We report on the experimental demonstration of self-sustaining feedback oscillators referenced to ultrathin molybdenum disulfide (MoS2) nanomechanical resonators vibrating in the ∼10 to 20 MHz range. Based on comprehensive open-loop characterization of MoS2 resonators with dynamic ranges up to 85 dB, self-sustaining oscillators are constructed by incorporating the MoS2 resonators into an optoelectronic feedback circuitry. The prototyped MoS2 self-sustaining oscillators generate stable radio frequency waveforms with frequency stability (measured in Allan deviation) down to ∼2 × 10−5 and phase noise mainly limited by electronic thermal noise. Beyond self-sustaining oscillations, we demonstrate feedback cooling of thermomechanical motion of a bilayer (2L) MoS2 resonator from 300 K to 255 K by tuning the phase in the feedback, suppressing or “squashing” the noise level of the system.

Vibrating micro/nanoelectromechanical systems (M/NEMS) are ubiquitous and essential for a range of important scientific explorations and critical technological applications. In fundamental studies, M/NEMS resonators have enabled probing sub-nanometer scale objects,1,2 leading to new discoveries and tremendous advances in scientific instrumentation and metrology. Toward energy-efficient radio frequency (RF) signal processing and communication technologies, resonant M/NEMS have been explored as ultralow-power frequency-selecting elements and other building blocks.3,4 Meantime, M/NEMS have prevailed in resonant sensing as their resonance frequencies are engineered to be exceptionally responsive to minuscule events or perturbations, enabling detection of ultrasmall quantities such as mass and force, attaining near yocto-gram (10−24 g)5 and single-spin6 sensitivities. Moreover, engineering M/NEMS in the quantum regime has been explored by cryogenic cooling to millikelvin7 or suppressing thermal noise, thus lowering effective mode temperature (e.g., via sideband cooling8) toward realizing functionalities of quantum devices (e.g., quantum memory9).

Advances in resonators, from MEMS to NEMS, have often been achieved by miniaturizing the devices and increasing resonance frequencies while engineering the tradeoffs to maintain levels of other key performance metrics.10 Recently, van der Waals (vdW) layered materials [e.g., graphene and molybdenum disulfide (MoS2)] have emerged as an atomically thin platform for creating highly miniaturized NEMS. In addition to excellent electrical (e.g., electron mobility of μ = 140 000 cm2 V−1 s−1 for graphene11 and μ = 850 cm2 V−1 s−1 for MoS212 at room temperature) and optical (e.g., unusual optical conductivity in graphene,13 bandgap and photon interactions depending on number of layers in MoS214) properties, such atomic layer crystals intrinsically possess excellent mechanical properties (e.g., Young's modulus EY ∼ 1 TPa for graphene15 and EY ∼ 270 GPa for MoS2,16 ultrahigh strain limit of ∼25% or the same order for many such atomic layer membranes), making them attractive candidates for vibratory NEMS. In particular, semiconducting layered materials such as MoS2 have adequate properties, including higher refractive index,17,18 piezoresistivity,19 and piezoelectricity,20,21 facilitating reading out ultrasmall motions of NEMS. In exploiting these advantages in vdW semiconductors, MoS2 NEMS resonators have been studied,22–26 exhibiting intriguing and encouraging performance including excellent frequency tunability of 25% and broad dynamic range (DR) up to 108 dB.23 Their potential for resonant sensing of physical stimuli, including pressure24,25 and γ-ray radiation,26 has been examined.

To advance MoS2 NEMS in their functionalities and applications, it is desirable to build self-sustaining oscillators and exploit advantages over passive resonators (e.g., real-time frequency tracking).27,28 With a proper feedback loop, mechanical motions of NEMS resonators become autonomous, enabling self-sustaining oscillators with stable alternating current (AC) output signals while only supplied by direct current (DC) power. In oscillators built on miniaturized resonators, characteristics such as DR,23,29 nonlinearity,30 additive noise, and frequency fluctuations31 have considerable effects on performance. Accordingly, careful monitoring of device motion from thermomechanical noise up to nonlinearity is required to understand such NEMS oscillators. Moreover, feedback control has also been employed to explore cooling of micro/nanomechanical resonant modes32–35 and to approach quantum ground state.34 

In this Letter, we describe self-sustaining MoS2 NEMS oscillators and feedback cooling by employing optoelectronic feedback control (Fig. 1). Prior to closed-loop implementation, comprehensive characteristics and performance of MoS2 resonators are investigated by utilizing ultrasensitive interferometric signal transduction.22,23 Upon realizing stable self-sustaining oscillators referenced to the MoS2 NEMS, frequency instability and phase noise performance are examined. Furthermore, feedback cooling of resonance mode is demonstrated for a bilayer (2L) MoS2 resonator.

