Abstract
Since the discovery of the electron, the accurate detection of electrical charges has been a dream of the scientific community. Owing to some remarkable advantages, micro/nanoelectromechanical systembased resonators have been used to design electrometers with excellent sensitivity and resolution. Here, we demonstrate a novel ultrasensitive charge detection method utilizing nonlinear coupling in two micromechanical resonators. We achieve singleelectron charge detection with a high resolution up to 0.197 ± 0.056 \({\mathrm{e}}/\sqrt {{\mathrm{Hz}}}\) at room temperature. Our findings provide a simple strategy for measuring electron charges with extreme accuracy.
Introduction
Highly sensitive electrometers have been a focus of research for more than a century and can be used in many diverse applications, such as mass spectrometry^{1}, surface charge analysis^{2}, particle detection of nuclear studies^{3}, and various applications of aerosol science^{4}. owing to their low cost, fast response, high accuracy, and ability to be manufactured in batches^{5}, micro/nanoelectromechanical system (M/NEMS)based resonators have been used to design electrometers with excellent sensitivity and resolution^{6}. To date, researchers have developed various highresolution charge sensors utilizing selfassembled quantum dots^{7}, MEMS vibrating reeds^{8}, carbon nanotubes^{9}, mode localization^{10}, etc. However, some limits, including the requisite ultralow environmental temperature^{6,7,9,11}, a complicated mechanical structure^{8}, the required linear dynamic response^{6,10,12} or unachievable realtime detection^{6,7,8,9,10,11,13}, will inevitably hinder their practical application. owing to the size effect^{14}, microresonators are more easily excited into the nonlinear regime^{15}. Coupling between individual resonators can also lead to complex nonlinear behavior^{16,17}. The exploitation of nonlinear phenomena^{18,19,20,21} to improve performance has recently attracted significant attention, including mass sensing in terms of coupled nonlinear MEMS resonators^{22} and novel signal amplification schemes through a bifurcation topology^{23}. Clearly, bifurcation exists widely in nonlinear systems^{24,25}, and it is worthwhile to excavate the potential of nonlinear phenomena for practical applications. The use of multiple resonators also has the advantage of improving commonmode rejection capabilities^{26}. Here, we show that the internal resonance of two coupled nonlinear microresonators can significantly enhance the resolution of an electrometer. Ultrasensitive charge detection with a resolution of 0.197 ± 0.056 \({\mathrm{e}}/\sqrt {{\mathrm{Hz}}}\) at room temperature is achieved. The proposed device has a simple structure and can provide realtime measurements.
Materials and methods
Fabrication of the microresonator
The basic structure of the electrometer proposed here consists of two identical siliconbased microresonators, as shown in Fig. 1a. Each microresonator is designed as a widely used structure called a doubleended tuning fork that is 350 μm long, 20 μm wide, and 25 μm thick. Specific sizes can be found in Supplement Material S1. A siliconbased chip is attached to a chip carrier. Figure 1b shows a comparison of a chip and a coin. The microresonators are fabricated through a commercial silicononinsulator (SOI)–MUMPs micromachining process^{27}, as shown in Fig. 1c.
Measurement scheme
All measurements are tested in a vacuum chamber at a pressure below 3 Pa at room temperature. The experimental environment is shown in Supplement Material S2.
The openloop measurement circuit shown in Fig. 2a is built to characterize this coupled system. As a common detection method, piezoresistive readout^{28,29,30} is used throughout all the tests because of its unique advantages, such as higher sensitivity and fewer commonmode signal components. An electric voltage \(V_{DC} + V_{AC}\,{\mathrm{cos}}({\Omega} \tilde t)\) is loaded to drive resonator R1 to vibrate. DC voltage ±V_{D} is applied to both sides of actuated resonator R1 to generate a motional current caused by a timevarying electrical resistance due to the piezoresistive effect. Thus, the transverse displacement of the resonant beam can be detected clearly. A network analyzer (Agilent E5061B) performs the frequency sweep operation by outputting sinusoidal motivation and collecting the vibration amplitude and phase. DC voltage V_{C} is loaded to one side of resonator R2, which generates a coupling voltage V_{C} across the coupling parallel plates. V_{C} is accurately controlled through a SourceMeter (KEITHLEY 2400) with a highstability output.
