Recently, considerable attention has been paid to the research of mechanical resonators, and their classical^[1–4] and quantum^[5,6] characteristics. Due to its low loss, superconductor material has been used to fabricate mechanical resonators.^[7–9] Though interesting quantum phenomena have been explored in such superconducting mechanical resonators, detailed exploration of their classical characteristics is lacking. Instead of solid material we choose the diluted S1813 photoresist as a sacrificial layer, which can be easily removed by acetone, and successfully fabricate a parallel-plate capacitor, in which the suspended top layer film acts as a membrane resonator. Because the device is measured in a vacuum, such a capacitor is usually named a vacuum-gap capacitor (VGC).^[10] Note that the structure here is quite different from that in previous work,^[11] in which the top layer film of VGC is fixed due to the posts used to support the top plate. Thus the vibration of top layer film is suppressed. Another difference is the method to stimulate the vibration of the mechanical resonator. Usually an electrostatic bias is used in previous work. Here the VGC together with a superconducting octagonal spiral inductor constitutes a superconducting microwave resonator, which is coupled to a coplanar waveguide transmission line. A microwave signal is used to measure the characteristics of the device and at the same time to stimulate the vibration of the membrane mechanical resonator. The vibration of the top layer membrane changes the distance of the gap in the capacitor and thus modulates the frequency of the microwave resonator, the characteristics of which can be detected with a network and a spectrum analyzer.

Our device is composed of an octagonal spiral inductor L and a parallel-plate capacitor C. Figure 1(a) shows the optical photograph of the LC resonator. The insets show the SEM of the corner part of VGC and the overlapping part of the inductance layer and the lower layer of VGC, from which the suspended structures are clearly demonstrated. Such an LC resonator is coupled to a coplanar waveguide transmission line, which is used to measure the resonant characteristics of the LC circuit. The value of the inductor is simulated with FastHenry. The capacitor here has a suspended structure and its value significantly depends on the area of the parallel-plate and the distance between the plates, which can be determined by the parallel-plate capacitor formula.

Fig. 1.

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Fig. 1. (color online) (a) Optical photograph of LC resonator coupled to a coplanar waveguide transmission line. The upper and lower insets show SEM images of the corner part of VGC and the overlapping part of the inductor and the lower electrode of VGC, respectively. (b) Fabrication procedure for VGC. Top view and cross-sectional view along the dashed line in the top view are presented, respectively.

Figure 1(b) shows the detailed fabrication procedure as follows.

(i) A 50-nm-thick aluminum film was deposited by electron beam evaporation on a high-resistivity silicon substrate and the lower electrode plate of VGC was fabricated by the lift-off process.

(ii) The diluted S1813 with a certain thickness is used to form a sacrificial layer, which was then patterned through UV photolithography.

(iii) Another aluminum film was deposited to form the rest of the circuit, including the top electrode plate of VGC, the octagonal spiral inductor and the coplanar waveguide transmission line by wet etching. Note that the sacrificial layer is completely covered except the four corners by the top capacitor plate after wet etching of aluminum. This is extremely important for removing the sacrificial layer with acetone in the final step.

A schematic diagram of our measuring circuit is shown in Fig. 2. At room temperature, the input microwave signal is divided into two parts by a splitter. One of them was fed into the cryostat and the other one was used as the local signal of a mixer, which was used as a down-converter of the amplified output microwave signal from the device. A network analyzer was used to measure the transmission characteristic and a spectrum analyzer was used to characterize the noise spectrum of the device. A computer was used to automatically record the data from the microwave source, network analyzer and spectrum analyzer. Our device was located in a dilution refrigerator with a base temperature lower than 20 mK. The detailed description of the measuring circuits in the cryostat has been given in Ref. [12].

Fig. 2.

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	Fig. 2. (color online) Measurement set-up for testing the transmission characteristic and down-converted sideband frequency of the device.

Figure 3 shows the transmission characteristic S₂₁ obtained with a vector network analyzer, in which a resonance dip appears as expected. This resonance dip comes from the coupling of the superconducting LC microwave circuit to the transmission line. Due to the structure of the resonator, we used to determine the resonant frequency, in which the inductance of the resonator was calculated by FastHenry and the parallel-plate capacitor was determined from C = S ⋅ ε₀/d with S = 45 μm × 45 μm area. Here ε₀ is the permittivity of a vacuum and d is the distance between the top and lower plates. With L = 1.399 nH and f = 5.778 GHz, we obtained an actual capacitance value of 0.54 pF of VGC and a distance of 33 nm between the capacitor plates.

Fig. 3.

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	Fig. 3. (color online) Plot of measured S21 versus frequency of the transmission line, where a resonance dip is due to the coupling of the superconducting microwave resonant circuit to the transmission line. The resonant frequency and quality factor are 5.788 GHz and 7542, respectively.

