Static and Dynamic Analyses of Electrostatically Actuated MEMS GO/CMUT Gas Sensor

Static and Dynamic Analyses of Electrostatically Actuated MEMS GO/CMUT Gas Sensor | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article Static and Dynamic Analyses of Electrostatically Actuated MEMS GO/CMUT Gas Sensor Tirad Owais, Mahmoud Khater, Hussain Al-Qahtani This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-7179088/v2 This work is licensed under a CC BY 4.0 License Status: Posted Version 2 posted You are reading this latest preprint version Show more versions Abstract This paper presents a detailed analytical investigation into the static and dynamic performance of a capacitive micromachined ultrasonic transducer (CMUT) functionalized with a graphene oxide (GO) sensing layer for gas detection applications. A bi-layer lumped-mass model is developed to capture the electromechanical response of the hybrid system, explicitly accounting for the mechanical stiffening effect of GO and its consequent influence on resonant characteristics. Static analysis reveals that the GO layer enhances operational stability by increasing the pull-in voltage from 143 V for a bare CMUT to 147.5 V. Dynamic assessments demonstrate that GO functionalization elevates resonance frequencies due to a stiffness-dominated regime, yielding a mass sensitivity of 24.19 Hz/pg at an 80 V DC bias. Frequency response analyses under combined DC/AC excitation highlight the system's linear dynamics, including distinct superharmonic resonances and bifurcation thresholds. The results underscore the role of GO in improving sensitivity, stability, and bandwidth, establishing GO/CMUTs as highly promising platforms for next-generation, high-performance gas sensors. MEMS GO/CMUT pull-in analysis mass sensitivity resonance frequency bifurcation analysis 1. Introduction Extensive quantities of organic vapors are generated within the industrial sector. It is crucial to monitor their production due to their toxicity, rapid evaporation, and potential for easy dissemination into the surrounding environment. As a result, the initial devices designed to identify such organic vapors emerged during the early 1920s, attributed to James Sumner's efforts in detecting methanol and formaldehyde [ 1 ]. Since that time, the need for dependable and easy-to-use gas detection solutions has increased, particularly for the detection of dangerous and toxic gases. Consequently, there has been a surge of interest in developing advanced microelectromechanical systems (MEMS) and nanoscale gas sensors. MEMS gas sensors have demonstrated exceptional reliability, minimal power consumption, seamless integration capabilities, and affordability in detecting minimal quantities of various gases in parts-per-million (ppm), positioning them as leading candidates for highly sensitive sensing applications [ 2 ]. Explicitly, resonant MEMS sensors have achieved significant success due to their extremely compact size and exceptional sensitivity, which enable them to be utilized for gas sensing applications [ 3 – 5 ]. These sensors operate based on how specific gas affects the resonance frequencies of the resonators in various ways. In the last decade, capacitive micromachined ultrasonic transducers (CMUTs) MEMS-based resonant sensors have been thoroughly researched for various fields of sensing, utilizing gravimetric detection [ 6 – 10 ]. When a sensing film is applied, a CMUT sensor can capture a specific analyte, which is detected by observing the shift in capacitance or in resonance frequency due to the added mass as illustrated in Fig. 1 . In the sensing field, CMUTs exhibit advantageous characteristics, including high mass sensitivity and reliability [ 11 – 13 ]. Additionally, CMUTs advantageously possess low detection limits, straightforward functionalization, and compact array designs suitable for multi-analyte detection. The detection capability of a resonant mass-sensitive sensor depends critically on both the sensor