^{1}, Arvind Raman

^{2}, Jeffrey Rhoads

^{2}and Ronald G. Reifenberger

^{1}

### Abstract

In this work, parametric noise squeezing and parametric resonance are realized through the use of an electronic feedback circuit to excite a microcantilever with a signal proportional to the product of the microcantilever's displacement and a harmonic signal. The cantilever's displacement is monitored using an optical lever technique. By adjusting the gain of an amplifier in the feedback circuit, regimes of parametric noise squeezing/amplification and the principal and secondary parametric resonances of fundamental and higher order eigenmodes can be easily accessed. The exceptionally symmetric amplitude response of the microcantilever in the narrow frequency bandwidth is traced to a nonlinear parametric excitation term that arises due to the cubic nonlinearity in the output of the position-sensitive photodiode. The feedback circuit, working in both the regimes of parametric resonance and noise squeezing, allows an enhancement of the microcantilever's effective quality-factor (Q-factor) by two orders of magnitude under ambient conditions, extending the mass sensing capabilities of a conventional microcantilever into the sub-picogram regime. Likewise, experiments designed to parametrically oscillate a microcantilever in water using electronic feedback also show an increase in the microcantilever's effective Q-factor by two orders of magnitude, opening the field to high-sensitivity mass sensing in liquid environments.

A.R. thanks the National Science Foundation (NSF) for partial financial support of this research under Grant No. CMMI- 0700289. We would like to thank Luis Colchero of Nanotec Electronica for the design and construction of the parametric circuit used throughout this study.

I. INTRODUCTION

II. THEORY

A. Parametric resonance

B. Parametric amplification

III. EXPERIMENTAL CONSIDERATIONS

IV. EXPERIMENTAL RESULTS

A. Characterization of microcantilever response in the parametric resonance and parametric noise squeezing regimes

B. Mass sensing in air

1. Mass sensing in the parametric resonance regime

2. Mass sensing in the parametric amplification regime

C. Parametric amplification in liquids

V. CONCLUSIONS

### Key Topics

- Normal modes
- 19.0
- Photodiodes
- 13.0
- Atomic force microscopy
- 9.0
- Microelectromechanical systems
- 8.0
- Optical resonators
- 7.0

## Figures

(a) A schematic of the circuit used to implement the parametric feedback for the microcantilever excitation. (b) The instability tongue for the parametrically excited microcantilever in the parameter space of *G/G* _{th} vs. Ω. The solid dots are values of *G* used in the computer simulations. The inset shows an enlarged view of the situation for *G* > *G* _{th}.

(a) A schematic of the circuit used to implement the parametric feedback for the microcantilever excitation. (b) The instability tongue for the parametrically excited microcantilever in the parameter space of *G/G* _{th} vs. Ω. The solid dots are values of *G* used in the computer simulations. The inset shows an enlarged view of the situation for *G* > *G* _{th}.

Numerical simulation of parametric resonance of a microcantilever at different feedback gain (*G*) values for *G* > *G* _{th}. These simulations include the nonlinear parametric term, *δ x* ^{3}cos (Ωt), with *δ* = 1 × 10^{−7} m^{−2}.

Numerical simulation of parametric resonance of a microcantilever at different feedback gain (*G*) values for *G* > *G* _{th}. These simulations include the nonlinear parametric term, *δ x* ^{3}cos (Ωt), with *δ* = 1 × 10^{−7} m^{−2}.

(a) Numerically simulated time history of the microcantilever fluctuations at different feedback gains, *G* = 0.0*G* _{th}, *G* = 0.5*G* _{th}, and *G* = 0.8*G* _{th} while excited by white noise. (b) Normalized PSD of thermal vibration of the cantilever “enhanced” by parametric amplification at different feedback gains below *G* _{th}. The maxima of peaks at *G* = 0.5*G* _{th} and *G* = 0.8*G* _{th} were 4.5 and 600 times larger than the maximum value of the peak at *G* = 0.0*G* _{th}, respectively.

(a) Numerically simulated time history of the microcantilever fluctuations at different feedback gains, *G* = 0.0*G* _{th}, *G* = 0.5*G* _{th}, and *G* = 0.8*G* _{th} while excited by white noise. (b) Normalized PSD of thermal vibration of the cantilever “enhanced” by parametric amplification at different feedback gains below *G* _{th}. The maxima of peaks at *G* = 0.5*G* _{th} and *G* = 0.8*G* _{th} were 4.5 and 600 times larger than the maximum value of the peak at *G* = 0.0*G* _{th}, respectively.

A schematic diagram of the circuit used to implement parametric excitation. The components are identified using their part number. The various filters were constructed using standard RC networks.

A schematic diagram of the circuit used to implement parametric excitation. The components are identified using their part number. The various filters were constructed using standard RC networks.