After knowing the resonant frequency of the superconducting microwave LC circuit, we measure the spectrum to obtain the effect of the mechanical resonator on the microwave resonator. As shown in Fig. 2, a microwave signal with the resonant frequency 5.788 GHz is applied to the device. A spectrum analyzer is used to obtain the spectrum. As is well known, when the frequency of the microwave source is equal to the resonant frequency of the microwave LC circuit, most of the input power will be coupled into the LC circuit. When the electric field energy stored in the VGC reaches a certain threshold, the electric force will trigger the vibration of the top layer of aluminum film. Such an electromechanical coupling together with mechanical motion results in the frequency conversion of the input microwave. As we can see from Fig. 4(a), when the nominal power of the microwave signal is −19 dBm, only one peak, which is corresponding to the resonance dip in S₂₁, occurs. As the power increases to −18 dBm, four more peaks appear. These peaks are located at regular intervals on both sides of the resonant peak. We gradually increase the microwave power and extract the position of the peaks from the measured spectrum. As shown in Fig. 4(b), with increasing power, the resonant peak remains unchanged while the sideband peaks move further away from the resonant peak.

Fig. 4.

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	Fig. 4. (color online) (a) Spectrum measured at different input microwave powers. When the input power is −19 dBm, there is only one peak (lower). When the input power increases to −18 dBm, four sideband frequencies appear (upper). (b) Positions of peaks versus the nominal input microwave power.

Homodyne technology is used to down-convert the microwave frequency. As shown in Fig. 5(a), several peaks are distributed equally, corresponding to the blue sideband frequencies in Fig. 4(b). Note that the lowest frequency is the fundamental frequency of the membrane mechanical resonator. The higher frequencies do not correspond to the high-order vibration modes of the mechanical resonator but the harmonics of the mechanical fundamental frequency. These harmonics are caused by the nonlinear transfer characteristic of the microwave resonator system.^[13] Again, by increasing the microwave power, the fundamental frequency and its harmonics will change. Figure 5(b) shows the extracted position of the fundamental frequency and its harmonics versus the nominal microwave power, which are fitted by parabolic functions as indicated by the solid lines. These are consistent with previous results,^[3] in which a static gate voltage is used to modify the resonant frequency of the mechanical resonator. Here when a microwave signal is used, the induced instant electric attractive force^[1,14] will draw the top layer of VGC downward and produce a tension, which is related directly to the stiffness of the vibrator. This tension is proportional to the square of the instant electric force thus the nominal power of the applied microwave signal. On the other hand, the shift of the fundamental frequency is proportional to the square root of the tension.^[15] So there is a parabolic relationship between the nominal applied power and the fundamental frequency.

Fig. 5.

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	Fig. 5. (color online) (a) Blue sideband frequencies down-converted by a mixer. Nominal input power is −13 dBm. (b) Fundamental frequency and its harmonics of the mechanical resonator. Parabolic relations between input power and resonant frequency are clearly demonstrated. Experimental and fitting data are indicated by symbols and solid lines, respectively.

In order to confirm the experimental results of our superconducting membrane mechanical resonator, we numerically simulate the vibration behavior of the suspended structure, by scanning the parameters of the pretension. As shown in Fig. 6, we find that the dependence of the pretension on the fundamental frequency of the vibrator also satisfies a parabolic function. Note that the simulated fundamental frequency without pretension in our device is 0.82 MHz, while the lowest one in the measured fundamental frequencies is about 3.35 MHz. Their difference may come from two reasons: one is that in the fabrication process of the device, the photoresist is baked at 115 Centigrade, which is equivalent to an annealing treatment of the device, thereby increasing the pretension of the suspended membrane and thus the mechanical vibration frequency;^[16,17] the other possible reason is that there is no static gate voltage to activate the mechanical resonator in our system. When the equivalent electric force induced by the input microwave power is less than a certain threshold, there is no obvious vibration which can be measured. Thus the lowest one in the measured fundamental frequencies is higher than the simulated one without pretension.

Fig. 6.

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	Fig. 6. (color online) Simulated relationship between pretension and fundamental frequency of the membrane mechanical resonator.

Using the diluted S1813 photoresist as a sacrificial layer instead of solid material, which needs a long reactive ion etching procedure, we have successfully fabricated a membrane mechanical resonator in a vacuum-gap capacitor. The fundamental vibration frequency and its harmonics of the mechanical resonator are measured through an LC microwave resonator. The input microwave power has a parabolic dependence on the fundamental vibration frequency, which accords well with the numerical simulation. Such devices promise to serve as a medium between microwave and mechanical signals and/or even optical signals. What is more, they can also be used to explore quantum behavior of the mechanical vibration, which is important for quantum information processing.