Figure 2b displays a closedloop circuit based on the phaselocked loop^{31} (PLL) to track the resonant frequency for realtime testing. A PLL keeps the output signal synchronized with a reference input signal in both frequency and phase. More precisely, the PLL is simply a servo system that controls the phase of its output signal in such a way that the phase error between the output phase and the reference phase reduces to a minimum. In this circuit, the transverse displacement of resonator R1 is first extracted by a differential circuit. After the differential operation and phase and amplitude control, the resulting signal is fed back to drive resonator R1 for vibration. The feedback driving force keeps resonator R1 in oscillation by compensating for the energy dissipation. In closedloop tests, the resonators are electrically actuated, sensed and embedded using a digital lockedin amplifier (LIA, Zurich Instruments HF2LI). The coupling voltage V_{C} loaded on the body of resonator R2 is a combination of the polarization voltage V_{P} and dynamic voltage signal V_{S}. Owing to the coupling interaction, the output frequency of R1 will shift as V_{C} varies, which implies a possible way to perform dynamic voltage signal detection by measuring the change in oscillation frequency of R1.
Feedthrough cancellation
Capacitive parasitic feedthrough is an impediment that is inherent to all electrically interfaced MEMS resonators^{32}. The feedthrough signal always distorts the real amplitude and phase response of a resonator. Hence, the feedthrough signal has to be properly eliminated.
In the openloop experiment, we collect a response signal mixed with a feedthrough signal (with V_{DC} on) and a pure feedthrough signal (with V_{DC} off) through a frequency sweep. The feedthrough signal can be eliminated by using the program codes provided in Supplement Material S7. In the closedloop tests, a differential operation is used to remove the feedthrough signal in real time.
Results
The basic mechanism of highresolution charge detection
In openloop measurement, we obtain the amplitudefrequency responses of R1 for varying coupling voltage V_{C}. The corresponding peak frequency with increasing V_{C} is plotted in Fig. 3a as the green dots. An 1895 Hz discontinuous jump of the peak frequency is observed owing to the nonlinear coupling of two resonators when V_{C} reaches a critical value V_{C} = 3.4 V. The inset of Fig. 3a shows the amplitudefrequency curves of R1 for V_{C} below and above the critical value, which further demonstrates this discontinuous phenomenon.
Figure 3b shows the amplitude–frequency curves for uniformly increasing V_{C} above the critical value. It is shown that the peak frequency of R1 varies continuously with V_{C} again once V_{C} crosses the critical value. Figure 3c plots the peak frequency of R1 as a function of V_{C}, which reveals an extremely linear relation between the peak frequency shift and V_{C} with a fitting R^{2} coefficient of 0.9999. Figure 3d shows the response curves of R1 when the coupling voltage varies with a very small step of 0.01 V, where linear variation can still be detected clearly. According to \({\mathrm{{\Delta} }}Q = C{\mathrm{{\Delta} }}V_C\), the equivalent variation in charge ΔQ for small voltage step ΔV_{C} = 0.01 V is calculated as ΔQ = 129.7E3 fC (~810 electrons), where C = 0.01297 pF is the capacitance between two sensing electrodes (the fringe effect is considered as shown in Supplement Material S3). The linear dependence of the frequency shift on the coupling voltage provides a highresolution method for charge detection. This device implements charge detection as follows: the input charges change the coupling voltage V_{C}, which leads to a linear shifting of the peak resonant frequency of R1 that can be tracked in real time by the PLL. Meanwhile, as shown in Fig. 3c, the coupling voltage V_{C} has a linear range of more than 11 V; thus, the electrometer has a broad charge sensing region of up to ~1000000 electrons according to \({\mathrm{{\Delta} }}Q = C{\mathrm{{\Delta} }}V_C\).
Numerical simulation
According to the measured response curves, R1 and R2 exhibit strong Duffinglike nonlinearity. Considering the coupling between these two resonators, a coupled Duffing model of the microsystem is formulated as follows^{33}:
where m_{1} and m_{2} are the equivalent masses of R1 and R2, respectively; \(\tilde x\) and \(\tilde y\) are the equivalent transverse displacements of R1 and R2, respectively; c_{1} and c_{2} are the equivalent viscous damping coefficients; k_{1} and k_{2} are the equivalent linear stiffness coefficients; Γ_{1} and Γ_{2} are the equivalent cubic nonlinear stiffness coefficients; S is the area of the driving or coupling electrode; \({\it{\epsilon }}_r\) is the dielectric constant; d is the initial spacing between electrodes; V_{DC} and V_{C} are the DC polarization voltages for driving and coupling, respectively; and V_{AC} and Ω are, respectively, the amplitude and frequency of the AC driving voltage.