Identification of the form of dominant nonlinearity in the system. (a) The amplitude of 1st harmonic response signal of a cantilever measured by the photodiode plotted as a function of the excitation amplitude applied to the dither piezo. The amplitude of the 1st harmonic signal is less than the direct proportionality represented by a dotted straight line. (b) Plot of the amplitudes of the 2nd and 3rd harmonic response signal vs. the amplitude of the 1st harmonic signal. Together these results clearly show that the photodiode provides a softening quadratic and cubic nonlinearity which when multiplied with a harmonic signal and fed back to the cantilever lead to a dominant cubic parametric term in the system which dominates the cantilever response in the parametric resonance regime.

Identification of the form of dominant nonlinearity in the system. (a) The amplitude of 1st harmonic response signal of a cantilever measured by the photodiode plotted as a function of the excitation amplitude applied to the dither piezo. The amplitude of the 1st harmonic signal is less than the direct proportionality represented by a dotted straight line. (b) Plot of the amplitudes of the 2nd and 3rd harmonic response signal vs. the amplitude of the 1st harmonic signal. Together these results clearly show that the photodiode provides a softening quadratic and cubic nonlinearity which when multiplied with a harmonic signal and fed back to the cantilever lead to a dominant cubic parametric term in the system which dominates the cantilever response in the parametric resonance regime.

In (a), the parametric resonance response of a microcantilever in air is compared to the conventional resonance peak obtained by driving the cantilever base using a dither piezo. In this experiment, the parametric gain is set to *G* = 1.03**G* _{th}. The effective Q-factor of the microcantilever is modified from a value of 350 to 3000. In (b), the Primary Parametric Resonance (PPR, blue), Secondary Parametric resonance (SPR, red) and Tertiary Parametric Resonance (TPR, green) peaks of a microcantilever excited in the first eigenmode are shown when the excitation frequency (*f* _{ d }) is set to be 2*f* _{0}, *f* _{0} and 2*f* _{0}/3, respectively. The gain for each peaks are different and set above the threshold values. The inset shows a zoomed in SPR peak. The natural frequency *f* _{0} of the first eigenmode is 156.45 kHz. In (c), the PPR (blue), SPR (red) and TPR (green) peaks of a microcantilever excited in the second eigenmode are shown when the excitation frequency (*f* _{ d }) is set to be 2*f* _{2}, *f* _{2} and 2*f* _{2}/3, respectively. The inset shows a zoomed in TPR peak. The natural frequency *f* _{2} of the second eigenmode of the microcantilever is 974.4 kHz. The gain for each peak is different and in each case was set above the threshold value

In (a), the parametric resonance response of a microcantilever in air is compared to the conventional resonance peak obtained by driving the cantilever base using a dither piezo. In this experiment, the parametric gain is set to *G* = 1.03**G* _{th}. The effective Q-factor of the microcantilever is modified from a value of 350 to 3000. In (b), the Primary Parametric Resonance (PPR, blue), Secondary Parametric resonance (SPR, red) and Tertiary Parametric Resonance (TPR, green) peaks of a microcantilever excited in the first eigenmode are shown when the excitation frequency (*f* _{ d }) is set to be 2*f* _{0}, *f* _{0} and 2*f* _{0}/3, respectively. The gain for each peaks are different and set above the threshold values. The inset shows a zoomed in SPR peak. The natural frequency *f* _{0} of the first eigenmode is 156.45 kHz. In (c), the PPR (blue), SPR (red) and TPR (green) peaks of a microcantilever excited in the second eigenmode are shown when the excitation frequency (*f* _{ d }) is set to be 2*f* _{2}, *f* _{2} and 2*f* _{2}/3, respectively. The inset shows a zoomed in TPR peak. The natural frequency *f* _{2} of the second eigenmode of the microcantilever is 974.4 kHz. The gain for each peak is different and in each case was set above the threshold value

(a) Power spectral density (PSD) plot of microcantilever intrinsic vibration without any parametric feedback (*G* = 0) in air. In (b) and (c), the PSD of the microcantilever's intrinsic vibrations at gain ratios (*G/G* _{th}) of 0.6 and 0.9, respectively. (d) Plot of effective Q-factor vs. gain ratio in the regime of parametric amplification. The solid line is a guide to the eye.

(a) Power spectral density (PSD) plot of microcantilever intrinsic vibration without any parametric feedback (*G* = 0) in air. In (b) and (c), the PSD of the microcantilever's intrinsic vibrations at gain ratios (*G/G* _{th}) of 0.6 and 0.9, respectively. (d) Plot of effective Q-factor vs. gain ratio in the regime of parametric amplification. The solid line is a guide to the eye.

(a) Power spectral density (PSD) plot of microcantilever intrinsic vibration without any parametric feedback (*G* = 0) in water. In (b) and (c), the PSD of the microcantilever's intrinsic vibrations at gain ratios (*G/G* _{th}) of 0.5 and 0.86, respectively. (d) Plot of effective Q-factor vs. gain ratio in the regime of parametric amplification. The solid line is a guide to the eye.

