1. Introduction

Photoacoustic imaging combines the advantages of optical absorption contrast and low scattering of ultrasound during propagation in tissues resulting in images with optical contrast and deep penetration depth [1]. Commonly-used photoacoustic imaging systems contain a pulsed laser that irradiates the tissues in a scanning mode. Tissues are locally heated at the irradiated locations and expand transiently producing photoacoustic signals in the MHz range. Then, an ultrasound transducer is used to receive the generated photoacoustic signals. Conventional ultrasonic transducers are not transparent, thus, for photoacoustic applications the laser path to the tissue must be different from the ultrasonic-reception path. There is also a blind spot in front of the transducer as the laser pulse cannot transmit through the transducer. In contrast, a transparent transducer would enable light delivery directly to the tissues with a common optical and ultrasonic path. This would provide considerable signal-to-noise ratio (SNR) advantages [2]. Such a setup also presents great market potential in developing miniatured and handheld imaging systems. However, there is limited work reported on ultrasonic transducers that are transparent in the visible range. Some of the most commonly used ultrasonic transducers are based on the piezoelectric effect, optical-based ultrasound sensing, and electrostatic-based sensors. Piezoelectric transducers have been well developed for years using piezoelectric ceramics, which are usually not transparent. Transparent optical-based ultrasound transducers include Fabry-Perot etalons, micro-ring resonators, among others [3–7]. Optical based sensing platforms, while extremely sensitive and broadband typically require optical wavelength tuning for each interrogation location, thus limiting readout speed [2,8,9]. Capacitive Micromachined Ultrasonic Transducers (CMUTs) have more flexibility in parallel readout, structure design and material selection but little work has been done on transparent variants.

In this paper, we focus on the design and fabrication of CMUTs due to their advantages in design flexibility, capability in array fabrication, and potentially the enhanced acoustic impedance matching compared with piezoelectric transducers [10,11].

The design of transparent CMUTs involves careful material selection and suitable fabrication methods. CMUTs can be made by either surface micromachining methods [12, 13] or wafer bonding methods [14–17]. Surface micromachining methods usually require 6 to 7 film deposition, lithography and etching processes [18], which makes the fabrication more complicated than the wafer bonding method. The concern in compatibility between device materials and etching solvents or gases also limits the material selection. Therefore, we selected a wafer bonding method to fabricate CMUTs for the purpose of producing transparent transducers. Silicon, silicon dioxide and silicon nitride are the three most often-used materials for CMUTs [16–18]. Other than those, glass, conductive oxide such as indium tin oxide (ITO) [15,19], and polymers such as SU-8 or benzocyclobutene (BCB) have also been applied for producing CMUTs [20–22]. Among these materials, only the polymers, ITO and silicon nitride provides enough transparency in the visible range, while silicon provides relatively good transparency in the infrared (IR) range. Silicon transparency also degrades with doping levels. Functioning CMUTs have been reported using polymers, though cavity sealing was not performed in [21] and membranes were not transparent in [20].

Recently, Pang et al., demonstrated transparent, flexible CMUTs using SU-8 polymer membranes [21]. While promising, these devices were meant for air operation and thus were not suitable for fluid-coupled photoacoustic applications. They had a low resonance frequency of ≈880 kHz in air. Some variants were not vacuum sealed thus were not applicable for immersion operation and roll-based fabrication methods were not done in a vacuum environment. Even if these membranes could be made to be vacuum sealed, they would suffer from gas diffusion through the polymer membranes, limiting the lifetime of their hermetic seal.

Polymers membranes present much lower Young’s Modulus than silicon-based materials resulting a limitation in operational frequencies. For instance, one of the reported CMUT designs used SU-8 as membrane with thickness of 3.07 μm and diameter of 50 μm giving an in-air resonance at 2.83 MHz [20]. In comparison, if the membrane can be made of silicon nitride, which offers good transparency in the visible range, the in-air resonance frequency can be increased to around 5 MHz with reduced membrane thickness down to 600 nm [14]. On the other hand, IR-transparent CMUTs with glass substrates and silicon membranes were recently reported showing functionality in photoacoustic imaging [19], but the transparency in the visible range was poor and operating frequencies in immersion were only ≈1.4 MHz. It would be useful to apply laser pulses in the visible range for applications such as the oxygen saturation imaging of hemoglobin where molar extinctions are highest in the visible range [23]. The purpose of this work was to develop a durable CMUT structure with good transparency in the visible range.

