Introduction

With the increasing global focus on marine resource development and deep-sea scientific research, underwater submersibles are playing an increasingly vital role in ocean environment monitoring, resource exploration, and seabed investigation1,2,3. In the high-pressure sea environment, the functionality and safety of underwater vehicles heavily depend on the support of various precision sensors. Among these, pressure sensors, often referred to as the “eyes” and “ears” of underwater vehicles4, are indispensable for pressure monitoring, maintaining dive depth, and navigation5,6,7,8,9. However, the high-pressure conditions in deep-sea environments impose stringent requirements on pressure sensors. These sensors must not only withstand tens of mega Pascals of hydrostatic pressure but also maintain high stability and long-term reliability under extreme conditions. Traditional ocean pressure sensors rely on rigid materials and high-pressure chambers to withstand deep-sea hydrostatic pressure10,11,12,13. While this design effectively resists high pressures, it presents several challenges. First, the sensors are typically bulky and heavy due to the need for thick enclosures and complex mechanical structures. Second, the use of high-strength materials complicates the manufacturing process, increases production and maintenance costs. Furthermore, traditional systems often require large power sources and occupy significant space. As a result, this approach limits system efficiency and integration, making it difficult to meet the modern demand for lightweight, integrated sensor systems. To address these challenges, researchers have proposed various innovative designs and manufacturing methods. R.B. McIntosh et al.14 presented a pressure sensor with curved electrodes with high sensitivity and linearity in the pressure range of 0–1.96 MPa. Jie Yu et al.15 developed a dual-resonant cavity pressure sensor based on micro-electromechanical systems (MEMS) technology, with a measurement range up to 70 MPa. The system requires a steel casing for protection, which adds to its overall size and complexity. Aravamudhan et al.16 proposed a dual-membrane resistive pressure sensor with excellent stability in a 7 MPa pressure environment. The system has a smaller measurement range, which may limit its suitability for applications requiring higher pressure thresholds. Haijun Wang et al.17 designed a capacitive pressure sensor based on 3D printing and verified its capability to measure pressures up to 100 MPa. The system, while showcasing impressive pressure measurement capabilities, has a relatively large volume, which may restrict its applicability in environments with space constraints or where a compact design is crucial.

This paper presents, for the first time, a pressure sensor based on the CMUT structure, which enables precise pressure monitoring through capacitance variation. The CMUT structural sensor, utilizing a multi-chamber array design, effectively addresses deformation issues in high-pressure environments, offering a highly reliable, low-power passive pressure measurement solution. Compared to traditional pressure sensors, the CMUT structure pressure sensor, fabricated using MEMS technology, offers advantages such as small size, lightweightness, low power consumption, wide measurement range, and ease of integration with backend circuitry18,19,20,21. These features make the CMUT pressure sensors particularly suitable for the requirements of modern miniaturized systems.

Results

Structural

To evaluate the chamber structure of the pressure sensors fabricated using wafer bonding technology, a single CMUT sensor was subjected to destructive characterization. A sample containing the cross-section of the chamber was selected and characterized using scanning electron microscopy (SEM). As shown in Fig. 1a, the gap height, Silicon membrane thickness, and silicon dioxide (SiO2) insulator layer thickness dimensions of the cross-sectional structure are consistent with the design specifications, verifying the precision and reliability of the fabrication process.