(a) Power spectral density (PSD) plot of microcantilever intrinsic vibration without any parametric feedback (*G* = 0) in water. In (b) and (c), the PSD of the microcantilever's intrinsic vibrations at gain ratios (*G/G* _{th}) of 0.5 and 0.86, respectively. (d) Plot of effective Q-factor vs. gain ratio in the regime of parametric amplification. The solid line is a guide to the eye.

The stability of the parametric electronic feedback system is demonstrated. In (a), a time dependent study of the resonance when the microcantilever is resonated in air. The abscissa represents half of the excitation frequency. The parametrically resonant microcantilever shows a drift of 3 Hz in 1 h. This drift is equivalent to the error of 6% in the mass sensitivity. In (b), a second experiment in which a cantilever is driven parametrically under water. The abscissa represents half of the excitation frequency. The data are for forward and backward sweeps and demonstrate the non-hysteretic behavior of the resonance.

The stability of the parametric electronic feedback system is demonstrated. In (a), a time dependent study of the resonance when the microcantilever is resonated in air. The abscissa represents half of the excitation frequency. The parametrically resonant microcantilever shows a drift of 3 Hz in 1 h. This drift is equivalent to the error of 6% in the mass sensitivity. In (b), a second experiment in which a cantilever is driven parametrically under water. The abscissa represents half of the excitation frequency. The data are for forward and backward sweeps and demonstrate the non-hysteretic behavior of the resonance.

In (a), an optical image of a hygroscopic particle of CaCl_{2} attached to the apex of a microcantilever. In (b), a comparison of the resonance peaks from a parametrically and conventionally resonating microcantilever, as humidity is increased by introducing air in the chamber originally filled with dry nitrogen. As air is introduced into the AFM chamber, a downshift in the parametric resonance peak is observed whereas there is no visible shift in the conventional resonance peak. For conventional excitation, the abscissa represents the excitation frequency whereas for parametric excitation, the abscissa represents half of the excitation frequency. The inset shows the measured frequency shift in more detail. In (c), the results of a second time-dependent study which illustrates the frequency shift of a microcantilever with an attached hygroscopic particle of CaCl_{2} (natural frequency 47.3 kHz) as the chamber is cycled between dry N_{2} and air.

In (a), an optical image of a hygroscopic particle of CaCl_{2} attached to the apex of a microcantilever. In (b), a comparison of the resonance peaks from a parametrically and conventionally resonating microcantilever, as humidity is increased by introducing air in the chamber originally filled with dry nitrogen. As air is introduced into the AFM chamber, a downshift in the parametric resonance peak is observed whereas there is no visible shift in the conventional resonance peak. For conventional excitation, the abscissa represents the excitation frequency whereas for parametric excitation, the abscissa represents half of the excitation frequency. The inset shows the measured frequency shift in more detail. In (c), the results of a second time-dependent study which illustrates the frequency shift of a microcantilever with an attached hygroscopic particle of CaCl_{2} (natural frequency 47.3 kHz) as the chamber is cycled between dry N_{2} and air.

The normalized PSD of a PDMS coated microcantilever before and after exposure to toluene vapor. The red data points highlight the measured response of a parametrically amplified microcantilever operating under nitrogen. The natural Q-factor of the microcantilever is 400. With *G* set to 0.3G_{th}, the effective Q-factor increases to 1000. The blue data points highlight the measured response of a parametrically amplified microcantilever after exposure to toluene vapor. The solid lines are best fit to the data.

The normalized PSD of a PDMS coated microcantilever before and after exposure to toluene vapor. The red data points highlight the measured response of a parametrically amplified microcantilever operating under nitrogen. The natural Q-factor of the microcantilever is 400. With *G* set to 0.3G_{th}, the effective Q-factor increases to 1000. The blue data points highlight the measured response of a parametrically amplified microcantilever after exposure to toluene vapor. The solid lines are best fit to the data.

Comparison of the resonance peaks of the parametrically (in black) and the conventionally (in blue) resonating microcantilever in water. The microcantilever is oscillated piezoelectrically at the base. For conventional excitation, the abscissa represents the excitation frequency whereas for parametric excitation abscissa represents half of the excitation frequency. Both the excitation peaks are also compared with the thermal spectrum shown in red solid circles. The conventional excitation shows multiple peaks whereas parametrically excited microcantilever shows a sharp resonance peak.

Comparison of the resonance peaks of the parametrically (in black) and the conventionally (in blue) resonating microcantilever in water. The microcantilever is oscillated piezoelectrically at the base. For conventional excitation, the abscissa represents the excitation frequency whereas for parametric excitation abscissa represents half of the excitation frequency. Both the excitation peaks are also compared with the thermal spectrum shown in red solid circles. The conventional excitation shows multiple peaks whereas parametrically excited microcantilever shows a sharp resonance peak.

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