The presented devices use ITO as both the top and bottom electrodes. Silicon nitride deposited by the Low Pressure Chemical Vapor Deposition (LPCVD) process is used as a membrane material. Glass is used as a device substrate and photosensitive polymer is used for the structural and insulating layers. Compared to previously reported CMUTs, the use of silicon nitride membrane provides good potential for cavity sealing and higher frequency performance compare to those CMUTs with polymer membranes. Furthermore, all materials applied to construct the CMUT are highly transparent as demonstrated with spectrophotometer measurements in the visible range. In this paper, we firstly describe the fabrication process in Section 2. Then, we provide characterization data to show the structural dimensions, transparency and the acoustic performance of the produced CMUTs with trans-illumination photoacoustic signal detection demonstrated in Section 3. This work represents the first demonstration of visible-range photoacoustic signal detection through a transparent ultrasound transducer. Future work will aim to further develop this technology to enable large arrays with high sensitivity.

2. Device Fabrication

The transparent CMUTs were fabricated based on an adhesive wafer bonding technique using Photo BCB as both adhesive and structural layers. The Photo BCB was purchased from Dow Chemical (Cyclotene 4022-25). An ITO coated glass wafer (ITO-Glass wafer) and a silicon wafer with LPCVD silicon nitride coating were used as the bottom wafer and the top wafer, respectively. The ITO-Glass wafer was purchased from University Wafer Company. The thickness of the ITO coating is 500 nm with sheet resistance of 9 Ω/sq. The silicon nitride films were deposited through low pressure chemical vapor deposition (LPCVD) with film thickness of 663.4 ± 37.3 nm. An illustration of the fabrication process is given in the Fig. 1.

 

Fig. 1 Fabrication process of transparent CMUTs.

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The first step is wafer cleaning. ITO-Glass wafers were cleaned with RCA1 solution, a mixture of ammonium hydroxide, hydrogen peroxide and deionized water in volume ratio of 1:1:5, for 15 minutes at 75 °C. LPCVD silicon nitride wafer was cleaned with Piranha solution, a mixture of sulfuric acid and hydrogen peroxide in volume ratio of 3:1, for 15 minutes.

The second step is adhesive coating and patterning, which is similar to previous work [14]. Briefly, adhesion promoter, AP 3000 from Dow Chemical, was firstly spin coated on both the ITO-Glass wafer and the nitride wafer with spinning speed of 3000 RPM for 30 seconds followed by a soft baking at 150°C for 1 minute. Then, a layer of Photo BCB was spin coated on both wafers at 5000 RPM for 30 seconds after an initial solution-spreading step at 500 RPM for 10 seconds. A soft bake at 60 °C for 90 seconds on a hotplate was applied after the spinning process. After the soft baking, the Photo BCB film on the nitride wafer was partially exposed to the UV with mask 1 and the Photo BCB film on the ITO-Glass wafer was fully exposed to the UV without applying a mask. A post-exposure bake at 50 °C for 60 seconds was applied for both wafers before the developing process. Then, the top nitride wafer was developed with developer, DS 2100 from Dow Chemical, through a 2 minutes puddle developing process by adding then spinning off the developer solution at 2000 RPM for 30 seconds. This second step produced an insulating layer coated on the ITO-Glass wafer with cavities patterned on the nitride wafer.

The third step involves applying thermal treatment to the coated and patterned Photo BCB films. Wafers were placed in a vacuum oven starting at room temperature then evacuating air to 625 Torr. Then, the temperature was elevated to 190 °C and maintained for an hour. The oven was then cooled to room temperature by turning off the oven and wafers were brought to the next step.

