Encapsulated bulk mode microresonators in the megahertz range are used in commercial timekeeping and sensing applications, but their performance is limited by the current state of the art of readout methods. We demonstrate a readout using dispersive coupling between a high-Q encapsulated bulk mode micromechanical resonator and a lumped element microwave resonator that is implemented with commercially available components and standard printed circuit board fabrication methods and operates at room temperature and pressure. A frequency domain measurement of the microwave readout system yields a displacement resolution of 522fm/Hz, which demonstrates an improvement over the state of the art of displacement measurement in bulk-mode encapsulated microresonators. This approach can readily be implemented in cryogenic measurements, allowing for future work characterizing the thermomechanical noise of encapsulated bulk mode resonators at cryogenic temperatures.

Micro- and nanoelectromechanical (M/NEM) resonators are widely used as timing references,1 inertial sensors,2,3 and mass sensors.4,5 Due to their small size, MEM resonators have high resonance frequencies in the kHz to MHz range and low Size, Weight and Power (SWaP) requirements. Resonators using the bulk modes of encapsulated silicon plates have several additional desirable properties, including mechanical quality factors above 1 × 106 (Ref. 6) and environmental isolation,7 making them extremely promising in precision sensing applications.

However, this encapsulation process imposes limitations on the resonator sensing methods that can be used. Piezoelectric sensing provides high signal-to-noise ratio but is challenging to incorporate with high-temperature epitaxial encapsulation and introduces significant losses to the resonator, degrading bandwidth and increasing phase noise.8 Optical sensing methods offer femtometer displacement resolution but require large, high power, external measurement equipment and are incompatible with encapsulated devices that are opaque to visible light.9,10 Capacitive detection relies on coupling between the resonator motion and the field in electrodes placed near the resonator. At higher mechanical frequencies, direct readout can suffer from feedthrough, reducing sensitivity and increasing the noise of the position detection.11 Optomechanical-style readout is an attractive alternative to direct readout, not only offering improvements in readout noise, but also harnessing decades of progress in the field of quantum optomechanics.12–14 

In optomechanics, a mechanically compliant structure modulates the resonant frequency of a higher-frequency electromagnetic mode, typically either a resonant microwave circuit or an optical cavity. This modulation upconverts the position information from the mechanical resonance frequency to the microwave/optical frequency, which has enabled position detection near15,16 and even beyond17 the quantum limit in several systems. The electromagnetic mode can also modify the dynamics of the mechanical resonator, which can cool it to its quantum mechanical ground state18 and generate entangled states shared between the electromagnetic and mechanical mode.19 

While resonator encapsulation is opaque to visible light, vias through the encapsulation layer allow for optomechanical coupling via lower frequency electromagnetic waves in the megahertz to gigahertz regime. This makes encapsulated bulk mode silicon MEMs devices an excellent basis for further cavity optomechanics experiments, offering extremely high, material-property-limited quality factor and frequency products (Qf)20 and easy integration into microwave circuits.

In this Letter, we study an encapsulated bulk mode resonator that is dispersively coupled to a lumped-element microwave resonator. In this configuration, motion of the high-Q Lamé mode resonator modulates the natural frequency of the microwave resonator, inducing sidebands in its frequency response. This system is analogous to the optomechanical system of a Fabry–Pérot cavity with a mechanically compliant mirror, which has been well characterized. Using a microwave homodyne receiver, we use these sidebands to measure the mechanical motion of the bulk mode resonator with state-of-the-art resolution. We derive a model for the dispersive interaction in our system using input–output theory that allows us to predict the sideband amplitude resulting from mechanical motion. We validate this model with experiments on our coupled system and use fitting to extract the key parameters of our system.