FIG. 1.

MoS2 NEMS self-sustaining oscillator and feedback cooling. (a) Optoelectronic feedback loop with a bilayer (2L) MoS2 NEMS resonator. White-dashed line circle in the optical microscopy image indicates suspended region of the MoS2 resonator. Scale bar is 3 μm. (b) Illustration of self-sustaining oscillation (blue line) and feedback cooling (red line) with different feedback phase conditions. Green-dashed line represents the thermomechanical noise spectrum of the resonator.

FIG. 1.

MoS2 NEMS self-sustaining oscillator and feedback cooling. (a) Optoelectronic feedback loop with a bilayer (2L) MoS2 NEMS resonator. White-dashed line circle in the optical microscopy image indicates suspended region of the MoS2 resonator. Scale bar is 3 μm. (b) Illustration of self-sustaining oscillation (blue line) and feedback cooling (red line) with different feedback phase conditions. Green-dashed line represents the thermomechanical noise spectrum of the resonator.

Close modal

Suspended MoS2 resonators are fabricated by mechanical exfoliation of bulk MoS2 crystal onto 290 nm SiO2-on-Si substrate with lithographically defined microtrenches.22 The devices are annealed at 250 °C for 0.5–3 h to clean surface adsorbates. Device #1 consists of a ∼60% covered MoS2 diaphragm with a diameter of d ≈ 5.3 μm and thickness of t ≈ 20 nm, while device #2 (d ≈ 5.3 μm and t ≈ 31 nm) and device #3 [d ≈ 3 μm and t ≈ 1.4 nm, bilayer (2L)] are in fully covered drumhead geometry. The thicknesses of devices #1 and #2 are confirmed by atomic force microscopy (AFM) measurements, while that of device #3 is determined by measured photoluminescence (PL) signatures of 2L MoS2.

The MoS2 resonators are first carefully characterized in open-loop scheme, by using an optical interferometry.23 For driven resonances, an amplitude-modulated 405 nm laser is employed for photothermal excitation of device resonance motion, with its ∼5 μm spot focused on the substrate at ∼5 μm away from the MoS2 device and with its modulation frequency and amplitude controlled by an RF signal (provided by a network analyzer). A 633 nm laser is focused onto the center of the device with a spot size of ∼1 μm to measure resonance motion. The laser power is limited below 0.25 mW for the 405 nm laser and 0.35 mW for the 633 nm laser to avoid laser heating. All measurements are performed in moderate vacuum (∼10 mTorr).

Figure 2 shows the measured resonance characteristics of the MoS2 resonators. Both undriven thermomechanical noise and driven resonances are carefully examined. Without external driving force, measured total displacement spectral density is

Sx,total1/2(ω)=(Sx(ω)+Sx,sys)1/2=[4kBTω0MeffQ1(ω2ω02)2+ω02ω2/Q2+Sx,sys]1/2.
(1)
FIG. 2.

Open-loop characteristics of MoS2 resonators. Both thermomechanical noise and photothermally driven resonances are investigated. (a) Driven resonance of device #1 with driving voltage of vdrv = 10 mV. Blue and green symbols are measured amplitude and phase, respectively. (b) Measured resonance of device #1 at vdrv = 110 mV, showing clear Duffing nonlinearity. Hysteresis and bifurcation are observed by sweeping frequency upward (blue line) and downward (red line). (c) Dynamic range (DR) calibration for device #1. Driven responses are measured through vdrv = 10 to 110 mV with a 10 mV step. Measured (d) driven (vdrv = 10 mV) resonance and (e) undriven thermomechanical noise of device #2. Red dashed lines in (a), (d), and (e) are fitting curves to a damped harmonic resonator model. (f) DR calibration of device #2, with driven resonances at vdrv = 10 to 90 mV with a 10 mV step.