The fourth step is the adhesive wafer bonding. A SUSS Wafer Bonder (ELAN CB6L) was used to achieve the wafer bonding. The nitride wafer was first placed on the chuck of the bonder and the ITO-Glass wafer was placed over the nitride wafer supported with a removable spacer. After vacuuming the chamber to 5 mTorr, the spacer was removed automatically by the wafer bonder letting the ITO-Glass wafer drop onto the nitride wafer. Then, a compressive pressure of 0.5 MPa was applied to ensure contact between two wafers at the interface. Lastly, to finish the bonding, wafers were heated to 150 °C and held for 15 minutes followed by another temperature elevation to 250 °C and held for 1 hour. Once the temperature was cooled down to less than 100 °C, wafers were unloaded from the wafer bonder to finish the wafer bonding step.

The fifth step is membrane release. By removing the backside nitride of the nitride wafer with a dry etching process (TRION Phantom RIE, CF₄ 45 sccm, Oxygen 5 sccm, pressure 150 mTorr, Power 125 W), the silicon substrate of the nitride wafer was removed by potassium hydroxide solution (concentration of 25%, solution temperature of 80 °C). Then, the nitride film is freed for vibration over the cavity regions.

The sixth step is top electrode deposition and patterning. ITO films of 300 nm in thickness were sputtered on the nitride films at room temperature with 50 sccm of Ar and no oxygen under chamber pressure of 6 mTorr. The deposited ITO films presented acceptable sheet resistance (50 Ω/sq) but the transparency was poor. Therefore, an annealing process was used in a vacuum oven. One-hour annealing at 245 °C in vacuum at 625 Torr was found to be sufficient to improve the transparency in the visible range. As can be found from Fig. 2, after the annealing process, the sheet resistance was found slightly enhanced from 50.83 to 34.5 Ω/sq and the transparency was greatly enhanced up to nearly 90% in the visible range. Transparency measurements were done using Perkin-Elmer NIR-UV Spectrophotometer and first tested on glass slides. The enhancement in both the film conductivity and the transparency is due to the crystalline structure rearrangement during the annealing process as can be demonstrated by the XRD measurement results given in Fig. 3. The ITO film was then patterned through a lithography process using Mask 2 and wet etching process using non-diluted 35% Hydrochloric acid for about two minutes.

 

Fig. 2 Effect of ITO annealing on transparency and sheet resistivity.

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Fig. 3 XRD results comparison of the ITO before and after annealing demonstrating the crystalline structure reorganization of the ITO film.

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The last two steps involve exposing the bottom electrode and depositing metals for wire bonding. RIE was used to etch through the nitride membrane and the Photo BCB layer. The recipe for etching the nitride layer was the same as the one used in the fifth step. The recipe for BCB etching is changed to CF₄ 20 sccm Oxygen 80 sccm Pressure 80 mTorr power 40 W. Then, a 20-nm thick layer of chromium and a 200-nm thick layer of aluminum were deposited and patterned on the ITO layers through a lift-off process to provide bond pads.

3. Device Characterization and Discussion

Some of the fabricated CMUTs are shown in Fig. 4. Each die is of size 6 mm by 6 mm and the active area (the area with cavities) is about 4 mm by 4 mm. Optical microscopy images with annotated membrane dimensions is also given in Fig. 4.

 

Fig. 4 Photos of the fabricated CMUTs. The photos on the top were taken by a camera showing the dimensions of the die and the active area. The transparency in visible range is also intuitively presented as the printed name of University of Alberta can be directly see through the CMUTs that were placed on the top. The bottom two optical microscopy images show a closer view of the cavities in the active area. Diameter of the cavities are shown, along with the cell-to-cell distance.

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Devices were characterized to demonstrate functionality by structural inspection, C–V testing, and receive sensitivity measurements. We also performed a photoacoustic test to demonstrate capacity for photoacoustic imaging applications.