Fig. 1: Sensor characteristics.
figure 1

a Cross-sectional SEM image of the CMUT-structured pressure sensor chamber. b Deflection of the 5×5 array in the CMUT pressure sensor under 3 V AC (at 5 MHz) excitation. c I–V curve characteristics of the membrane and wire bonding pad in CMUT structure pressure sensor. d C–V characteristic curve between the two electrodes of the CMUT-structured pressure sensor

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Deflection

In the small-chamber array structure designed in this study, the consistency of membrane deflection plays a crucial role in overall performance. Furthermore, the silicon film thickness, the thickness of the remaining 100 nm of SiO2 after RIE etching of 200 nm SiO2, and the uniformity of the radius size also play pivotal roles. To assess the deflection consistency of the arrays in response to external forces, the deflection of the membranes was measured using an MSA-600 (Polytech) at 3 V AC (at 5 MHz). The 5 MHz excitation frequency is chosen because the “in air” resonance frequency of the membranes is 5 MHz. Hence, maximum deflection of the membranes is expected to happen at the “in air” resonance frequency of the membranes. As shown in Fig. 1b, the deflection of the membrane array exhibits high consistency, with a maximum deviation of 8.2%.

Electrical

Electrical characterization is one of the key aspects of evaluating performance of the CMUT sensor. To assess the impedance between the silicon membrane and the wire bonding pad, a 4200-SCS semiconductor analyzer (KEITHLEY) was used to measure the resistance. The voltage range was set from −5 V to 5 V. The test results, as shown in Fig. 1c, indicate that when the absolute value of the voltage exceeds 3 V, a current greater than 0.1 A is generated, resulting in current overflow shown as horizontal straight lines in positive voltage as well as negative voltage directions of the voltage axis. Within the voltage range of ± 3 V, the I-V curve exhibits a linear slope through the origin. Calculations show a resistance value of 30 Ω, indicating good ohmic contact between the membrane and the wire bonding pad.

Furthermore, to evaluate the capacitance response of the sensor to variations in the plate gap, a DC bias voltage ranging from −40 V to 40 V was applied to the top electrode, superimposed with an AC signal of 500 kHz and 100 mV, while the bottom electrode was grounded. Figure 1d illustrates the relationship between capacitance and DC bias voltage. When the DC bias is 0 V, the sensor exhibits a capacitance of 527 pF. The capacitance variation between −40 V and 40 V DC bias voltage follows a U-shaped trend, indicating high sensitivity to changes in the plate gap.

Pressure measurement in conventional mode

Figure 2a shows the simulation, experimental, and fitted curve results within the pressure range of 0–1.1 MPa. The experimental data were fitted using a cubic model \(y=21.65{x}^{3}-15.02{x}^{2}+13.15x+261.01\), and the coefficient of determination \(R^{2}=0.99952\) was obtained. The results show a strong correlation between the sensor’s response to pressure and the fitted curve within this range. However, as the pressure approaches 1.1 MPa, more discrepancies arise between the simulation results and the experimentally measured capacitance values. This is primarily due to the more complex behavior exhibited by the CMUT pressure sensor as it approaches the collapse state.

Fig. 2: Sensor pressure measurement.
figure 2

a Simulation, experimental, and fitting curve results in the 0–1.1 MPa pressure range. b Comparison of initial experimental results with those after three months, along with the deviation, in the 0–1.1 MPa pressure range. c Simulation, experimental, and fitting curve results in the 0–20 MPa pressure range. d Comparison of initial experimental results with those after three months, along with the deviation, in the 0–20 MPa pressure range. e Repetition tests of the sensor at 1.1 MPa pressure. f Capacitance response experimental results under 0–20 MPa pressure range for three sensors (A, B, C) manufactured from the same wafer. g Linearity and sensitivity for 0–20 MPa range segments

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Pressure measurement in collapse mode

Figure 2c shows the simulation, experimental, and fitted curve results within the pressure range of 0–20 MPa. The experimental data were fitted using a cubic model \(y=0.017{x}^{3}-0.75{x}^{2}+12.95x+276.34\), and the coefficient of determination \(R^{2}=0.9979\) was obtained. The correlation between the simulation, experimental, and fitted curves is relatively high. At a pressure of 1.07 MPa, both the simulation and experimental capacitance curves exhibit a sharp change, indicating that the CMUT structure pressure sensor transitions from the conventional mode to the collapse mode Table 1. Numerical comparison shows that the experimental value is 340.68 pF higher than the simulated value. This is primarily because, during the simulation process, the model focused only on the collapsed membrane region to improve computational efficiency, omitting the fixed capacitance in the edge regions. Additionally, the cables used in the experimental setup introduced extra capacitance, resulting in a fixed value deviation.