FIG. 2.

Open-loop characteristics of MoS2 resonators. Both thermomechanical noise and photothermally driven resonances are investigated. (a) Driven resonance of device #1 with driving voltage of vdrv = 10 mV. Blue and green symbols are measured amplitude and phase, respectively. (b) Measured resonance of device #1 at vdrv = 110 mV, showing clear Duffing nonlinearity. Hysteresis and bifurcation are observed by sweeping frequency upward (blue line) and downward (red line). (c) Dynamic range (DR) calibration for device #1. Driven responses are measured through vdrv = 10 to 110 mV with a 10 mV step. Measured (d) driven (vdrv = 10 mV) resonance and (e) undriven thermomechanical noise of device #2. Red dashed lines in (a), (d), and (e) are fitting curves to a damped harmonic resonator model. (f) DR calibration of device #2, with driven resonances at vdrv = 10 to 90 mV with a 10 mV step.

Close modal

Here, kB is the Boltzmann's constant, T is the temperature, Meff is the effective mass, ω0 is the angular frequency, Q is the quality factor, Sx is the power spectral density of thermomechanical motion, and Sx,sys is the equivalent displacement noise power of the measurement system. Device #1 shows a fundamental resonance frequency of f0 = ω0/(2π) ≈ 9.78 MHz with Q ≈ 520, while device #2 has f0 ≈ 19.95 MHz with Q ≈ 430. At sufficiently high driving voltages, the MoS2 resonators exhibit stiffening Duffing nonlinearity with hysteresis [Fig. 2(b)]. The measured resonance in the voltage domain is converted to the displacement domain by using a transduction responsivity of the optical interferometry obtained by calibrating the measured thermomechanical noise spectrum.22 From the measured linear to nonlinear behavior of the devices [Figs. 2(c) and 2(f)], the critical amplitude23,29 is determined to be ac ≈ 2.9 nm for device #1 and ac ≈ 2.1 nm for device #2. We note that all the measured amplitude signals in Fig. 2 are root mean square (RMS) values, so are the ac values determined from the measurements above. The intrinsic DR of the resonator, set by its onset of nonlinearity (1 dB compression point below ac) over its thermomechanical noise (in dB), is directly obtained by using the analytical model DR20log(0.75ac/SxΔf) (Refs. 23 and 29), where Sx1/2 is the measured thermomechanical noise [blue curves at the bottom of Figs. 2(c) and 2(f)] and Δf = 1 Hz is the measurement bandwidth. The measured intrinsic DR values are DR ≈ 76 dB for device #1, and DR ≈ 85 dB for device #2.

Based on the open-loop calibration and quantification, we build self-sustaining MoS2 NEMS oscillators using an optoelectronic feedback circuitry [Fig. 3(a)]. In addition to the apparatus used in the open-loop two-port calibration, we have introduced electronic components in the loop. The 633 nm optical signal modulated by the resonator motion is first transduced to the electronic domain by a photodetector (PD), and it passes through a low noise amplifier (LNA), a voltage-controlled phase shifter, and a bandpass filter; all of which carefully calibrated before being incorporated into the feedback. The signal is then converted back to the optical domain with the amplitude modulated 405 nm laser and fed into the MoS2 resonator to complete the closed-loop feedback system. To satisfy the Barkhausen criterion, gain of the LNA, phase ϕps of the phase shifter, and modulation depth of the 405 nm laser are carefully adjusted. Initially, the amplified thermomechanical noise [green curves in Figs. 3(b) and 3(c)] is measured after the splitter without feeding the signal into the 405 nm laser, and it is ∼46 dB above the original thermomechanical noise [blue curves in Figs. 3(b) and 3(c)] measured after the photodetector. This comparison provides the total gain of the electronic feedback, which includes the gain of the LNA and insertion losses from the phase shifter and the bandpass filter.

FIG. 3.