Utilizing Taylor series expansion and ignoring highorder coupling terms, the nondimensional form of Eq. (1) is obtained^{26}:
where \(x = \frac{{\tilde x}}{d}\) and \(y = \frac{{\tilde y}}{d}\) are the nondimensional displacements; \(\omega _1 = \sqrt {k_1/m_1}\) and \(\omega _2 = \sqrt {k_2/m_2}\) are the natural resonant frequencies, with \(\omega _1 \approx \omega _2\); \(p = \omega _2/\omega _1 > 1\); \(t = \omega _1\tilde t\); \(Q = \frac{{c_1}}{{m_1\omega _1}} \approx \frac{{c_2}}{{m_2\omega _1}}\) is the quality factor; \(\gamma = \frac{{{\Gamma} _1d^2}}{{m_1\omega _1^2}} \approx \frac{{{\Gamma} _2d^2}}{{m_2\omega _1^2}}\) is the normalized cubic nonlinear stiffness coefficient; \(f = \frac{{S{\it{\epsilon }}_rV_{AC}V_{DC}}}{{m_1\omega _1^2d^3}}\) is the normalized amplitude of the electrostatic excitation; \(\omega = \frac{{\Omega} }{{\omega _1}}\) is the normalized electrostatic excitation frequency; and \(\alpha = \frac{{S{\it{\epsilon }}_rV_C^2}}{{m_1\omega _1^2d^3}} \approx \frac{{S{\it{\epsilon }}_rV_C^2}}{{m_2\omega _1^2d^3}}\) is the strength of the electrostatic coupling.
Generally, Eq. (2) cannot be solved analytically owing to its complicated solution branches. Thus, numerical simulations of the amplitude–frequency curves of this system are obtained and plotted in Fig. 4 through a numerical method called the timedomain collocation method^{34} with parameters extracted from the experiments (details of the procedure are displayed in Supplement Material S5). A bifurcation point P_{1} is observed when the coupling voltage V_{C} reaches a threshold, which can exactly explain the discontinuous jump phenomenon of the peak frequency in Fig. 3a. The bifurcation point is owing to the internal resonance between these two resonators. With increasing coupling strength, more energy is transferred from actuated R1 to coupled R2. When the coupling voltage reaches a threshold, R1 does not have enough energy to maintain a large amplitude, thus causing the amplitude of R1 to drop to the lower branch, which leads to the discontinuous phenomenon. The inset (1) of Fig. 4 shows that the frequency at the bifurcation point decreases linearly as the coupling voltage increases with a fitting R^{2} coefficient of 0.997, which agrees with the experimental results in Fig. 3c.
In the above studies, the capacitance C between electrodes is assumed to be constant. However, owing to the relative motion of the two sensing electrodes, the capacitance C may vibrate. The effect of the vibration of C on charge detection has been investigated analytically and proved to be negligible (see Supplement Material S4).
Realtime charge detection method
A closedloop circuit is set up for realtime charge detection, utilizing the above linear dependence of the peak frequency on the coupling voltage, as shown in Fig. 2b. Here, coupling voltage V_{C} is a combination of polarization voltage V_{P} and dynamic voltage signal V_{S}. V_{P} is a fixed voltage that is set above the critical value as V_{P} = 4 V (>3.4 V) to ensure that the resonators work in the linear region, whereas V_{S} is a step voltage signal used to imitate the changing voltage caused by external charge. In Fig. 5a, we plot the step response when the coupling voltage V_{C} varies with a tiny step 0.001 V (equivalent variation in charge is 12.97E3 fC). The variation in the peak frequency of R1 is clearly identified, which indicates a much higher (10fold) resolution for charge variation than in the openloop measurement. The timedomain response for the step voltage signal can be seen in Supplement Material S6.
The resolutions of charge detection are normally calculated in two different ways by using the Allan deviation or noise floor. To obtain the Allan deviation and noise floor, we need to track a time series of the peak frequency of the amplitudefrequency curves of R1. The frequency fluctuation data are recorded at a fixed sample time of 1/225 s for a duration of 300 s using an LIAbased PLL.
The Allan deviation σ_{A} is a common indicator to evaluate the frequency stability^{35}, which is given by the frequency fluctuations averaged over an integration time τ and can be expressed as^{21}
where \(\overline {f_p} _i\) are the relative frequency fluctuations averaged over the ith discrete integration of τ. Figure 5b plots the Allan deviation σ_{A} of the tracked peak frequency of R1 of one test as a function of the integration time τ, from which the minimum Allan deviation is observed to be 2.89 ppb. Then, the resolution R of charge detection is calculated to be 2.19E4 fC according to \(R = C \cdot \delta f/K_{sf}\) (\(\delta f\) is the frequency fluctuation of resonator R1, which equals the Allan deviation \(\sigma _A\) multiplied by the characteristic frequency (i.e., f_{p}); K_{sf} represents the scaling factor of the peak frequency shift to coupling voltage V_{C}, which is 40.60 Hz/V from Fig. 5a), which is equivalent to the charge provided by ~1.3 electrons. To pursue a more credible presentation with a statistical property, multiple independent tests were completed with a mean value of 4.66 ppb and a standard deviation of 1.96 ppb of the minimum Allan deviation. Thus, the resolution R is calculated to be 3.53E4 ± 1.45E4 fC, which equals the charge provided by ~2.1 ± 0.9 electrons.