Figure 1(a) depicts a schematic for the coupled resonator system. The surface-mount air core inductor on the printed circuit board (PCB) seen in Fig. 1(c) forms a lumped element microwave resonator with the parasitic capacitance of the coil and the encapsulated MEM device connected in parallel. This microwave resonator is capacitively coupled to an on-board coplanar waveguide forming a transmission mode resonator. Figure 1(d) shows the amplitude response of the microwave resonator. Fitting this response to a Lorentzian allows us to extract the resonator natural frequency, ωrf/2π=631.6MHz, intrinsic quality factor, QI=38.7, and loaded quality factor QL=15.0. In this configuration, the resonant frequency of the microwave resonator, ωrf, is a function of the total capacitance between the sense terminal of the mechanical device and ground, Ct, and the total inductance, Lr, and is given by

ωrf=1LrCt(x).
(1)

The total capacitance has a static contribution from the parasitic capacitance of the inductor, circuit board, and device interconnects, Cr, and a position dependent contribution due to the MEM resonator capacitance, Cm(x). The total capacitance can be written as

Ct(x)=Cr+Cm(x).
(2)
FIG. 1.

(a) A schematic of the coupled resonator system and microwave homodyne readout. Motion of the Lamé mode plate resonator, shown in dotted lines, modulates the capacitance between the sense electrode connected to node Vr and ground. This modulation induces sidebands in the resonator response to a probe voltage, Vp, which are measured using a doubly balanced mixer as a phase detector in a homodyne configuration. The amplified mixer output voltage, Vout, is proportional to the mechanical resonator's displacement. A small portion of the resonator signal, Vcal, is measured directly for readout calibration purposes using a directional coupler. (b) A false color micrograph of the Lamé mode resonator showing the drive electrodes in blue and sense electrodes in red. (c) A picture of the printed circuit board (PCB) containing the encapsulated mechanical resonator and lumped element microwave resonator. The two coaxial connector ports on the upper portion of the board correspond to ports one and two in the schematic. One of the three lower ports is used to apply the drive signal. The purple outline denotes the PCB extents in the schematic. (d) A plot of the measured microwave resonator transmission amplitude, |S21|, vs frequency. We plot the measured data (blue dots) vs a fit to a Lorentzian (black dotted line) to determine the natural frequency, ωrf=631.6MHz, intrinsic quality factor, QI=38.7, and loaded quality factor, QL=15.0, of the microwave resonator.

FIG. 1.

(a) A schematic of the coupled resonator system and microwave homodyne readout. Motion of the Lamé mode plate resonator, shown in dotted lines, modulates the capacitance between the sense electrode connected to node Vr and ground. This modulation induces sidebands in the resonator response to a probe voltage, Vp, which are measured using a doubly balanced mixer as a phase detector in a homodyne configuration. The amplified mixer output voltage, Vout, is proportional to the mechanical resonator's displacement. A small portion of the resonator signal, Vcal, is measured directly for readout calibration purposes using a directional coupler. (b) A false color micrograph of the Lamé mode resonator showing the drive electrodes in blue and sense electrodes in red. (c) A picture of the printed circuit board (PCB) containing the encapsulated mechanical resonator and lumped element microwave resonator. The two coaxial connector ports on the upper portion of the board correspond to ports one and two in the schematic. One of the three lower ports is used to apply the drive signal. The purple outline denotes the PCB extents in the schematic. (d) A plot of the measured microwave resonator transmission amplitude, |S21|, vs frequency. We plot the measured data (blue dots) vs a fit to a Lorentzian (black dotted line) to determine the natural frequency, ωrf=631.6MHz, intrinsic quality factor, QI=38.7, and loaded quality factor, QL=15.0, of the microwave resonator.

Close modal

The mechanical resonator is a square, 400 μm wide, 43 μm thick, plate resonator fabricated from single crystal silicon and suspended at the corners with a compliant structure designed to reduce clamping loss via mechanical impedance mismatching.21,22 The resonator is fabricated in a wafer scale encapsulation process that leads to oxide-free, particle-free, low pressure cavities enabling single crystal silicon resonators with high quality factors.23 All measurements are performed with the encapsulated mechanical resonator die, lumped-element microwave resonator, and readout electronics inside a temperature-stabilized oven at 25°C and atmospheric pressure. In this work, we study the resonator's Lamé mode, which due to its volume-conserving (isochoric) property exhibits low thermoelastic dissipation.20 The lumped mass of this mode is 7.96 μg. By applying a periodic drive voltage, Vd, offset by a DC bias, Vb, through a bias tee, we induce a periodic displacement, xm, in the mechanical resonator given by