Demonstration of self-sustaining MoS2 NEMS feedback oscillators. (a) Optoelectronic feedback circuitry. (b) and (c) Undriven thermomechanical noise after the photodetector (PD) (blue curves) and after the power splitter (green curves) in the open loop, and self-oscillation spectra in the closed loop (red curves), for (b) device #1 and (c) device #2, respectively; all measured by the spectrum analyzer in (a). Time-domain waveforms of (d) device #1 and (e) device #2. Red-dashed lines are fitting curves to sinusoidal waveforms.

FIG. 3.

Demonstration of self-sustaining MoS2 NEMS feedback oscillators. (a) Optoelectronic feedback circuitry. (b) and (c) Undriven thermomechanical noise after the photodetector (PD) (blue curves) and after the power splitter (green curves) in the open loop, and self-oscillation spectra in the closed loop (red curves), for (b) device #1 and (c) device #2, respectively; all measured by the spectrum analyzer in (a). Time-domain waveforms of (d) device #1 and (e) device #2. Red-dashed lines are fitting curves to sinusoidal waveforms.

Close modal

Upon satisfying the Barkhausen criterion and closing the feedback loop, prominent signatures of self-sustaining oscillators are observed: linewidth narrowing and amplitude boosting [Figs. 3(b) and 3(c)]. Output of the oscillators is measured in both frequency domain and time domain. The thermomechanical noise of the resonators (measured in open loop) and their corresponding self-sustaining oscillator output spectra (measured in closed loop) are plotted together using the same Δf = 1 Hz in Figs. 3(b) and 3(c). Following the same convention,27,28 we obtain a linewidth narrowing (compression) ratio of Δoscres ≈ 1/8 for device #1, where Δres and Δosc are the linewidth values (i.e., full-width at half-maximum (FWHM) in power signal) for the resonance and the oscillator output spectrum, respectively. This measured linewidth compression corresponds to effective Q of Qeff = ω0/(2πΔosc) ≈ 4100. For device #2, Δoscres ≈ 1/9 and Qeff ≈ 3800 are attained.

The measured results are further analyzed to find the operation points of the MoS2 resonators in their oscillators. Panels (d) and (e) of Fig. 3 show the measured time-domain oscillation waveforms from two NEMS oscillators built upon devices #1 and #2, respectively. The stable time-domain sinusoidal waveforms are recorded using an oscilloscope [Fig. 3(a)]. The RMS signal amplitude of the waveform from the oscillator referenced to device #1 is vosc ≈ 177 mV, at ∼73 dB higher signal level compared with that of its thermomechanical noise. Since the DR of device #1 is ∼76 dB, the oscillation is sustained at ∼3 dB below the onset of nonlinearity of device #1, translating to x ≈ 1.6 nm RMS displacement. This operation point in the displacement domain corresponds to a power required to sustain resonator device #1's motion, P=2πf0keffx2/Q=0.64 pW, where keff is the effective stiffness. This suggests the required RF signal power to sustain the resonator's stable oscillation at the signal ceiling (onset of nonlinearity) of its intrinsic DR would be PC = 1.28 pW. Meantime, for the oscillator, the amplitude limit or overall gain control is provided by the optoelectronic feedback loop.

The performance of the MoS2 NEMS oscillator is then characterized by measuring its frequency stability. Figure 4(a) displays the “instantaneous” fluctuations and drift of the output frequency of the oscillator referenced to device #1 measured by using a frequency counter [see Fig. 3(a)] with a time interval of τ = 10 ms. In addition to short-term frequency fluctuations, the oscillation frequency slowly drifts downward. This long-term trend might be the result of possible noise processes such as temperature variations and surface adsorbate fluctuations on the resonator, consistent with previous observations in MoS2 resonators.26 Based on the measured frequency traces, Allan deviation36 is calculated using σA(τA)=[12(N1)i=1N((f¯i+1f¯i)fosc)2]1/2, where f¯i is the measured average frequency in the ith discrete time interval of τA, to evaluate the frequency stability of the MoS2 oscillator. As shown in Fig. 4(b), for the oscillator referenced to device #1, σA2×105 is attained in averaging time τA < 20 ms and σA7×105 for averaging time τA > 200 ms.

FIG. 4.