The noise floor here is experimentally achieved by fast Fourier transformation analysis of the time series data of the recorded frequency^{36}. Figure 5c is the noise floor of the peak frequency of one test with a maximum value of 0.215 \({\mathrm{e}}/\sqrt {{\mathrm{Hz}}}\). Similarly, multiindependent tests are completed, and a mean value of 0.197 \({\mathrm{e}}/\sqrt {{\mathrm{Hz}}}\) and standard deviation of 0.056 \({\mathrm{e}}/\sqrt {{\mathrm{Hz}}}\) of the maximum noise floor are obtained. Therefore, a resolution of 0.197 ± 0.056 \({\mathrm{e}}/\sqrt {{\mathrm{Hz}}}\) is obtained. Singleelectron charge detection at room temperature is thus achievable using this device.
Conclusions
In summary, we presented a new charge detection method with ultrahigh resolution utilizing two coupled nonlinear MEMS resonators. This kind of detection concept has not been reported in previous studies. Beyond unfavorable effects, nonlinearity can greatly enhance the resolution of an electrometer, which sheds light on the considerable potential of nonlinear applications. We explored the complex dynamic response of the aforementioned system through a numerical method. These numerical results demonstrated the emergence of the discontinuous phenomenon in this system and further showed the linear relation between the peak frequency shift and the coupling voltage. We also built a realtime closedloop measurement circuit for charge detection.
References
Taylor, S., Tindall, R. F. & Syms, R. R. A. Silicon based quadrupole mass spectrometry using microelectromechanical systems. J. Vac. Sci. Technol. 19, 557–562 (2001).
Horenstein, M. N. Measuring isolated surface charge with a noncontacting voltmeter. J. Electrost. 35, 203–213 (1995).
Krueger, F. & Larson, J. Chipmunk IV: development of and experience with a new generation of radiation area monitors for accelerator applications. Nucl. Instrum. Methods Phys. Res. Sect. A: Accelerators, Spectrometers, Detect. Assoc. Equip. 495, 20–28 (2002).
Han, B. et al. A novel bipolar charger for submicron aerosol particles using carbon fiber ionizers. J. Aerosol Sci. 40, 285–294 (2009).
Wang, X. F. et al. Effect of nonlinearity and axial force on frequency drift of a Tshaped tuning fork microresonator. J. Micromech. Microeng. 28, 125012 (2018).
Cleland, A. N. & Roukes, M. L. A nanometrescale mechanical electrometer. Nature 392, 160 (1998).
Kiyama, H. et al. Singleelectron charge sensing in selfassembled quantum dots. Sci. Rep. 8, 13188 (2018).
Lee, J., Zhu, Y. & Seshia, A. Room temperature electrometry with SUB10 electron charge resolution. J. Micromech. Microeng. 18, 025033 (2008).
Zhou, X. & Ishibashi, K. Single charge detection in capacitively coupled integrated single electron transistors based on singlewalled carbon nanotubes. Appl. Phys. Lett. 101, 123506 (2012).
Zhang, H. et al. A highsensitivity micromechanical electrometer based on mode localization of two degreeoffreedom weakly coupled resonators. J. Microelectromech. Syst. 25, 937–946 (2016).
Bunch, J. S. et al. Electromechanical resonators from graphene sheets. Science 315, 490–493 (2007).
Lee, J. E. Y., Bahreyni, B. & Seshia, A. A. An axial strain modulated doubleended tuning fork electrometer. Sens. Actuators A: Phys. 148, 395–400 (2008).
Chen, D. et al. Sensitivity manipulation on micromachined resonant electrometer toward high resolution and large dynamic range. Appl. Phys. Lett. 112, 013502 (2018).
Agarwal, M. et al. Scaling of amplitudefrequencydependence nonlinearities in electrostatically transduced microresonators. J. Appl. Phys. 102, 074903 (2007).
Kaajakari, V. et al. Nonlinear limits for singlecrystal silicon microresonators. J. Microelectromech. Syst. 13, 715–724 (2004).
Westra, H. J. R. et al. Nonlinear modal interactions in clampedclamped mechanical resonators. Phys. Rev. Lett. 105, 117205 (2010).