3.1. Device Structural Dimensions

A focused ion beam was used to create a small opening through the CMUT exposing the cross-sectional structure. Helium ion microscopy images were then taken to inspect the structure, shown in Fig. 5. In order to minimize the damage from the ions during cutting with the ion beam, a small beam dose was chosen and the insulating Photo BCB layer was not cut through.

 

Fig. 5 Helium ion microscopy image for cross-sectional structure inspection. Insulating layer made of Photo BCB was not cut through with ion beam.

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As a result, the thickness of the top ITO layer (top electrode), t_ITO, and the total thickness of the top ITO layer, nitride layer and the two Photo BCB layers (insulating Photo BCB and structural Photo BCB constructing cavities), t_all, were measured with a contact profilometer (Alpha-Step IQ). Knowing the thickness of the top ITO layer, the thicknesses of the nitride and cavity depth (structural Photo BCB layer) can be obtained through the helium ion microscopy image having t_SiN and t_cavity, respectively. Then, the insulating Photo BCB layer can be calculated by t_BCB = t_all − t_SiN − t_cavity. Results are summarized in Table 1.

Table 1. Structural dimensions.

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3.2. C–V Testing

A Keithley 4200-Semiconductor Characterization System was used to perform the C–V testing in order to inspect the functionality of the produced CMUTs. As the curve in the solid line shows in Fig. 6, a continuous and smooth increase in the capacitance is observed with the increase of the applied bias voltage from 0 to 200 V with 0.5 V increments. In order to validate the measurement data, theoretical calculations were performed based on theoretical models. The calculation details are given in the Appendix. The calculation results are shown as the dashed curve in Fig. 6. The maximum difference between the calculation results and the experimental results is found to be only 0.2% at a bias voltage of 200 V showing a good agreement between the fabrication and design.

 

Fig. 6 C–V curve of the fabricated CMUT with DC voltage applied to the top and bottom electrodes. Capacitance increased by the increase of voltage from 0 to 200 V. The measurement result is compared with the values obtained through theoretical calculation using the structural dimensions in Table 1.

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3.3. Device Transparency

The fabricated CMUTs contain five layers, which are ITO as the top electrode, LPCVD silicon nitride as the membrane, Photo BCB layer for both insulating and structural material, another ITO layer as the bottom electrode and the glass substrate. We characterized the transparency after each step. Some steps including thermal treatment and ITO annealing may impact transparency between steps. Therefore, we present the transparency spectrum of the final device using Perkin-Elmer NIR-UV Spectrophotometer in Fig. 7. It is found that in the wavelength range from 504 nm to 605 nm, the CMUT has more than 70% transparency. The transparency is still more than 30% in the remaining visible range, which is higher than the latest reported transparent CMUTs using thin silicon membranes [19]. Much of the losses in transparency are likely reflective rather than absorptive given the excellent visible transparency of devices, shown in Fig. 4.

 

Fig. 7 Transparency measurement by spectrophotometer of the fabricated CMUT.

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3.4. Receive Sensitivity

Receive sensitivity of a CMUT can be evaluated by the ratio of the received sound signal to the corresponding acoustic pressure. As illustrated in Fig. 8, receive sensitivity tests were performed in a tank with vegetable oil. Measurement 1 and 2 show the acoustic signal generated by a piezoelectric transducer and detected by the CMUT and a hydrophone (Onda Corporation), respectively. A piezoelectric transducer (5MHz, Olympus Panametrics-NDT V310-SM 0.25″) driven by a signal generator was used to generate pulses of ultrasound. Each CMUT was wire bonded to a printed circuit board (PCB) with a transimpedance amplifier setup. The setup used an amplifier OPA 354 with open-loop gain of 90 dB under supply voltage of 5 V. The cut-off frequency of the circuit was 10.6 MHz determined by the feedback resistor (25 kΩ) and capacitor (0.6 pF). The CMUT-mounted PCB was placed in front of the piezoelectric transducer to detect soundwaves. The PCB amplified and converted the signal from the CMUT to voltage signals, which were further recorded with an oscilloscope. In order to acquire the acoustic pressure at the surface of the CMUT, a hydrophone was used. By analyzing the hydrophone data, acoustic pressure was obtained and the receive sensitivity of the CMUT was calculated.