Frequency stability and phase noise performance of the MoS2 feedback oscillator referenced to resonator device #1. (a) Real-time oscillator output frequency recorded by a frequency counter with τ = 10 ms. (b) Frequency stability expressed in Allan deviation computed from the data trace in (a). (c) Measured phase noise of the MoS2 NEMS oscillator. Red-dashed line indicates the 1/f2 power law of one segment of the data. Green-dashed line shows calculated phase noise lower limit set by thermomechanical noise and intrinsic DR of the MoS2 resonator. Purple-dashed line is estimated from the operation point of the resonator and its thermomechanical noise.

FIG. 4.

Frequency stability and phase noise performance of the MoS2 feedback oscillator referenced to resonator device #1. (a) Real-time oscillator output frequency recorded by a frequency counter with τ = 10 ms. (b) Frequency stability expressed in Allan deviation computed from the data trace in (a). (c) Measured phase noise of the MoS2 NEMS oscillator. Red-dashed line indicates the 1/f2 power law of one segment of the data. Green-dashed line shows calculated phase noise lower limit set by thermomechanical noise and intrinsic DR of the MoS2 resonator. Purple-dashed line is estimated from the operation point of the resonator and its thermomechanical noise.

Close modal

We then examine phase noise performance of the MoS2 oscillator using a phase noise measurement module in the spectrum analyzer. As shown in Fig. 4(c), three distinct behaviors are observed in the phase noise spectrum of the oscillator referenced to resonator device #1: phase noise slowly decreases in the offset frequency range of ∼102 to ∼103 Hz; it decreases with a 1/f2 power law from ∼103 to ∼105 Hz, suggesting contribution from thermal (white) noise; it flattens out as offset frequency goes beyond 105 Hz. The lower limit of phase noise [green dashed line in Fig. 4(c)] is set by thermomechanical noise and intrinsic DR of the MoS2 resonator,27L(f)=10log[(kBT/(2PCQ2))(fosc/f)2]. Compared with the phase noise estimated from the operation point [purple-dashed line in Fig. 4(c)] and thermomechanical noise of the resonator, the measured phase noise is mostly ∼6–15 dBc/Hz higher in ∼103 to ∼105 Hz offset frequency range, which is attributed to the noise contribution from the feedback circuitry, additive noise and frequency fluctuations in the resonator beyond its thermomechanical noise.31 Further engineering of the feedback circuitry and improvement in the MoS2 NEMS resonators should enable better phase noise performance of the feedback oscillators.

Furthermore, we turn to show cooling of the MoS2 resonator mode via the feedback control. Feedback cooling in optomechanical resonators has been demonstrated to reduce their resonance mode temperatures and to suppress or “squash” noise levels in the resonance band.32–35 To explore such effects on the MoS2 resonators, we first measure the closed-loop oscillation spectra [with the spectrum analyzer and closed-loop system in Fig. 3(a)] with varying feedback phase setting ϕps via the voltage-controlled phase shifter in the feedback circuit. Panels (a) and (b) in Fig. 5 show the measured output of the feedback loop referenced to resonator device #1 by changing the feedback phase. We first set ϕps = 254.5°, which meets the Barkhausen criterion, thus establishing the self-sustaining oscillator. When ϕps = 162.0° or 7.8°, shifted ∼±90° away from the self-sustaining oscillation condition, half side of spectra exhibits a peak, while the other half shows a dip [Fig. 5(a)], demonstrating the effects of vector addition of two phasors with varying phase difference, between the amplified thermomechanical noise and the background level of the closed-loop system. At ϕps = 88.4°, about 180° opposite phase of that in the self-sustaining oscillation condition [Fig. 5(c)], the spectrum exhibits a “squashing” dip into below the background level.

FIG. 5.

MoS2 oscillator with feedback cooling. (a) Spectra measured from the closed-loop feedback system referenced to resonator device #1 with different feedback phase settings. (b) Color map of measured spectra with varying feedback phase. The phase shifter set values of the measured spectra in (a) are marked using dashed lines. (c) Feedback cooling of the closed-loop system referenced to device #3 (2L MoS2). Blue data trace is open-loop thermomechanical noise, with blue dashed line showing the fitting to Eq. (1). Green data trace is closed-loop spectrum; green-dashed line is the fitting to Eq. (2), yielding feedback gain g = 0.27 and mode temperature Tmode = 255 K. (d) Estimated mode temperature. Blue dashed line in (d) shows the calculation result from Eq. (3), and green symbol marks the measured feedback gain and temperature extracted from (c).

