Radio-frequency optomechanical characterization of a silicon nitride drum

Abstract

On-chip actuation and readout of mechanical motion is key to characterize mechanical resonators and exploit them for new applications. We capacitively couple a silicon nitride membrane to an off resonant radio-frequency cavity formed by a lumped element circuit. Despite a low cavity quality factor (QE ≈ 7.4) and off resonant, room temperature operation, we are able to parametrize several mechanical modes and estimate their optomechanical coupling strengths. This enables real-time measurements of the membrane’s driven motion and fast characterization without requiring a superconducting cavity, thereby eliminating the need for cryogenic cooling. Finally, we observe optomechanically induced transparency and absorption, crucial for a number of applications including sensitive metrology, ground state cooling of mechanical motion and slowing of light.

Introduction

Cavity optomechanics boasts a number of tools for investigating the interaction between radiation and mechanical motion and enables the characterization and development of highly sensitive devices1. Silicon nitride membranes have been fabricated to exhibit very high tensile stress, resulting in high quality factors, and have been used for a number of applications including measurement of radiation pressure shot noise2, optical squeezing3, bidirectional conversion between microwave and optical light4, optical detection of radio waves5,6, microkelvin cooling7 and cooling to the quantum ground state of motion8,9,10.

Radio-frequency (rf) cavities allow for sensitive mechanical readout on-chip5,11,12,13,14. We characterize a silicon nitride membrane at room temperature making use of an off-resonant rf cavity13,14. In our approach, the use of lumped elements greatly simplifies the detection circuit in terms of fabrication and allows the integration on chip with the mechanical oscillator. Our circuit has a lower operation frequency than microwave cavities11,14, and allows for a larger readout bandwidth than previous works13. Also, our cavity allow us to inject noise, effectively increasing the mechanical mode temperature. We are able to detect several modes and extract the quality factor and cavity coupling strength for each of them. When the membrane is driven, we are able to resolve the membrane’s motion in real time. We achieve a sensitivity of \(0.4\,{\rm{pm}}/\sqrt{{\rm{Hz}}}\). A sensitivity of \(4.4\,{\rm{pm}}/\sqrt{{\rm{Hz}}}\) was reported in ref. 14, although it must be noted that these sensitivities cannot be easily compared, due to the much smaller size of their mechanical resonator. We observe optomechanically induced transparency (OMIT) and optomechanically induced absorption (OMIA) on-chip and deep in the unresolved sideband regime, allowing for the characterisation of the membrane’s motion under radiation pressure. OMIT and OMIA are an unambiguous signature of the optomechanical interaction15 and they can be used to slow or advance light16. OMIT has also been proposed as a means to achieve ground state cooling of mechanical motion in the unresolved sideband regime17,18.

Experiment

Our device consists of a high-stress silicon nitride membrane which is 50 nm thick; it has an area of 1.5 mm × 1.5 mm and 90% of this area is metalized with 20 nm of Al. We suspend this membrane over two Cr/Au electrodes patterned on a silicon chip. A dc voltage Vdc = 15 V is applied to electrode 1, with electrode 2 grounded. Measurements were performed at room temperature and at approximately 10−6 mbar. For optomechanical readout and control, electrode 1 is connected to an effective rf cavity. The cavity is realised using an on-chip inductor L and capacitors CD and CM mounted on the sample holder19, in addition to the capacitance formed by the membrane CC. The circuit behaves similarly to a simple LC resonator with total capacitance CT = CC + CP, where CP accounts for the capacitance between the two sides of the antenna and other parasitic capacitances. This circuit acts as a cavity and can be driven by injecting an rf signal to the input (port 1) via a directional coupler in order to induce an optomechanical interaction between the cavity signal and the mechanical motion. In addition port 3 allows injection of an ac signal to directly excite the membrane’s motion. The entire setup forms a three-terminal circuit with input ports 1 and 3 and output port 2 (Fig. 1(a)). We used a vector network analyzer to measure scattering parameters and a spectrum analyzer to measure power spectra. Figure 1(b) shows the scattering parameter (|S21|) as a function of cavity probe frequency fP. The cavity resonance is evident in reflection as a minimum in |S21| with quality factor QE ≈ 7.4.