Table 1 Linearity and Sensitivity
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Sensor repeatability and durability

To evaluate the long-term reliability of the CMUT-based pressure sensor, the sensor remained in the pressure chamber after the initial data acquisition experiment, and data was recollected three months later. During this period, multiple pressurization and depressurization operations were performed, causing the internal pressure of the chamber to fluctuate within the range of 0 to 20 MPa. The capacitance changes of the sensor within the 0–20 MPa pressure range were re-collected, as shown in Fig. 2b, d. The results demonstrate that, after three months of continuous hydrostatic pressure fluctuations, the CMUT-based pressure sensor still exhibits excellent reliability. Compared to the initial test results, the maximum deviation was 1.97% in the 0–1.1 MPa pressure range, and 0.65% in the 1.1–20 MPa pressure range. To emphasize the reliability of the sensor in collapse mode under high hydrostatic pressure, we recorded the capacitance values of the sensor 100 times at 1.1 MPa, when the membrane was just beginning to collapse. The results, shown in Fig. 2e, indicate a maximum deviation of 12 fF. Compared to the test results from three months ago, the subsequent tests showed a shift of 0.33 ± 0.02 pF, which was primarily caused by unstable connections between the sensor and the impedance analyzer. The test results demonstrate that the manufactured sensor exhibits high repeatability and measurement stability.

Sensor consistency

In order to evaluate the consistency of the sensors, two other sensors diced from the same wafer were covered by PDMS and the same experiment was performed in the pressure chamber, and the results are shown in Fig. 2f. After compensating for the fixed capacitance offset (primarily caused by the cables connected to the pressure chamber), the capacitance response curves of the sensors exhibit excellent consistency. The experimental results demonstrate the high consistency of the CMUT structure pressure sensors fabricated based on the direct wafer bonding process between silicon and SiO2 surfaces.

Sensitivity and linearity

Capacitive pressure sensors exhibit certain limitations in linearity, particularly over a large measurement range, which is a common challenge for such designs. In this study, a segmented range approach was adopted to evaluate the sensor’s linearity and sensitivity characteristics, as shown in Fig. 2g. The results indicate that the maximum nonlinearity error (MNE) primarily occurs at the segmentation points. Additionally, the sensor demonstrates peak sensitivity in the operational ranges immediately before and after the collapse mode, with an average sensitivity of 5.29 pF/MPa across the entire measurement range.

The MEMS-based CMUT structure pressure sensor proposed in this paper offers advantages over high-range pressure sensors, including a compact size, no need for steel casing protection, and significant potential for integration with backend circuitry. Moreover, within the same size category, this sensor features a wider measurement range. Overall, the CMUT structure pressure sensor, fabricated using MEMS technology, combines the benefits of small size, lightweight design, low power consumption, wide measurement range, and ease of integration with backend circuits, making it particularly suitable for the needs of modern miniaturized deep-sea submersibles. (Table 2).

Table 2 Other Relevant Studies
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Discussion

This study proposes a pressure sensor based on a CMUT structure, featuring compact size and a wide measurement range. Through theoretical analysis and finite element simulations, the effects of key structural parameters on membrane deflection and capacitive response were systematically investigated. Based on MEMS fabrication technology, highly consistent sensor devices were successfully manufactured. High-pressure test results show that the sensor can stably measure hydrostatic pressures up to 20 MPa, demonstrating excellent reliability and long-term stability. At the same time, due to the sensor’s natural frequency of 5 MHz, it exhibits a response time in the microsecond range.

