Fundamental sensitivity limitations of nanomechanical resonant sensors due to thermomechanical noise

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.