Figure 1
figure1

(a) Experimental setup. A metalized silicon nitride membrane is suspended over two metal electrodes. Electrode 2 is grounded and electrode 1 is connected to a radio-frequency tank circuit acting as a readout cavity. The cavity is formed from an inductor L = 223 nH and 10 pF fixed capacitors CD and CM. Parasitic capacitances contribute to CM. The role of CM is to improve on the impedance matching between the circuit and the 50 Ohm line. CD controls the coupling to the cavity, i.e. the number of photons entering the cavity. Because these capacitors are larger than CT, they do not strongly affect the cavity frequency. Parasitic losses in the circuit are parameterized by an effective resistance R. To probe the cavity a radio-frequency signal is injected at port 1, passed via a directional coupler, and after reflection received at port 2. The signal is measured using a vector network analyser or spectrum analyser. The membrane’s motion is excited via port 3. Bias resistors allow a dc voltage Vdc to be added to the membrane drive signal. (b) Reflection as a function of frequency. The black curve is a fit to a circuit model. (c) Power spectrum of the signal at port 2 showing the carrier peak at the cavity resonance frequency, applied via port 1, and sidebands corresponding to a frequency modulated tone applied via port 3. The light blue and light green curves correspond to the power spectrum for an excitation fE below (60 kHz) and above (100 kHz) f0 respectively. Sidebands appear due to non-linearities in the circuit. The orange curve corresponds to fE ~ f0. In this case, the sidebands are larger. (d) Higher frequency sidebands in (c) as a function of fE. When fE is close to a mechanical mode of the membrane, a mechanical sideband appears. The four visible mechanical modes are indicated with arrows. Because higher modes produce fainter features, they are not displayed in this graph. Note that the colour scale displays a range of only 25 dBm in order to reveal the weaker features. Insets: zoom-in of the two highest frequency modes displayed. We observe a splitting of the mechanical feature, indicating nearly degenerate modes.

The dependence of the capacitor formed between the electrodes and the metalized membrane CC on the mechanical displacement u leads to coupling between the cavity and the mechanical motion. This coupling is given by \(\frac{d{f}_{{\rm{C}}}}{du}\approx 1/(4\pi \sqrt{{L}{{C}}_{{\rm{T}}}^{3}})\frac{\partial {C}_{{\rm{C}}}}{\partial u},\) where fC is the cavity resonance frequency, CT the total circuit capacitance and u the membrane’s displacement from its equilibrium position. The coupling causes phase shifts of the cavity reflection, allowing us to monitor the membrane’s motion in the unresolved sideband limit. The single-photon coupling strength, which measures the interaction between a single photon and a single phonon, is therefore,

$$\frac{{g}_{0}}{2\pi }\approx \frac{1}{4\pi \sqrt{{L}{{C}}_{{\rm{T}}}^{3}}}\frac{\partial {C}_{{\rm{C}}}}{\partial u}{u}_{{\rm{ZP}}},$$
(1)

where uZP is the amplitude of the membrane’s zero-point motion.

Using a simple circuit model (See Supplemental Material for details of the circuit simulation), we fit |S21| and extract CC ≈ 1.6 pF. Within the parallel plate capacitor approximation, \({C}_{{\rm{C}}} \sim \frac{{\varepsilon }_{0}a}{d},\) where ε0 is the vacuum permittivity, a is the metallised area of the membrane and d is the membrane-electrode gap. From this expression, we extract d ~ 9 μm.

Results

To find the mechanical resonances, we drove the cavity on resonance (fC ≈ 209.2 MHz) with input power PC = 5 dBm at port 1. Meanwhile, we excited the membrane’s motion via port 3, using a sinusoidal signal at frequency fE and with amplitude VM = 3.6 Vrms at electrode 1. In order for the mechanical response to appear broader in the frequency spectrum, facilitating the detection of the mechanical modes, the excitation frequency at port 3 was modulated with a white noise pattern with a deviation of 200 Hz. The power spectrum P of the reflected signal at port 2 shows sidebands at fC ± fE due to non-linearities of the rf circuit giving rise to frequency mixing. The mechanical signal appears when fE is close to a mechanical resonance, and is evident as a pronounced increase in sideband amplitude and width (Fig. 1(c)).