mechanism and the functional sensing layer. Consequently, careful consideration must be given to the choice of sensing material. Optimal materials should exhibit several key characteristics: significant mass sensitivity, fast response and recovery kinetics, excellent reversibility, and sustained stability under operational conditions. Currently, polymeric compounds and metal oxides dominate practical applications, though novel alternative materials continue to be actively investigated in research settings. Researchers have sought in the last decade to come up with effective sensing materials that respond well to gas molecules. Consequently, graphene-based materials were discovered, showing remarkable abilities for gas sensing applications [ 14 ]. Graphene, widely recognized for its remarkable mechanical and electrical properties, has demonstrated exceptional sensitivity to external stimuli. In fact, it has the capability to detect even individual molecules within diluted gas samples, as initially demonstrated by the team led by Novoselov in 2007 [ 15 ]. Graphene material is a thin layer composed of a lattice of carbon atoms arranged in a honeycomb-like pattern and serving as the fundamental unit for carbon-based substances, has garnered significant attention due to its immense potential for diverse applications. Graphene-based materials have lately garnered significant interest as contributors to gas sensing applications, primarily because of the ultra-thin, two-dimensional nature of graphene sheets and their exceptional properties [ 16 – 19 ]. Numerous applications of graphene depend on graphene oxides (GO), which possess active functional groups (additives) on their surfaces. The functional groups grant them additional benefits, including excellent dispersibility and straightforward functionalization [ 20 , 21 ]. Herein, our goal involves presenting a bi-layer lumped-mass analytical model, employing GO as a typical sensing layer. We aim to evaluate the static and dynamic characteristics of the integrated GO/CMUT to capture the gas surface interactions, electrical behavior, and mechanical responses of the integrated MEMS movable gas sensors. Static and dynamic modeling of GO-integrated CMUTs is crucial for advancing gas sensing applications. Static modeling helps optimize GO’s gas adsorption properties, ensuring high sensitivity and selectivity by analyzing steady-state changes in electrical parameters upon gas exposure. Meanwhile, dynamic modeling captures real-time response kinetics, enabling fast detection and recovery by assessing transient behaviors, resonance shifts, and GO-CMUT interactions under varying conditions. Without accurate analytical modeling, GO/CMUT-based gas sensors risk sluggish response, poor stability, or unreliable operation, underscoring the necessity of comprehensive static and dynamic modeling for developing high-precision, and durable sensing system. The developed analytical approach maintains validity for any gas-sensing material combination where adsorption phenomena occur. 2. Analytical Bi-Layer Lumped-Mass Model Due to the intricate nature of analytical modeling for GO/CMUT, a bi-layer lumped-mass model is employed to simplify the analysis of CMUT by representing a spatially distributed system with discrete elements. The bi-layer lumped-mass system generally consists of an effective mass, an effective spring, and an effective damper, as depicted in Fig. 2 . This model is commonly used to estimate the behavior of complex systems by consolidating various physical, mechanical, electrical, and other properties into these discrete components, thereby reducing overall complexity. The accuracy and relevance of the results hinge on how effectively the physics of the individual elements are represented. Figure 3 illustrates the CMUT featuring a deformed top bi-layer plate. The top bi-layer plate of the CMUT is modelled in a circle shape, as described in Fig. 4 , for the reason that a circular top plate experiences the