonators for the antisymmetric mode. The dots indicate the maximum displacement for each input rf power level, while the dashed line is a fit to the theoretical backbone curve given in Eq. (3).

Figure 5

The experimental frequency and phase responses of a pillar pair excited by a longitudinal SAW. The gap distance is $1.5\mu \mathrm{m}$. (a) The response of resonator L1. (b) The response of resonator L2. (c) Experimental phase maps at 12 dBm for the lower-frequency mode (iii) at 71.9 MHz (1) and for the higher-frequency mode (iv) at 73.4 MHz (2); and at 20 dBm for the lower-frequency mode (iii) at 71.24 MHz (3) and for the higher-frequency mode (iv) at 72.9 MHz (4).

Figure 6

The hysteresis cycle. (a) The frequency response of resonator $L1$ for varying input powers. (b) A comparison of up and down frequency sweeps for the first resonance of pillar $L1$ (closest to the excitation source) at a rf input power of 20 dBm. The frequency shift linked to the hysteresis cycle appears at about 70.9 MHz.

Figure 7

Typical phase states for the two proposed excitation schemes. The measurements are taken at a drive power of 20 dBm. (a) The symmetric mode at 68.3 MHz and (b) the antisymmetric mode at 71.5 MHz for a transverse excitation. (c) An illustration of rotating polarization states obtained for one of the two resonators in the case of a longitudinal excitation ($f=71.24\mathrm{MHz}$) and (d) for both resonators, with opposite handedness, for an excitation frequency of 72.9 MHz. Movies of the corresponding out-of-plane resonator displacements are provided as Supplemental Movies 1–4 in the Supplemental Material [48].