Although the proposed CMUT-based pressure sensor exhibits outstanding performance in high-pressure measurements, its sensitivity and linearity still require further improvement. Future research may focus on the following aspects: (1) employing higher-precision micro/nano fabrication techniques to develop larger-scale array structures, thereby enhancing sensitivity without compromising measurement range; (2) optimizing structural design to improve linear response characteristics; and (3) designing readout circuits tailored to pressure response behavior to enhance signal-to-noise ratio and measurement accuracy.

This work confirms the feasibility of CMUT structures in high-pressure environments and highlights their promising potential in applications such as deep-sea exploration. It provides a solid theoretical foundation and experimental validation for future engineering applications.

Materials and Methods

Theoretical analyses

The CMUT-structured pressure sensor comprises hundreds or thousands of chamber arrays, each of which can be simplified into a configuration consisting of clamped membranes, an insulation layer, a substrate, and capacitive gaps.

Under varying pressure levels, the membrane deflection exhibits two modes: conventional mode and collapse mode. In the conventional mode, the membrane does not contact the insulation layer, and its maximum deflection remains within the height of the cavity gap. As the hydrostatic pressure applied to the membrane continues to increase, the membrane will displace further. Once the pressure reaches a certain threshold, the center of the membrane will come into contact with the insulating layer situated above the bottom electrode. At this juncture, the pressure sensor will transition into collapse mode. The working schematic diagrams of the conventional mode and collapse mode are shown in Fig. 3a, b, respectively.

Fig. 3: Schematic diagram of the CMUT structural pressure sensor cell.
figure 3

a Operating in conventional mode. b Operating in collapse mode

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In the conventional operation mode, the membrane undergoes deflection under uniformly distributed pressure, and the relationship between the deflection L and the radius x of the circular clamped membrane can be derived using plate and shell theory. Based on Timoshenko’s research22, this relationship is expressed as:

$$L(x)=\frac{P}{64D}{({r}^{2}-{x}^{2})}^{2}.$$
(1)

here, \(P\) represents the external pressure applied to the membrane, \(r\) is the radius of the membrane, \(D\) is the bending stiffness of the membrane, representing the ability of the membrane to resist bending deformation. \(D\) is a critical parameter influencing the relationship between membrane deflection and applied hydrostatic pressure \(D\) is defined as:

$$D=\frac{E{t}_{{\rm{m}}}^{3}}{12(1-{\nu }^{2})}.$$
(2)

here, E is the Young’s modulus, v is the Poisson’s ratio, and tm is the thickness of the membrane.

In collapse mode, after the membrane contacts the underlying insulation layer, the mechanical behavior of the membrane changes significantly. At this stage, a different set of boundary conditions are required to calculate the deflection of the membrane. Selim Olcum et al.22 building on Timoshenko’s research23, derived the general solution for the deflection of circular plates under uniform pressure in collapse mode and proposed boundary conditions specific to this mode.

$$\begin{array}{ll}L(x)={C}_{1}+{C}_{2}\,{\mathrm{ln}}\,x+{C}_{3}{x}^{2}+{C}_{4}{x}^{2}\,{\mathrm{ln}}\,x+\frac{{x}^{4}}{64D}P,\\ {\rm{for}}\,{\rm{c}}\le x\le r.\end{array}$$
(3)
$$L(r)=0,L(c)={t}_{{\rm{g}}}.$$
(4)
$$\frac{{\rm{d}}L(x)}{{\rm{d}}x}{|}_{x=r}=0,\frac{{\rm{d}}L(x)}{{\rm{d}}x}{|}_{x=c}=0.$$
(5)
$${M}_{x}(c)=-D(\frac{{{\rm{d}}}^{2}L(x)}{{\rm{d}}{x}^{2}}+\frac{\nu }{x}\frac{{\rm{d}}L(x)}{{\rm{d}}x}){|}_{x=c}=0.$$
(6)

here, c represents the contact radius, and Mx is the radial bending moment on the membrane. The contact radius c is determined by solving Eq. (6), while the four unknown constants in Eq. (3) are determined by solving the four boundary conditions described in Eqs. (4) and (5).