Figure 1(d) shows the sideband at fC ± fE as a function of fE. The fundamental mode frequency f1,1, which we will call f0, is ~77.9 kHz, giving an unresolved sideband ratio of 2πf0/κ ~ 3 × 10−3, with κ = 2πfC/QE the cavity linewidth. As well as the fundamental mode, we observe less strong harmonics fi,j near the expected frequencies. The expected frequencies for higher harmonics theoretically satisfy the ratios \({f}_{i,j}/{f}_{0}=\frac{1}{\sqrt{2}}\sqrt{{i}^{2}+{j}^{2}}\) for integers i and j with symmetric roles as expected for a square membrane. Two of the sidebands are double peaked, evidencing nearly degenerate mechanical modes (insets Fig. 1(d)). The broken degeneracy could be due to imperfections in how the membrane was fabricated and fixed in place or uneven binding/deposition of the Al layer.

The entire set of mechanical resonances can be observed in a single measured power spectrum by driving the cavity at fC and injecting white noise via port 3. White noise excites the motion of the membrane at all frequencies, which is equivalent to raising the effective mechanical temperature. In this way, the root mean square (rms) displacement is increased, thereby facilitating the detection of mechanical modes. The noise signal has a bandwidth of 1 MHz, larger than the spectral width of the mechanical modes, and an amplitude VM = 2.7 Vrms. Figure 2(a) shows the mechanical sidebands at fC + fi,j. To distinguish mechanical sidebands from other parasitic signals, we increase PC until we observe a frequency shift (See Supplemental Material for further details). The fundamental mode of the membrane is at f0 = 78.573 ± 0.002 kHz. We fit each mechanical sideband with a Lorentzian (Fig. 2(b–d)). As in Fig. 1(d), we observe double peaked sidebands (Fig. 2(c,d)).

Figure 2
figure2

(a) Power spectrum as a function of frequency for the higher-frequency sidebands. A tone at fC is applied with input power PC = 5 dBm and white noise is injected via port 3 at VM = 2.7 Vrms at electrode 1. We observe several mechanical modes. The four mechanical modes visible in Fig. 1(d) are indicated with arrows. The mode at ~157 kHz, labelled (2,2), is not visible in Fig. 1(d) due to a higher noise floor than in (a), which arises from a higher resolution bandwidth. Higher modes are not labelled because the deviation from theoretical ratios makes their identification ambiguous. (bd) Zoom in showing the fundamental mode, and selected nearly degenerate modes. The black curves are Lorentzian fits. (e) Demodulated output signal as a function of time for the fundamental mode. The demodulation circuit is shown in the inset. (LO: local oscillator; LPF: low-pass filter). Input power is PC = 12 dBm at fand the white noise is VM = 3.6 Vrms at electrode 1.

The broad cavity bandwidth allowed us to measure the actuated membrane’s motion in real time. We excite the membrane with white noise whilst driving the cavity at fC. In order to record the motion in real time, the cavity output signal is mixed with a local oscillator. The output signal (Fig. 2(e)) shows clear sinusoidal oscillations evidencing the membrane’s motion.

We plot fi,j/f0 as a function of fi,j in Fig. 3(a) for all mechanical resonances observed in Figs. 1(d) and 2(a). Lower frequency modes show better agreement with theoretical ratios than higher frequency modes. From the Lorentzian fits of each mechanical sideband, we extract the mechanical quality factors Qi,j and single photon coupling strengths g0 (Fig. 3(b)). These values of Qi,j, measured in the spectral response, are sensitive both to dissipation and dephasing20. The fundamental mode has a quality factor Q0 = (1.6 ± 0.1) × 103 and the highest quality factor measured was (20 ± 4) × 103 for the mode at 241 kHz. Different modes and even nearly degenerate mechanical modes have significant differences in their values of Qi,j, as previously reported21,22. The values of Qi,j vary slightly as a function of VM ranging from 0.1 to 3.6 Vrms, but they do not show a specific trend (See Supplemental Material for data on the mechanical quality factors as a function of VM). The error bars correspond to this deviation in the values of Qi,j.