 

Fig. 8 Illustration of receive sensitivity measurement setup.

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The signal that was used to drive the piezoelectric transducer is shown in Fig. 9(a); the signal received by the hydrophone is given in Fig. 9(b); and the signals received by the CMUT with bias voltages of 50 V and 100 V are given in Fig. 9(c). The measured peak-to-peak values (Vpp) of the signals, hydrophone sensitivity based on the calibration data, calculated receive sensitivity, signal to noise ratio (SNR) and the noise equivalent sensitivity are summarized Table 2.

 

Fig. 9 Results of receive sensitivity measurements. (a) signal from signal generator for driving the piezo transducer generating acoustic waves; (b) hydrophone signal corresponding to the acoustic wave from the piezo transducer; (c) CMUT signals when it is biased at 50 and 100 V and positioned at the same position of the hydrophone.

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Table 2. Receive sensitivity test results.

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3.5. Generation and Detection of Photoacoustic Signals

The setup of the photoacoustic test was illustrated in Fig. 10. It is also performed in vegetable oil. A hole was drilled on the PCB to provide a window for laser light delivery through the CMUT. The active area of the CMUT is about 4 mm × 4 mm, thus the diameter of the hole was made to 4.5 mm. Laser light from a 532-nm pulsed 8-ns Nd:YAG laser (Surelight III, Continuum SLIII-10) was delivered through the CMUT using an optical fiber bundle (CeramOptec) at 10 Hz repetition rate. Measured laser powers of 23 mW and 50 mW emerging from the fiber bundle were tested. CMUTs were biased at 50 V and 100 V without effects of breakdown or pull-in. A piece of aluminum foil was placed in front of the CMUT being perpendicular to the laser beam. Once the laser irradiated the aluminum foil, a local spot on the foil produced photoacoustic signals on both sides of the foil. A hydrophone was also placed in the back of the aluminum foil. The thickness of the aluminum foil is 300 μm which was measured by a micrometer. Thus, the generated acoustic waves on both sides of the foil were deemed similar. As a result, the characteristics of the photoacoustic signal detected by the CMUT can also be analyzed by the hydrophone.

 

Fig. 10 Illustration of photoacoustic test setup.

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The detected CMUT signals and the hydrophone signals are given in Fig. 11. The top two plots Fig. 11(a) and Fig. 11(b) are the signals detected by the CMUT. There are four signals in each plot: Sig. 1 occurred when the laser transmitted through the CMUT. Heat was generated inside the CMUT at this moment inducing membrane vibration (Sig. 1). Acoustic waves were also generated traveling toward the aluminum foil. The echo signal is the Sig. 1e. Sig. 2 represents the photoacoustic signal from the aluminum foil. This signal represents the acoustic wave generated by the heated aluminum foil when it was hit by the laser pulse. When this wave was reflected twice by the CMUT surface and the aluminum foil, a reverberation echo signal Sig. 2e was also detected. In contrast, the bottom two plots (c and d) are the signals detected by the hydrophone. Signals of Sig. 1e, Sig. 2 and Sig. 2e can also be found in both the plots except the signal Sig. 1. Moreover, a temporal ringing signal can be found between the Sig. 1 and Sig. 2. It may be due to in part to substrate ringing but needs to be further investigated in future work.

 

Fig. 11 Photoacoustic signals detected by CMUT (a and b) and a hydrophone (c and d) with different laser power applied.

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The maximum signal response from CMUT was found when it was biased at 100 V with the laser power set to 50 mW. A frequency analysis, as can be found in the Fig. 12, was performed to analyze the Sig. 2 of this maximum signal response and the corresponding hydrophone signal. The CMUT signal had a center frequency of 2 MHz with a −6-dB fractional bandwidth of 52.3%. On the other hand, the hydrophone signal appeared a center frequency of 4.1 MHz with −3-dB fractional bandwidth of 133.8%. However, the peak frequency also happens around 2 MHz.