The calculation formula for the capacitance based on the deflection displacement of the membrane is:

$$C({L}_{x})={\int }_{\!0}^{r}\frac{2\pi {\varepsilon }_{0}x}{{t}_{G}-L(x)}{\rm{d}}x.$$
(7)

here, \({t}_{G}={t}_{g}+{t}_{i}/{\varepsilon }_{r}\) is the effective gap height, tg is the initial vacuum gap, ti is the thickness of the insulation layer, and \({\varepsilon }_{r}\) is the relative permittivity of the membrane material.

Based on the relationship between membrane deflection and capacitance in Eq. (7), combined with the displacement-pressure relationships in the conventional mode (Eq. 1) and collapse mode (Eq. 3), we can derive the capacitance-pressure relationships for the CMUT structure in both the conventional mode and collapse mode.

Finite element simulation

In this section, we used COMSOL finite element modeling (FEM) simulation software to construct a high-precision simulation model. We evaluated the response of key structural parameters of the sensor under 0–20 MPa pressure range, revealing the performance characteristics of the pressure sensor and potential design optimization directions. The CMUT structure typically consists of an array of hundreds or thousands of identical chambers (composed of flexible membranes (top electrodes), gaps, rigid insulating layers, and rigid bottom electrodes). Therefore, an efficient simulation approach involves creating an equivalent model for a single element.

Structural analysis

For the small membrane structure proposed in this paper, the key structural parameters include the membrane radius, thickness, and gap between the electrodes. Following the principle of varying a single parameter while maintaining all the other parameters as constants, the impact of each parameter on the capacitance-pressure response was simulated and analyzed, as shown in Fig. 4a–c. As the membrane radius increases, the full-scale sensitivity increases, the device volume becomes larger, Brownian noise increases, collapse pressure decreases, and the measurement range decreases. When the membrane thickness increases, the full-scale sensitivity decreases, collapse pressure increases, the measurement range expands, and Brownian noise decreases. An increase in the cavity gap leads to a decrease in full-scale sensitivity, an increase in collapse pressure, and an expansion of the measurement range. The above results reveal the inherent trade-off between sensitivity and measurement range of structural parameters. These results highlight the inherent trade-off between sensitivity and measurement range determined by the structural parameters. Therefore, design parameters should be selected based on specific application requirements.

Fig. 4: CMUT structural pressure sensor cell simulation.
figure 4

a Pressure vs. Capacitance response of a CMUT sensor for 3 different radii values. b R Pressure vs. Capacitance response of a CMUT sensor for 3 different membrane thickness values. c Pressure vs. Capacitance response of a CMUT sensor for 3 different gap values. d Deflection of the membrane in the pressure range of 0–0.5 MPa. e Deflection of the membrane in the pressure range of 0–30 MPa. f Capacitive response over a pressure range of 0–30 MPa. g Deflection of the sensor membrane under standard atmospheric pressure. h Membrane stresses under a pressure of 0.5 MPa (Conventional mode). i Membrane stresses under a pressure of 20 MPa (Collapse mode)

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For low-pressure applications, high sensitivity is typically a priority, whereas for high-pressure applications, sacrificing some sensitivity to extend the measurement range is acceptable. Notably, large-scale, high-consistency membrane arrays fabricated using MEMS technology can effectively enhance sensitivity. Based on the simulation results and manufacturing considerations, the finalized membrane structural parameters are: a radius of 34 µm, a thickness of 2 µm, and a gap of 0.2 µm.