Figure 3
figure3

(a) Frequency ratios of the mechanical resonances extracted from Fig. 1(d) (circles) and Fig. 2(a) (triangles). Grid lines indicate theoretical ratios. Errors in frequency and frequency ratios are smaller than symbols. Insets: zoom-in showing selected nearly degenerate modes. (b) Triangles: Quality factors extracted from Lorentzian fits as in Fig. 2 for each mechanical mode. Error bars were obtained by combining fit results for different values of VM. Squares: Single-photon coupling strength calculated for each mechanical mode. Circles: Single-photon coupling strength estimated from a parallel capacitor model, corrected for different mode profiles, for fi,j obtained from theoretical ratios.

We extract g0 for each mechanical mode from the effective thermomechanical power, i.e. from the area of the corresponding sideband in Fig. 3(b) (See Supplemental Material for details of the estimation of the single-photon coupling from noise measurements). For the first mechanical mode, we obtain g0/2π = 1.9 ± 0.1 mHz. The cavity drive was 5 dBm at port 1, yielding a multiphoton coupling strength \(g/2\pi =\sqrt{{n}_{{\rm{C}}}}{g}_{0}/2\pi \sim 4.4\) kHz, where nC is the mean cavity photon number. As expected, the first mode couples more strongly than higher frequency modes, due to its mode profile and larger zero point motion (See Supplemental Material for details).

These values of g0 can be compared with the predictions of Eq. 1. Taking CT and d from the circuit model, and using that \(\frac{\partial {C}_{{\rm{C}}}}{\partial u} \sim \frac{{\varepsilon }_{0}a}{{d}^{2}}\) for a parallel-plate capacitor (with a known prefactor to account for the mechanical mode profile of the membrane), gives a coupling strength g0/2π ≈ 2 mHz for the fundamental mode18. The estimated values of g0 are similar to the ones extracted from the sideband powers.

We can estimate the vibrational amplitude sensitivity Su given an amplifier voltage sensitivity \({S}_{v}=\sqrt{{k}_{{\rm{B}}}{T}_{{\rm{N}}}{Z}_{0}},\) where kB is the Boltzmann constant, Z0 ≡ 50 Ω and TN ~ 293 K the system noise temperature. We can write \({S}_{u}={S}_{v}/(\frac{\partial |{S}_{21}|}{\partial {C}_{{\rm{T}}}}\frac{\partial {C}_{{\rm{C}}}}{\partial u}{V}_{{\rm{in}}})\) where Vin is the voltage at electrode 1 corresponding to PC. Taking \(\frac{\partial |{S}_{21}|}{\partial {C}_{{\rm{T}}}}\) from the circuit model and \(\frac{\partial {C}_{{\rm{C}}}}{\partial u} \sim 1.7\times {10}^{-7}\) F/m extracted from the sideband’s area corresponding to the fundamental mode, we obtain \({S}_{u}=0.4\,{\rm{pm}}/\sqrt{{\rm{Hz}}}\). This value is comparable to that obtained for a suspended carbon nanotube device at cryogenic temperatures12.

Finally, we measure optomechanically induced transparency (OMIT) and absorption (OMIA), which are signatures of optomechanical coupling and demonstrate that the membrane’s motion can be actuated by radiation-pressure alone. OMIT and OMIA are characterized by the emergence of a transparency or absorption window in |S21| when a strong drive tone (fD) and a weak probe tone (fP) are injected into the cavity, and the frequency difference between these tones δf = fP − fD coincides with the frequency of a mechanical mode15. When this condition is met, the beat between the drive and the probe field excites the membrane’s motion and the destructive (constructive) interference of excitation pathways for the intracavity probe field results in a transparency (absorption) window in |S21|.