FIG. 5.

MoS2 oscillator with feedback cooling. (a) Spectra measured from the closed-loop feedback system referenced to resonator device #1 with different feedback phase settings. (b) Color map of measured spectra with varying feedback phase. The phase shifter set values of the measured spectra in (a) are marked using dashed lines. (c) Feedback cooling of the closed-loop system referenced to device #3 (2L MoS2). Blue data trace is open-loop thermomechanical noise, with blue dashed line showing the fitting to Eq. (1). Green data trace is closed-loop spectrum; green-dashed line is the fitting to Eq. (2), yielding feedback gain g = 0.27 and mode temperature Tmode = 255 K. (d) Estimated mode temperature. Blue dashed line in (d) shows the calculation result from Eq. (3), and green symbol marks the measured feedback gain and temperature extracted from (c).

Close modal

The mode temperature of the MoS2 resonator is further analyzed according to the equipartition theorem. Figure 5(c) shows measured thermomechanical noise spectrum density of the 2L MoS2 resonator (device #3) without (blue data trace) and with feedback cooling (green data trace), which has f0 ≈ 18.62 MHz and Q ≈ 50 calibrated in open-loop measurement before applying the feedback. With feedback cooling, the resonator's displacement spectral density measured in closed loop is32,33

Sx+sys1/2(ω)={4kBTω0QMeff+[(ω02ω2)2+ω02ω2/Q2]Sx,sys(ω02ω2)2+(1+g)2ω02ω2/Q2}1/2,
(2)

where g is the feedback cooling gain. By fitting measured noise spectral density with feedback cooling [green trace in Fig. 5(c)] to Eq. (2), the feedback cooling gain is determined to be g = 0.27. The mode temperature Tmode of the 2L MoS2 resonator is then32,33

TmodeT1+g+Meffω034kBQ(g21+g)Sx,sys.
(3)

Assuming T is the room temperature (∼300 K), and the mode temperature of device #3 with g = 0.27 is Tmode = 255 K [Fig. 5(d)]. We find that the minimum achievable mode temperature of device #3 is Tmode,min = 252 K at g = 0.38. Above this gain, the noise from the measurement system Sx,sys is fed back to the resonator, which elevates the mode temperature. Based on Eq. (3), it is clear that smaller Meff and ω0, and higher Q, help reduce heating effects from Sx,sys. Feedback cooling has mainly been demonstrated in optomechanical resonators thus far, operating in kHz to low MHz frequency range32–35 to lower Tmode,min. In the 2L MoS2 NEMS, thanks to ultrasmall Meff originated from atomically thin structure, feedback cooling is still effective in higher frequency range. With much higher Q factors of 2D NEMS at cryogenic temperatures,37,38 feedback cooling of 2D NEMS can be more effective, and it could help prepare 2D NEMS in quantum ground state.

In conclusion, we have demonstrated self-sustaining MoS2 NEMS oscillators using optoelectronic feedback circuitry. By satisfying the Barkhausen criterion, ∼10-fold enhancement of Q factors and stable sinusoidal output signals around ∼10 MHz and ∼20 MHz have been achieved. The frequency stability of the oscillators, measured in Allan deviation, falls in the 10−5 to 10−4 range over 10 ms to 100 s averaging time. The measured phase noise performance is limited by extrinsic noise sources in the optoelectronic feedback circuitry. Feedback cooling of the resonance mode is achieved by tuning the feedback phase, resulting in suppression of the resonance motion from 300 K down to 255 K. These findings herein may help facilitate further engineering of 2D NEMS for real-time sensing and feedback control in emerging classical and quantum applications.

We thank the financial support from the NSF CAREER Award (Grant No. ECCS-2015708) and EFRI ACQUIRE Award (Grant No. EFMA-1641099).

The authors have no conflicts to disclose.

The data that support the findings of this study are available from the corresponding author upon reasonable request.

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