Deflection and capacitance

The capacitive response of the sensor is directly affected by the membrane deflection. A uniformly distributed pressure load is applied on the basis of the initial deflection of the membrane, and the response characteristics of the diaphragm deflection and its capacitance to the applied pressure are obtained by simulation. In the pressure range of 0–0.5 MPa, the membrane is deflected in the conventional mode, and the displacement curves are in high agreement with the theoretical calculations, as shown in Fig. 4d. In the pressure range of 0–30 MPa, when the applied pressure reaches 1.07 MPa, the membrane contacts the bottom insulating layer and shifts to the collapsed mode, showing excellent pressure resistance, as shown in Fig. 4e. The trend of capacitance change with pressure is consistent with the deflection of the membrane, as shown in Fig. 4f.

Stress analysis

The chamber of the sensor is in vacuum, so the initial state of the membrane will be deflected under standard atmospheric pressure (101 kPa), and the simulation results are shown in Fig. 4g, with an initial deflection of 17 nm. The stress in the membrane during membrane deflection is closely related to the long-term reliability and stability of the sensor. Therefore, representative pressure values were selected to simulate the stress distribution in the membrane at 0.5 MPa (conventional mode) and 20 MPa (collapse mode), respectively. Under 0.5 MPa pressure, the stress of the membrane is mainly concentrated in the edge region and the center region of the membrane, and the maximum stress in the edge region is 1.89e7 N/m2, as shown in Fig. 4h. Under 20 MPa pressure, the maximum stress in the edge region is 9.41e8 N/m2, and the stress in the center region is 2.45e7 N/m2, as shown in Fig. 4i. The maximum stress is much smaller than the Young’s modulus (170 GPa) of Silicon, which indicates that the membrane remains reliably safe even in the face of a high hydrostatic pressure of 20 Pa.

Sensor fabrication and packaging

The CMUT-structured pressure sensor is fabricated based on wafer bonding technology. This process involves fewer manufacturing steps and ensures high uniformity in the fabricated membrane and cavity structures. The core challenge of this method lies in the Si to SiO2 bonding technique, as the bonding quality directly determines the final performance of the sensor. A schematic diagram of the fabrication process is shown in Fig. 5a–e, using a single-chamber structure for the sake of illustration.

Fig. 5: Process flow diagram for pressure sensor fabrication based on silicon to SiO2 bonding technology.
figure 5

a Wafer preparation. Silicon to SiO2 bonding process requires very clean and flat surfaces. In the preparation process, low resistivity (<0.03 ohm-cm) silicon wafers were used with SiO2 layer thickness of 300 nm on both sides. The device layer of the SOI wafer had a thickness of 2 µm (±0.5um) and a resistivity of less than 0.03 ohm-cm. b CMUT gap definition. After photolithography with the first mask (Mask #1), a mask layer was created, and the gap structure was defined on the SiO2 layer of the oxidized wafer using reactive ion etching (RIE) technology. The RIE process etched 200 nm of the SiO2 layer to form the gap height, leaving the remaining 100 nm of the SiO2 layer as the insulation layer in the gap region of the CMUT. c Silicon to SiO2 bonding. The unetched surface of the oxidized wafer with etched gaps was bonded to the device layer of the SOI wafer using the direct wafer bonding process with a wafer bonding machine (EVG 610). d Wafer thinning. A wafer thinning machine (LP50) was used to remove most of the silicon handle layer of the SOI wafer, leaving approximately 100 µm of handle layer. The remaining silicon handle layer was then removed using tetramethylammonium hydroxide (TMAH) etchant. Finally, buffered oxide etchant (BOE) was used to remove the buried oxide (BOX) layer of the original SOI wafer and the SiO2 layer on the other side of the 300 nm oxidized silicon wafer, with silicon serving as the etch stop layer. e Defining the electrodes. A magnetron sputtering system (EXPLORED) was used to sputter 30 nm of Cr and 300 nm of Au as blanket layers on both sides of the remaining substrate. A second photoresist mask (Mask #2) was used to cover the entire metal layer, leaving openings only in the alignment mark areas (8 mm × 5 mm) to expose the alignment mark location (Please note that, 8 mm × 5 mm is a relatively wide area that allows for alignment with large alignment error tolerance). Hence, the second mask is solely dedicated for revealing the alignment marks that were manufactured with the 1st mask. Subsequently, the metal layers (Cr and Au) in the open area (the open area that was developed with the second mask) are selectively removed using wet etching processes, and the device layer silicon is further etched by an RIE process to achieve a clear exposure of the alignment markers that were manufactured with the first mask before the wafer bonding process. Photolithography with the third mask (Mask #3) was then performed to create another mask layer. Using the third mask, the top electrode was patterned using wet etching to define the geometries of Cr and Au metal layers. f Wafer dicing. A dicing machine was used to singulate the wafer into individual CMUT structure pressure sensors. g Pressure sensor covered with PDMS