To show optomechanical control, we measured OMIT and OMIA by injecting a strong tone at frequency fD and a weak tone at frequency fP through a directional coupler at port 1 (Fig. 4(a)). We injected three different drive frequencies (Fig. 4(b)); fD ~ fC − κ/2 (red detuned), fD ~ fC (resonant) and fD ~ fC + κ/2 (blue detuned). When fD ~ fC, a peak (dip) is observed at fD − (+)f0 (Fig. 4(c)). When fD ~ fC ± κ/2, we observe Fano-like features at fD ± f0 (Fig. 4(d,e)). We do not expect complete extinction or revival of the reflected signal as the system is well within the unresolved sideband limit.

Figure 4
figure4

(a) Schematic of signal injection and detection for OMIT/OMIA measurements. A tone with frequency fD and a tone with frequency fP are injected through a directional coupler via port 1. We monitor |S21| with a vector network analyzer. (b) Schematic showing each of the drive frequencies injected and a cartoon of the mechanical sidebands appearing at fD ± f0. Drive frequencies were fD ~ fC (green), fD ~ fC − κ/2 (red) and fD ~ fC + κ/2 (blue). The corresponding mechanical sidebands are shown in panels (ce) respectively. (ce) Measured |S21| as a function of δf showing mechanical sidebands at δf ~ −f0 (left panel) and δf ~ f0 (right panel) for each of the values of fD considered in (b). At port 1, the drive power was 5 dBm and the probe power −27.5 dBm. Black curves are a fit to the data.

We fitted OMIT and OMIA features by modelling the transmission of the probe field (See Supplemental Material for details of the estimation of the single-photon coupling from OMIT measurements). From the fit of the spectral features, we extract f0 = 77.2 ± 0.1 kHz, Q0 = (1.2 ± 0.1)× 103 and g0/2π = 2.3 ± 0.3 mHz. Error bars were obtained by combining fit results of the six curves in Fig. 4(c–e). The values obtained for f0 and Q0 are similar to those extracted from the Lorentzian fits as in Fig. 2(b). The value of g0 is in agreement with that extracted from Fig. 2(a) and estimated from the parallel plate capacitor approximation.

Discussion

To conclude, we have characterized several modes of a silicon nitride membrane with an off-resonant rf circuit at room temperature, deeper in the unresolved sideband regime than previously explored. Our cavity allows for the injection of noise to actuate the motion of the membrane and effectively increase its mechanical mode temperature. We achieve a sensitivity of \(0.4\,{\rm{pm}}/\sqrt{{\rm{Hz}}}\), a tenfold improvement to that reported in ref. 14 although the smaller size of their mechanical resonator makes direct comparison difficult. Our results show that our on-chip platform can be used for membrane actuation and characterization. The readout circuit operates at a convenient frequency and does not require the cavity to be tuned into resonance with the membrane as in other approaches. It therefore has applications in mechanical sensing and microwave-to-optical conversion. Thanks to the large bandwidth of the cavity we have also measured the actuated membrane’s motion in real time. Finally we have observed OMIT and OMIA, from which we obtained a separate measure of the frequency, quality factor and coupling strength of the fundamental mode.

Data availability

The data analysed during the current study are available from the corresponding author on reasonable request.

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Acknowledgements

We acknowledge discussions with G. J. Milburn and support from EPSRC (EP/J015067/1), the Royal Society, the Royal Academy of Engineering and Templeton World Charity Foundation. The opinions expressed in this publication are those of the authors and do not necessarily reflect the views of Templeton World Charity Foundation. K.E.K. acknowledges travel funding from the Australian Research Council (CE110001013), and thanks the Oxford Materials Department for hospitality during the initial stages of this work.

Author information

A.N.P. and N.A. performed the experiment, analysed the data and prepared the manuscript with contributions from E.A.L. K.E.K. contributed to the theory, M.M. helped with device preparation. N.A. conceived the experiment. A.N.P., K.E.K., M.M., E.A.L., G.A.D.B. and N.A. discussed results and commented on the manuscript.

Correspondence to N. Ares.

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Pearson, A.N., Khosla, K.E., Mergenthaler, M. et al. Radio-frequency optomechanical characterization of a silicon nitride drum. Sci Rep 10, 1654 (2020). https://doi.org/10.1038/s41598-020-58554-x

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