Nonlinear resonators for ultrasensitive mass detection

Nonlinear nanomechanical resonators have been explored to enable the ultrasensitive mass detection that is of particular interest in chemical and biological sensing. The operating principle is based on mass or force loading of mechanical devices driven into nonlinear oscillations with large amplitudes. The sensing part of the transducers is an on-chip array of batch-fabricated, singly clamped beams excited into highly nonlinear resonance by mounting them on a miniature piezoelectric element. By combining such resonators with an experimental setup that does not require complex electronic components, we have been able to detect femtogram-level mass changes. (This level of performance was previously achievable only with much smaller resonators that are very difficult to employ in a practical device.)

Resonating nanoscale and microscale structures (similar to the microfabricated cantilevers used in atomic force microscopy) can detect external stimuli, such as mass or force, with high precision and sensitivity (see Figure 1). Conventional implementation of this concept, including earlier work by us,¹ relies on determining changes in the resonance frequency caused by loading, which can be closely approximated by a harmonic (i.e., linear) oscillator. Because this approach limits the maximum amplitudes at which the device can operate below the onset of nonlinearity, it means measuring and analyzing oscillation amplitudes often as small as 10⁻¹⁰m. This requirement in turn entails sophisticated low-noise optical and electronic components, such as position-sensitive detectors, lock-in amplifiers, and phase-locked loops. It also impedes practical application of the devices. Finally, microscale and nanoscale mechanical resonators are susceptible to energy dissipation and to thermal noise, both of which tend to randomize their activity in a manner similar to Brownian motion. As a result, their resonances are characterized by a relatively wide ‘bell’-shaped (Lorentzian) curve with a frequency uncertainty of the center position typically in the range of Δf/f~10⁻³to 10⁻⁵.

Figure 1. (a) Optical microscopy, (b) scanning electron microscopy, and (c) optical profilometry images of cantilever-array structures used as mass-sensitive resonators operating in a nonlinear regime. Stacks of silicon nitride and silicon oxide used as resonator structural materials provide high optical reflectivity, mechanical robustness, and controllable intrinsic stress.

Large oscillation amplitudes, in contrast, can be measured more accurately using simple means. However, at these amplitudes, all mechanical systems become nonlinear, and their resonance behavior differs drastically from that of linear systems. In particular, mass and force loading of nonlinear resonators cannot be analyzed using the conventional methods and instruments established for their linear counterparts. Importantly, mass loading of a nonlinear resonator cannot be determined by fitting its resonance curve to a Lorentzian distribution or by measuring the output frequency of self-oscillating circuitry based on a phase-locked loop. The method we present here takes advantage of nonlinear resonance and provides a very efficient way to detect mass or force loading of a mechanical resonator without needing to accurately measure its oscillation amplitude or any analog signal processing.

We mount a chip arrayed with cantilever resonators on a piezotransducer and drive it to amplitudes of up to several tens of micrometers, sufficient to deflect a laser beam focused on the resonator by an angle of tens of degrees. This corresponds to an ~2cm span of the laser spot projected on a screen positioned 5cm from the oscillating structure. A sweep generator is programmed to make a continuously repeated linear frequency sweep in the vicinity of the bifurcation point. Every time the frequency reaches that point, a sharp drop in the oscillation amplitude is observed and detected with a simple spot photodetector. By continuously repeating frequency sweeps and using a sharp change in the photodiode current as a digital time mark to log when the amplitude collapsed, we are able to monitor the bifurcation frequency with accuracy better than 50mHz for a structure resonating at ~140kHz. This corresponds to relative frequency accuracy of 3.5×10⁻⁷, an unprecedented value for linear oscillators with similar parameters. Using our system and experimental setup, we have detected femtogram-level mass changes caused by interaction with a model analyte compound. This level of performance was previously achievable only by using much smaller resonators and sophisticated readout of the nanoscale beam displacements.

In summary, we have described a practical approach to ultrasensitive mass detection with improved immunity to fundamental noise sources as well as to external disturbances. These enhancements should enable low-cost portable devices. Our ongoing and future work will focus on elucidating basic mechanisms of the behavior and rational design of nonlinear oscillators in a wide range of operating frequencies and sizes.