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The CMUT-structured pressure sensor designed and manufactured in this study is intended for high hydrostatic pressure measurements, requiring effective encapsulation to protect the sensor and ensure efficient pressure transmission in underwater environments. Polydimethylsiloxane (PDMS) was selected as the ideal encapsulation material due to its outstanding properties. PDMS exhibits excellent elasticity, with a Young’s modulus of only 0.75 MPa, significantly lower than Young’s modulus of silicon (approximately 170,000 MPa), allowing efficient transmission of external pressure to the sensor24,25. The chemical inertness and thermal stability of PDMS further ensures long-term reliability in harsh environments26. Based on these advantages, PDMS was selected to encapsulate the pressure sensor.

The process of covering the sensor with PDMS is as follows: First, a conductive curing paste is used to bond the diced individual pressure sensor (as shown in Fig. 5f) to the printed circuit board (PCB). This curing paste also serves as an electrical connector between the bottom electrodes of the CMUT structure and PCB. Next, gold wire is used to connect the top gold electrode of the sensor to the gold pads on the PCB. Then, black and red cables (5 cm) are connected to the PCB pads. Finally, a mold is used as a container to cover the sensor with approximately 1 mm thickness of PDMS to complete the encapsulation. To ensure the PDMS covering on the sensor is free of air bubbles, the PDMS and curing agent are degassed twice in a vacuum chamber, once after mixing and once after covering the sensor. The encapsulated pressure sensor is shown in Fig. 5g.

Experimental setup

The encapsulated CMUT capacitive pressure sensor (for capacitance measurements), and the piezometer (for pressure measurements) are placed in a high hydrostatic pressure chamber, the chamber is filled with tap water and the chamber is pressurized by adding a pump. The top and bottom electrodes of the CMUT-structured pressure sensor were connected to the internal wiring of the chamber. Then, these wirings are linked to an impedance analyzer outside the chamber for real-time monitoring of capacitance changes. The experimental setup is illustrated in Fig. 6a.

Fig. 6: Experimental setup.
figure 6

a Schematic diagram of the pressure testing method for CMUT-structured pressure sensors in high pressure environments. b Experimental site for high hydrostatic pressure environments

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Due to significant pressure fluctuations during pressurization by the booster pump and the consequent difficulty in achieving precise pressure control, the experiment adopted a depressurization-based approach for pressure regulation. The specific steps are as follows. First, the pressure chamber was pressurized to 20 MPa (limited by experimental equipment, while 30 MPa was used in simulations). After pressure stabilization, the CMUT sensor capacitance values measured by the impedance analyzer, and the pressure readings by the piezometer were recorded simultaneously. Then, the pressure was gradually reduced through the pressure relief valve. By repeating this operation approach, the CMUT sensor capacitance values from the impedance analyzer and pressure values from the piezometer at each stable pressure level (hence, all the pressure vs. capacitance data) over the entire 0–20 MPa range are obtained. Data recording during the pressure relief process was conducted in steps of 0.1 MPa. The capacitance was measured in real-time using the impedance analyzer (E4990A), with an AC signal of 100 mV amplitude and 500 kHz frequency applied during the measurement. The experimental setup and on-site layout are shown in Fig. 6b.