least stress at its edges compared to other shapes [ 22 ]. Under the assumption of a conductive top movable plate and no external load, the governing equation of motion for the lumped-mass system is given by: $$\:\begin{array}{c}{m}_{bi}\ddot{w}+{b}_{bi}\dot{w}+{k}_{bi}w=\:{F}_{es}\#\left(1\right)\end{array}$$ where m bi represents the modified effective mass accounting for the additional mass contribution from the GO coating can be determined by combining the mass of the GO film with that of the CMUT's top plate, b bi represents the effective damping, k bi represents the effective stiffness, F e s represents the attractive or electrostatic force generated at applied DC voltage, and w represents the bi-layer plate displacement. The top bi-layer plate's effective mass ( m bi ) can be expressed as [ 23 ]: $$\:\begin{array}{c}{m}_{bi}=1.84\:\left({m}_{m}+{m}_{s}\right)=1.84\:\left({\rho\:}_{m}\:{A}_{m}\:{t}_{m}+{\rho\:}_{s}\:{A}_{s}\:{t}_{s}\right)\#\left(2\right)\end{array}$$ This formulation accounts for the mechanical properties of both constituent layers: the silicon (Si) structural top plate of the CMUT and the GO sensing element. The Si component is characterized by its mass density ( ρ m ), cross-sectional area ( A m = πa 2 ), and vertical dimension ( t m ), while the GO layer is similarly defined by its density ( ρ s ), active area ( A s = A m =A ), and thickness profile ( t s ). The expression of the linear effective stiffness of the equivalent spring force is given by [ 23 ]: $$\:\begin{array}{c}{k}_{bi}=\frac{192\:\pi\:\:{D}_{bi}}{{a}^{2}}\#\left(3\right)\end{array}$$ where a represents the top bi-layer plate radius. The ultrathin GO coating combined with the top Si surface layer of the CMUT forms a composite top plate structure, which adds mechanical rigidity to the top layer of the CMUT, altering its static deflection profile. The flexural rigidity ( D o ) of the uncoated circular top plate of the CMUT is given by [ 24 ]: $$\:\begin{array}{c}{D}_{o}=\frac{{E}_{m}\:{{t}_{m}}^{3}}{12\left(1-{{\nu\:}_{m}}^{2}\right)}\#\left(4\right)\end{array}$$ However, when the sensing GO layer is introduced on top of the circular top plate of the CMUT, the bi-layer flexural rigidity ( D bi ) can be calculated using the following parameters [ 25 ]: $$\:\begin{array}{c}{D}_{bi}=\frac{\varLambda\:\lambda\:-{\chi\:}^{2}}{\varLambda\:}\#\left(5\right)\end{array}$$ where, $$\:\begin{array}{c}\varLambda\:=\frac{{E}_{m}\:{t}_{m}}{1-{{\nu\:}_{m}}^{2}}+\frac{{E}_{s}\:{t}_{s}}{1-{{\nu\:}_{s}}^{2}}\#\left(6\right)\end{array}$$ $$\:\begin{array}{c}\lambda\:=\frac{{E}_{m}\:{t}_{m}^{3}}{3\left(1-{{\nu\:}_{m}}^{2}\right)}+\frac{{E}_{s}\left({\left({t}_{m}+{t}_{s}\right)}^{3}-{t}_{m}^{3}\right)}{3\left(1-{{\nu\:}_{s}}^{2}\right)}\#\left(7\right)\end{array}$$ $$\:\begin{array}{c}\chi\:=\frac{{E}_{m}\:{t}_{m}^{2}}{2\left(1-{{\nu\:}_{m}}^{2}\right)}+\frac{{E}_{s}\left({\left({t}_{m}+{t}_{s}\right)}^{2}-{t}_{m}^{2}\right)}{2\left(1-{{\nu\:}_{s}}^{2}\right)}\#\left(8\right)\end{array}$$ where E m , ν m , E s and ν s represent the Young’s modulus and Poisson’s ratio of the GO sensing layer material and top Si layer of the CMUT, respectively. The electrostatic force input F es used to drive the bi-layer parallel plate capacitor can be obtained by applying Castigliano's Theory, which involves taking the derivative of the potential energy U with respect to the top plate deflection w as: $$\:\begin{array}{c}{F}_{es}\text{=}\text{}\text{}\frac{\partial\:U}{\partial\:w}=\frac{{\epsilon\:}_{o\:}\text{}A\text{}{V}^{\text{2}}}{\text{2}{\text{(}{d}_{o}\text{-\:w)}}^{\text{2}}}\#\left(9\right)\end{array}$$ where d o denotes the gap height and ε o denotes the vacuum permittivity. The value of ε o is equal to 8.8542×10 − 12 F/m. The pull-in voltage of the GO/CMUT is a critical parameter that defines the voltage at which the electrostatic force F es acting on the top plate overcomes the mechanical restoring force F s =k eq w , causing the top plate to collapse into the gap and become in