xm=|Xm(ω)|cos(ωt+ϕm),
(3)

where |Xm(ω)| is the magnitude of the resonator's amplitude response due to the applied drive signal and ϕm is the phase of the resonator. Figure 2 shows |Xm(ω)| and ϕm as the drive signal is swept through the resonant frequency of the mechanical mode. Fitting a Lorentzian to the response yields ωm/2π=10.088MHz and Qm=2.2×106. The motion of the mechanical resonator results in time variance of the capacitance between the sense terminal and ground of the mechanical resonator, Cm, which is given by

Cm(x)=ε0Agγx,
(4)

where ε0 is the permittivity of free space, A is the capacitor area, g is the gap size, and γ is the mode shape transduction factor.24 This modulation of the capacitance results in modulation of the resonant frequency of the microwave resonator, since ωrf depends on Cm, as can be seen in Eq. (1). The magnitude of the modulation is characterized by the single-photon coupling strength, g0, defined as

g0=ωrfx|x=0xzpf,
(5)

where xzpf is the zero point fluctuation of the mechanical mode. The resonant frequency modulation results in a signal at the microwave resonator output at the sum and difference of the frequencies of the microwave resonator and mechanical resonator drive frequencies. These sideband amplitudes from the capacitive modulation are calculated from the current through the microwave resonator capacitance. The current in the microwave resonator, Ir, due to the time variance of the microwave resonator voltage, Vr, and the capacitance, Ct, is

Ir=ddt(CtVr).
(6)

Expanding the total voltage and total capacitance in terms of a sum of the unperturbed value and the slow perturbation induced by the motion of the mechanical resonator:

Ct=C+δC,Vr=Vr0+δV,
(7)

where C is the constant capacitance and δC is the portion of the capacitance varying at ωm. Vr0 is the voltage at ωrf due to microwave drive power and δV is the perturbation at ωm due to the capacitive modulation. Writing the resonator current using these perturbative expansions gives

Ir=ddt(CVr0(t)+CδV(t)+Vr0(t)δC(t)+δV(t)δC(t)).
(8)

The first term describes the current in the resonator at ωrf due to the applied drive power. The second term describes the current induced at ωm due to the perturbation δV interacting with the constant capacitance. The final term is the product of two small modulation amplitudes and will result in current at ωrf±2ωm, which we will ignore in this analysis. The third term creates currents at ωrf±ωm due to the product of modulation of the capacitance and the cavity drive. This current is the sideband signal of interest, and results in forward traveling waves at the output port of the resonator, b2̃. A derivation of this signal can be seen in more detail in the supplementary material. The magnitude of b2̃ is given by

|b2̃(ωrf±ω)|=|S23(ωrf±ωm)|g0|Vr|Lrωrf±ωmωrf3|Xm(ω)|xzpfRr2,
(9)

where Rr=ωrfLrQint is the equivalent loss resistance, and |S23| is the magnitude of the transmission ratio between the MEM device capacitance and microwave resonator output. We sample the resonator response containing these sidebands directly using a directional coupler. By calibrating the coupler, we refer the measured voltage at the coupler port, Vcal, to resonator output power, Pres. Figure 3(a) shows the spectrum of Pres, which displays the carrier tone due to the applied microwave drive at ωrf, and the two sideband signals with amplitude given by Eq. (9) at ωrf±ωm. The total sideband voltage, Vsb, is

Vsb=1Z0(|b2̃(ωrf+ωm)|+|b2̃(ωrfωm)|).
(10)

