Fundamental Sensitivity Limitations of Nanomechanical Resonant Sensors Due to Thermomechanical Noise

Abstract

Nanomechanical resonators are used as high performance sensors of physical stimuli such as force and mass changes. Any such physical stimulus produces a shift in the resonance frequency of the nanomechanical structure, which can be measured accurately by using a feedback system that locks the frequency of a signal generator to the resonance. Closed-loop frequency tracking is the most prevalent technique in the fields of nanomechanical sensors and non-contact atomic force microscopy. Ultimate performance of sensors is limited by various nonideal effects such as temperature variations, radiation, electromagnetic interference, and noise arising from inherent physical mechanisms. Here, we consider the noise performance of nanomechanical resonant sensors, which has so far eluded explanation with conflicting results reported in the literature. We present a precise theory for these ubiquitous sensors based on nanomechanical resonators under feedback in order to decipher the fundamental sensitivity limitations due to thermomechanical noise. The results we obtain, when the performance is limited by the thermomechanical noise of the resonator, are in complete agreement with the ones from stochastic simulations. Our findings shed light on recent results in the literature and resolve a critical problem regarding the frequency noise of nanomechanical sensors under feedback. Our results have applications in nanomechanics, atomic force microscopy, microwave and suspended microchannel resonators.

Frequency fluctuations in nanomechanical silicon nitride string resonators

“…Finally, closed-loop measurements are performed, where it can be shown that τ r can be increased through variation of the loop bandwidth. All Allan deviation measurements are corroborated by computations based on a theoretical model [26] and good agreement is observed both for open loop and closed loop.…”

“…This was attributed to a flattening of the phase noise spectrum at low frequencies. A thorough theoretical investigation of the same scenario by Demir et al however found no Q dependence of the frequency stability [26].…”

Method to Determine the Closed-Loop Precision of Resonant Sensors From Open-Loop Measurements

“…Four imprecision sources affecting the system are modelled through different inputs: signal source noise V src (s), thermomechanical noise F thm (s), resonance frequency fluctuations Ω n (s) and detector noise V msr (s), whose descriptions can be found in Appendix II. To simplify the analysis, all the signals are considered approximately harmonic, and the system is linearized around its operating point ω a = ω n [21], [24]. This results in the phase-space system shown in Fig.…”

“…a closed-loop scheme) is needed to continually adjust the driving frequency, such that it stays near the resonance frequency. Two types of closed-loop schemes for driving resonant sensors can be found in the literature: direct feedback oscillators [18], [19], and phase-locked loops (PLL) [20], [21]. The former need automatic gain controllers to avoid non-linear behaviour at large amplitudes that would be detrimental for the frequency precision, or even more sophisticated controllers to operate the resonator at optimal points of the non-linear regime [22], [23].…”

“…The former need automatic gain controllers to avoid non-linear behaviour at large amplitudes that would be detrimental for the frequency precision, or even more sophisticated controllers to operate the resonator at optimal points of the non-linear regime [22], [23]. The latter rely on phase detection to control the driving frequency, requiring fine tuning the phase detection bandwidth and the controller parameters [21], [24]. Despite requiring extra complexity in the implementation, closed-loop schemes are preferred in most practical sensing scenarios, where significant resonance frequency drifts occur over time, and/or a high number of sensing events (e.g.…”