 

Fig. 12 Frequency analysis of the photoacoustic signal when CMUT was biased at 100 V and the laser power was set to 50 mW.

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3.6. Device Durability

The amplitudes of the photoacoustic signal shown in Fig. 11 remained stable over 3 hours or repeated testing with a 10 Hz repetition rate laser. The devices have been tested 5 months after fabrication with no apparent degradation in performance, indicating durability and good hermetic sealing. Future work should further investigate long-term reliability.

4. Conclusion

We presented a fabrication process for a CMUT that is transparent in the visible range with transmission rate of up to 82%. Having a membrane structure that contains 704 nm of nitride and 300 nm of ITO, the produced CMUT showed a centre frequency at 2 MHz in immersion demonstrating the capability of detecting photoacoustic signal at the same time of letting laser pulse transmitted through the transducer. A receive sensitivity of 65.507 μV/Pa was obtained when the CMUT was biased at 100 V on a PCB with a transimpedance amplifier setup. Such setup also showed a noise equivalent sensitivity of 95 Pa. As a future work, array fabrication is planned toward achieving a photoacoustic imaging probe.

Appendix

This Appendix explains the calculation process for validating the C–V test results in section 3.2. As shown in Fig. 13, the cross-sectional structure of a CMUT cell is explained with thickness of each layer labelled. In the figure, notations are explained in the Table 3. Material properties of the nitride membrane and the ITO top electrode were estimated using data from a material database of MIT (silicon rich LPCVD silicon nitride and ITO, respectively) [24]. The material property of Photo BCB was estimated using the data in the technical specification from Dow Chemical [25].

 

Fig. 13 An illustration of the CMUT structure for C–V calculation.

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Table 3. Explanation of notations in the Fig. 13

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The capacitance of device is contributed from two regions. The first region is covered by the top electrode, but outside the cavities, as labeled as A₁. The capacitance in this area is not changed by the change of bias voltage. The other region contains all the cavity regions and each cavity region is named A_2i. There are N=6248 cavities in each die and the capacitance in the cavity regions change with bias voltage. As a result, the entire structure is treated as multiple paralleled connected capacitors and the total capacitance can be calculated by adding the capacitance of each capacitor. Calculation process can be explained in the following 7 steps based on the theoretical models described in [26,27].

• Step 1: Calculate the capacitance in A₁ by:

  (1)$$C_1=\epsilon \frac{S_1}{g_{01}}$$

  Where S₁ = l₁l₂ − Nπa² is the area of A₁, ε is the dielectric constant of vacuum; g₀₁ is the equivalent electrode distance in area of A₁, which is calculated by:

  $$g_{01}=\frac{\left(t_{\mathit{Cavity}}+t_{\mathit{BCB}}\right)}{{\epsilon }_{\mathit{BCB}}}+\frac{t_{\mathit{Nitride}}}{{\epsilon }_{\mathit{Nitride}}}$$

  Parameters ε_BCB and ε_Nitride are the relative dielectric constant of the Photo BCB and the silicon nitride, respectively.
• Step 2: Assign values of displacement at the membrane centre, u_P, and calculate corresponding average membrane displacement, u_a.

  Assume that the membrane has a displacement of u_P at the centre. Then, the average membrane displacement can be calculated by:

  $$u_a=\frac{u_p}{3}$$

• Step 3: Calculate the mechanical force on individual membranes of each cell, F_mi, resulting from the membrane deflection.

  Based on the material properties of the membrane, the spring constant of the membrane k can be calculated by:

  (2)$$k=\frac{192\pi D}{a^2}=\frac{F_{mi}}{u_a}$$
  where F_mi is the mechanical force from the deformed membrane, D is the transformed flexural rigidity of the bi-layer membrane that consist of ITO and nitride. Having the Young’s Modulus, Poisson’s ratio and the layer thickness of the silicon nitride and the Photo BCB, D can be calculated by:
  $$\begin{array}{ll}D=\hfill & \frac{E_{\mathit{ITO}}{t}_{\mathit{ITO}}^3}{3\left(1-{\nu }_{\mathit{ITO}}^2\right)}+\frac{E_{\mathit{Nitride}}\left[{\left(t_{\mathit{ITO}}+t_{\mathit{Nitride}}\right)}^3-t_{\mathit{ITO}}^3\right]}{3\left(1-{\nu }_{\mathit{Nitride}}^2\right)}\hfill \\ \hfill & -\frac{{\left\{\frac{E_{\mathit{ITO}}{t}_{\mathit{ITO}}^2}{2\left(1-{\nu }_{\mathit{ITO}}^2\right)}+\frac{E_{\mathit{Nitride}}\left[{\left(t_{\mathit{ITO}}+t_{\mathit{Nitride}}\right)}^2+t_{\mathit{ITO}}^2\right]}{2\left(1-{\nu }_{\mathit{Nitride}}^2\right)}\right\}}^2}{\frac{E_{\mathit{ITO}}{t}_{\mathit{ITO}}}{1-{\nu }_{\mathit{ITO}}^2}+\frac{E_{\mathit{Nitride}}{t}_{\mathit{Nitride}}}{1-{\nu }_{\mathit{Nitride}}^2}}\hfill \end{array}$$

  Where E_ITO, E_Nitride, ν_ITO and ν_Nitride are the Young’s modulus and the Poisson’s ratio of ITO and silicon nitride, respectively.

• Step 4: Calculate the capacitance of each cell, C_2i, corresponding to the assigned displacement at the membrane centre u_P.

  (3)$$C_{2i}=\frac{C_{2i0}{\text{tanh}}^{-1}\left(\sqrt{\frac{u_p}{g_{02}}}\right)}{\sqrt{\frac{u_P}{g_{02}}}}$$

  where g₀₂ is the equivalent electrode distance when the membrane is not deflected.

  $$g_{02}=t_{\mathit{Cavity}}+\frac{t_{\mathit{BCB}}}{{\epsilon }_{\mathit{Nitride}}}+\frac{t_{\mathit{Nitride}}}{{\epsilon }_{\mathit{Nitride}}}$$

  where C_2i0 is the capacitance when the membrane is not deflected.

  $$C_{2i0}=\frac{\epsilon \pi a^2}{g_{02}}$$
• Step 5: Calculate the first order derivative of with respective to u_a.

  (4)$$\frac{d{C}_{2i}}{d{u}_a}=\frac{\epsilon \pi a^2}{2g_{02}{u}_a\left(1-\frac{u_p}{g_{02}}\right)}-\frac{C_{2i}}{2u_a}$$
• Step 6: Calculate the bias voltage V.

  The electrostatic force generated on each membrane due to the applied bias voltage V can be calculated by:

  (5)$$F_{\mathit{ei}}=-\frac{1}{2}\frac{d{C}_{2i}}{d{u}_a}{V}^2$$
  The C–V test was performed in air. Thus, the atmosphere pressure also applied a force, F_pi, to the top surface of the membrane must be included. Then, the relationship between the mechanical force and the electrostatic force can be setup by:
  (6)$$\sum F=F_{mi}+F_{ei}+F_{pi}=F_{mi}+\left(-\frac{1}{2}\frac{d{C}_{2i}}{d{u}_a}{V}^2\right)+\left(-p_0\pi a^2\right)=0$$
  where p₀ is the atmosphere pressure.

  Then, the bias voltage V corresponding to each displacement at the membrane centre can be calculated based on Eq. (2) and Eq. (4)–(6).

• Step 7: Calculate the total capacitance C of the one-element CMUT. The capacitance of the entire device can be calculated based on Eq. (1) and Eq. (3).

  $$C=C_1+N{C}_{2i}$$

  Finally, we can plot calculated C–V curve based on the results obtained in the Step 6 and 7.

Funding

Natural Sciences and Engineering Research Council (RGPIN-2018-05788, STPGP 494293-16, RGPIN 355544); Canadian Institutes of Health Research (PS 153067).

Acknowledgments

We are grateful for resources supplied by the CMC Microsystems Canada.

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