Fitting to the expression for Vsb in Eq. (10) for various levels of MEM resonator amplitude gives an estimate of the single-photon coupling strength, g0/2π=35.3mHz. This estimation of the single-photon coupling strength is extremely sensitive to parasitic inductance in the PCB and wirebonds used to connect to the MEM resonator, which is not included in the model. Parasitic inductance can modify the transmission of modulated signals from the MEM resonator to the output and can modify the relative circulating power in the capacitor. The homodyne receiver mixes both of the sideband signals down to low frequency using a doubly balanced mixer as a phase detector and amplifies the resulting voltage. The final output voltage, Vout, is given by

|Ṽout(ω)|=GrRr2Z0[|S23(ωrf+ωm)|+|S23|(ωrfωm)]g0|Vr|ωrf2Lr|Xm(ω)|xzpf,
(11)

where Gr is the net gain of the homodyne detector and amplifier. Figure 4 shows a spectrum of the output voltage given by Eq. (11) when a coherent response is induced in the mechanical resonator by applying a bias voltage of Vb=5V and a drive voltage of Vd=30mV at the mechanical resonance frequency. This response is calibrated to units of meters using the known applied drive signal and resonator properties, which result in a displacement amplitude of 4.1 nm. For the details of this calibration, see the supplementary material, Sec. I A. The white noise floor gives the displacement resolution of the readout as 522fm/Hz.

FIG. 2.

The driven amplitude-frequency (a) and phase-frequency (b) response of the Lamé mode mechanical resonator measured using the optomechanical readout depicted in Fig. 1 overlaid with a least squares fit to a Lorentzian response for a drive voltage of 10 mV and a bias voltage of 5 V as a function of offset frequency, Δω=ωωm. The good agreement with a Lorentzian model demonstrates the lack of feedthrough present in the measurement. This fitting allows us to extract the resonant frequency of the mechanical device as ωm/2π=10.1MHz and the mechanical quality factor as Qm=2.2×106. The amplitude is calibrated to be in units of meters using the model detailed in the supplementary material.

FIG. 2.

The driven amplitude-frequency (a) and phase-frequency (b) response of the Lamé mode mechanical resonator measured using the optomechanical readout depicted in Fig. 1 overlaid with a least squares fit to a Lorentzian response for a drive voltage of 10 mV and a bias voltage of 5 V as a function of offset frequency, Δω=ωωm. The good agreement with a Lorentzian model demonstrates the lack of feedthrough present in the measurement. This fitting allows us to extract the resonant frequency of the mechanical device as ωm/2π=10.1MHz and the mechanical quality factor as Qm=2.2×106. The amplitude is calibrated to be in units of meters using the model detailed in the supplementary material.

Close modal
FIG. 3.

(a) The resonator power plotted vs frequency, ω/2π for a mechanical drive voltage of 30 mV and a bias voltage of 5 V that results in an estimated of displacement amplitude of 4.1 nm. The resonator power was sampled with a resolution bandwidth of 5.1 kHz through a directional coupler and referred back to the microwave resonator output port by applying a known calibration. Motion of the mechanical resonator induces sidebands at ±ωm about the carrier tone proportional to the mechanical resonator amplitude. (b) The total sideband amplitude, Vsb, plotted vs normalized mechanical resonator amplitude, |Xm(ω)|/xzpf, for various levels of mechanical resonator drive amplitude. The measured data (open orange circles) are plotted vs a least squares fit to the model in Eq. (10) (solid gray line), which is used to extract an estimated of the single-photon coupling strength, g0/2π=35.3mHz.

FIG. 3.

(a) The resonator power plotted vs frequency, ω/2π for a mechanical drive voltage of 30 mV and a bias voltage of 5 V that results in an estimated of displacement amplitude of 4.1 nm. The resonator power was sampled with a resolution bandwidth of 5.1 kHz through a directional coupler and referred back to the microwave resonator output port by applying a known calibration. Motion of the mechanical resonator induces sidebands at ±ωm about the carrier tone proportional to the mechanical resonator amplitude. (b) The total sideband amplitude, Vsb, plotted vs normalized mechanical resonator amplitude, |Xm(ω)|/xzpf, for various levels of mechanical resonator drive amplitude. The measured data (open orange circles) are plotted vs a least squares fit to the model in Eq. (10) (solid gray line), which is used to extract an estimated of the single-photon coupling strength, g0/2π=35.3mHz.

Close modal
FIG. 4.