Lifshitz, R. & Cross, M. C. Response of parametrically driven nonlinear coupled oscillators with application to micromechanical and nanomechanical resonator arrays. Phys. Rev. B 67, 134302 (2003).
Boales, J. A., Mateen, F. & Mohanty, P. Optical wireless information transfer with nonlinear micromechanical resonators. Microsyst. Nanoengineering 3, 17026 (2017).
Leuthold, J., Koos, C. & Freude, W. Nonlinear silicon photonics. Nat. Photonics 4, 535 (2010).
Murali, K. et al. Reliable logic circuit elements that exploit nonlinearity in the presence of a noise floor. Phys. Rev. Lett. 102, 104101 (2009).
Antonio, D., Zanette, D. H. & López, D. Frequency stabilization in nonlinear micromechanical oscillators. Nat. Commun. 3, 806 (2012).
Baguet, S. et al. Nonlinear dynamics of micromechanical resonator arrays for mass sensing. Nonlinear Dyn. 95, 1203–1220 (2019).
Karabalin, R. B. et al. Signal amplification by sensitive control of bifurcation topology. Phys. Rev. Lett. 106, 094102 (2011).
Kacem, N. & Hentz, S. Bifurcation topology tuning of a mixed behavior in nonlinear micromechanical resonators. Appl. Phys. Lett. 95, 183104 (2009).
Kacem, N. et al. Stability control of nonlinear micromechanical resonators under simultaneous primary and superharmonic resonances. Appl. Phys. Lett. 98, 193507 (2011).
Thiruvenkatanathan, P. et al. Enhancing parametric sensitivity in electrically coupled MEMS resonators. J. Microelectromech. Syst. 18, 1077–1086 (2009).
Cowen A, et al. SOIMUMPs design handbook. MEMSCAP Inc. 2011.
Li, C. S. et al. Differentially piezoresistive sensing for CMOSMEMS resonators. J. Microelectromech. Syst. 22, 1361–1372 (2013).
Lin, A. H. et al. Methods for enhanced electrical transduction and characterization of micromechanical resonators. Sens. Actuators A: Phys. 158, 263–272 (2010).
Zhang, W., Zhu, H. & Lee, J. E. Y. Piezoresistive transduction in a doubleended tuning fork SOI MEMS resonator for enhanced linear electrical performance. IEEE Trans. Electron Devices 62, 1596–1602 (2015).
Hsieh, G. C. & Hung, J. C. Phaselocked loop techniquesA survey. IEEE Trans. Ind. Electron. 43, 609–615 (1996).
Lee, J. E. Y. & Seshia, A. A. Parasitic feedthrough cancellation techniques for enhanced electrical characterization of electrostatic microresonators. Sens. Actuators A: Phys. 156, 36–42 (2009).
Agrawal, D. K., Woodhouse, J. & Seshia, A. A. Modeling nonlinearities in MEMS oscillators. IEEE Trans. Ultrason. Ferroelectr. Frequency Control 60, 1646–1659 (2013).
Dai, H. H., Schnoor, M. & Atluri, S. N. A simple collocation scheme for obtaining the periodic solutions of the duffing equation, and its equivalence to the high dimensional harmonic balance method: subharmonic oscillations. Comput. Model. Eng. Sci. 84, 459 (2012).
Sansa, M. et al. (2016). Frequency fluctuations in silicon nanoresonators. Nat. Nanotechnol. 11, 552 (2016).
Zhao, C. et al. (2017). Experimental observation of noise reduction in weakly coupled nonlinear MEMS resonators. J. Microelectromech. Syst. 26, 1196–1203 (2017).
Acknowledgements
This work was financially supported by the National Natural Science Foundation of China (11772293, 51575439, 51421004), National Key R & D Program of China (2018YFB2002303), Key Research and Development Program of Shaanxi Province (2018ZDCXLGY0203), and 111 Project (B12016). We also appreciate the support from the Collaborative Innovation Center of HighEnd Manufacturing Equipment and the International Joint Laboratory for Micro/Nano Manufacturing and Measurement Technologies.
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X. Wang developed the concept, performed the experiments, and wrote the manuscript. R. Huan developed the analytical model and cowrote the manuscript. D. Pu and X. Wei verified the experiments, provided technical guidance, and commented on the manuscript. All the authors contributed through scientific discussions.
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Wang, X., Wei, X., Pu, D. et al. Singleelectron detection utilizing coupled nonlinear microresonators. Microsyst Nanoeng 6, 78 (2020). https://doi.org/10.1038/s41378020001924
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DOI: https://doi.org/10.1038/s41378020001924
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