Phase Stochastic Resonance in a Forced Nanoelectromechanical Membrane

Stochastic resonance is a general phenomenon usually observed in one-dimensional, amplitude modulated, bistable systems. We show experimentally the emergence of phase stochastic resonance in the bidimensional response of a forced nanoelectromechanical membrane by evidencing the enhancement of a weak phase modulated signal thanks to the addition of phase noise. Based on a general forced Duffing oscillator model, we demonstrate experimentally and theoretically that phase noise acts multiplicatively, inducing important physical consequences. These results may open interesting prospects for phase noise metrology or coherent signal transmission applications in nanomechanical oscillators. Moreover, our approach, due to its general character, may apply to various systems.

Figure 1

(Top left) A scanning electron microscopic view of the device shows the membrane (thickness of 260 nm and a $10\times 20\mu {\mathrm{m}}^2$ surface) forming the mechanical oscillator (purple) and the interdigitated electrodes (yellow) underneath, at a distance of about 400 nm. (Top right) Finite element model (F.E.M.) simulation of the fundamental mechanical mode under study with enhanced out-of-plane displacement for clarity. (Bottom left) Amplitude $R$ and (bottom right) phase $\mathrm{\Phi }$ spectra of the driven oscillator response in a frequency sweep-up and sweep-down experiment with $V_{\mathrm{ac}}=9\mathrm{V}$, $\mathrm{\Delta }\varphi =0$, and ${\xi }_{\mathrm{RMS}}=0$; the theoretical response is displayed in red dashed lines in both spectra. The driving frequency ${\mathrm{\Omega }}_d/2\pi =2.824\mathrm{MHz}$ (dashed lines) lies close to the linear fundamental mechanical resonance at ${\mathrm{\Omega }}_0/2\pi =2.822\mathrm{MHz}$ (i.e., zero normalized detuning). Normalized detuning is defined as $({\mathrm{\Omega }}_d-{\mathrm{\Omega }}_0)/{\mathrm{\Omega }}_0$.

Figure 2

Simulated (open squares) and experimental (red triangles) polar angle difference between the two stable states for increasing noise strengths. Inset: Experimental polar plots with experimental values of ${\xi }_{\mathrm{RMS}}=0.52\mathrm{rad}$ and ${\mathrm{\Omega }}_d/2\pi =2.824\mathrm{MHz}$.

Figure 3

The response of the system, now driven by a force combining a weak phase modulation and an increasing phase noise is shown. (From left to right) Experimental time traces recorded on a time scale of 300 s of the amplitude $R$ and phase $\mathrm{\Phi }$ of the fundamental mode for increasing noise strengths, with associated experimental and theoretical polar plots. (From upper to lower lines) Evolution of these four panels for increasing standard deviation ${\xi }_{\mathrm{RMS}}$.

Figure 4

(a) Experimental (red triangles) and theoretical (open black squares) spectral amplification in amplitude $R$ as a function of phase noise strength. (b) Experimental (blue circles) and theoretical (open black stars) spectral amplification in phase $\mathrm{\Phi }$ as a function of phase noise strength. Theoretical curves have been obtained by use of Eq. (4). (c) Experimental polar plot for ${\xi }_{\mathrm{RMS}}=0.44\mathrm{rad}$ (maximum amplification), highlighting the shape of the two stable points as well as the directions imprinted by the modulation on the input phase or amplitude. The two dashed gray lines are a guide for the eye, indicating the two distinct amplitude states, and the dotted white line highlights the threshold.