The amplitude spectral density (ASD) of the driven mechanical resonator as a function of frequency, ω/2π, for an applied resonant drive of 30 mV measured using our microwave homodyne detector. The response is calibrated to units of meters using the calibration detailed in the supplementary material, Sec. I A. Fitting to the white noise floor gives a displacement resolution of 522fm/Hz.

FIG. 4.

The amplitude spectral density (ASD) of the driven mechanical resonator as a function of frequency, ω/2π, for an applied resonant drive of 30 mV measured using our microwave homodyne detector. The response is calibrated to units of meters using the calibration detailed in the supplementary material, Sec. I A. Fitting to the white noise floor gives a displacement resolution of 522fm/Hz.

Close modal

Using a Zurich instruments HF2LI lock-in amplifier, we can sweep the frequency of the MEM resonator drive voltage, Vd, and measure the amplitude and phase of the output of the microwave homodyne receiver, Vout. Calibrating this response to units of meters using the known applied drive signal gives the amplitude response of the mechanical resonator, which can be seen in Fig. 2. This response, unlike that as measured with direct readout, lacks a direct contribution from the applied drive tone, thereby increasing the stability of clocks and the sensitivity of resonant sensors.

This work demonstrates that encapsulated bulk mode resonators are compatible with an optomechanical-style readout, which offers better noise performance than has been demonstrated with direct readout of encapsulated bulk mode resonators,25 and comparable noise performance to unencapsulated in-plane optical measurement of RF MEMS10 and optomechanical measurements of mg-scale silicon resonators.26 Additionally, these resonators are promising for future cavity optomechanics experiments. The Qf product of 2.2×1013 Hz is comparable with several other leading optomechanics platforms, including SiN nano-resonators27,28 and aluminum drumhead resonators,18 and is over the Qf6×1012Hz minimum requirement for room-temperature quantum optomechanics. In these experiments, the mechanical resonance frequency is slightly smaller than the relaxation rate of the microwave resonator, γrf given by ωrf/2πQL=34.9MHz, putting this system in the unresolved sideband regime. However, further optimization of the microwave circuit such as by improving the microwave performance of the vias through the encapsulation layer and including the use of superconducting microwave resonators, can substantially improve the microwave relaxation rate, potentially placing the system into the resolved sideband regime, an important criteria for exploration of quantum phenomena. These improvements would also reduce the system's sensitivity to parasitic inductance and capacitances and allow for more accurate system characterization. Cryogenic operation also allows for greatly improved rf homodyne detection that will allow for near quantum limited displacement sensitivity.15 

See the supplementary material for a detailed derivation of the sideband signal amplitude, microwave homodyne receiver behavior, and the method used for calibrating the MEM resonator response to units of displacement.

This work was supported by the National Science Foundation (NSF) Collaborative Research Program under Grant Nos. 1662464 and 1662500. Fabrication was performed in nano@Stanford labs, which are supported by the NSF as part of the National Nanotechnology Coordinated Infrastructure under Award No. ECCS-1542152, with support from the Defense Advanced Research Projects Agency's Precise Robust Inertial Guidance for Munitions (PRIGM) Program, managed by Ron Polcawich and Robert Lutwak.

The authors have no conflicts to disclose.

Nicholas E. Bousse: Conceptualization (lead); Data curation (lead); Formal analysis (lead); Investigation (lead); Methodology (equal); Software (lead); Supervision (lead); Validation (lead); Visualization (lead); Writing – original draft (lead); Writing – review and editing (lead). Stephen E. Kuenstner: Formal analysis (supporting); Methodology (equal); Resources (supporting); Writing – original draft (supporting); Writing – review and editing (supporting). James M. Lehto Miller: Writing – review and editing (supporting). Hyun-Keun Kwon: Resources (supporting). Gabrielle D. Vukasin: Resources (supporting). John Daniel Teufel: Methodology (supporting); Supervision (supporting); Writing – review and editing (supporting). Thomas W. Kenny: Funding acquisition (lead); Project administration (lead); Resources (supporting); Supervision (lead); Writing – review and editing (supporting).

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

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Supplementary Material