contact with the substrate due to pull-in limit point instability condition. The pull-in voltage investigation focuses on the electromechanical response of the deformable electrode under gradually increasing electrostatic potentials, where the applied DC voltage induces electrostatic attraction, causing structural deformation. The maximum deflection occurs at the center of the top resonating plate, reaching 1/3 of the gap height do before the system transitions into an unstable regime, marked by pronounced linear or nonlinear dynamics [ 26 ]. Understanding this phenomenon is essential for optimizing the design and performance of GO/CMUTs in gas sensing applications. In electrostatically actuated MEMS, pull-in limit point instability is a distinct feature that holds significant consequences. It frequently results in device failure, encompassing issues like stiction (adhesion), electrostatic discharge, and dielectric charging [ 27 – 29 ]. The pull-in voltage can be determined by [ 30 ]: $$\:\begin{array}{c}{V}_{pi}=\sqrt{\frac{\text{8}{k}_{bi}{{d}_{o}}^{\text{3}}\:}{27\:{\epsilon\:}_{o\:}\text{}A}}\#\left(10\right)\end{array}$$ The Capacitance-Voltage (C-V) characteristic, essential for defining the CMUT operational range and pull-in voltage, was modeled by solving the electromechanical force equilibrium. The plate displacement w ( V DC ) was obtained by numerically integrating the resulting ordinary differential equation from 0 to V pi . Capacitance was calculated as: $$\:\begin{array}{c}C({V}_{DC}\text{)}\text{}\text{=}\frac{{\epsilon\:}_{o\:}\text{}{A}_{m}\text{}}{{d}_{o}\text{-\:w(}{V}_{DC}\text{)}\:}\#\left(11\right)\end{array}$$ This approach efficiently generates the full C-V curve, which reveals a maximum stable capacitance of \(\:{C}_{pi}=1.5{C}_{o}\) . The derivative of this curve, d C/ d V , provides the small-signal sensitivity, which peaks at the optimal bias point for maximizing electromechanical coupling while ensuring stability. In the context of GO/CMUTs, the eigenvalue analysis involves solving the system's equation of motion that is described in Eq. (1) to find the natural frequencies \(\:\left({f}_{bi}\right)\) . This is crucial for understanding the device's resonance behavior. Herein, as we developed the system based on the lumped-mass model, only the linear center resonance frequency at which the system tends to oscillate in the absence of damping and driving forces is investigated as a function of the varying static DC voltage. Thus, the eigenvalue as a function of DC bias voltage of the integrated GO/CMUT system, accounting for changes in both effective mass and stiffness can be obtained by taking the Taylor series of Eq. (1), which yields to: $$\:\begin{array}{c}{\omega\:}_{bi}=\:\pm\:\sqrt{\frac{\left({k}_{bi}-{\epsilon\:}_{o\:}\text{}A\text{}{V}_{DC}^{2}\right)}{{\:m}_{bi}\:{d}_{o}^{3}}}\#\left(12\right)\end{array}$$ and the corresponding shift in the linear natural frequency can be obtained by: $$\:\begin{array}{c}{f}_{bi}=\:\frac{{\omega\:}_{bi}}{2\pi\:}\#\left(13\right)\end{array}$$ The balance between the modified mass and modified stiffness effects in the resonance frequency is quantified by the dimensionless parameter: $$\:\begin{array}{c}\phi\:=\frac{{k}_{bi}\:{t}_{s}^{2}\:}{{E}_{m}\:{t}_{m}^{2}}\#\left(14\right)\end{array}$$ For dimensionless parameter values φ 1, mass-dominated effects ( \(\:{m}_{bi}\) -driven) prevail, leading to a decreased resonant frequency ( \(\:{f}_{bi}\) ↓). Unlike Si and SiO 2 , whose properties have been thoroughly characterized [ 31 ], GO demonstrates considerable variation in its mechanical and dielectric behavior depending on synthesis conditions. This inherent variability necessitates careful compilation of GO parameters from different references [ 32 – 34 ]. Thus, Table 1 and Table 2 provide information on the parameters and properties of the selected materials that are utilized for GO/CMUT analytical modeling. Table 1 Material characteristics of SiO2, Si and GO [ 31 – 34 ]. Parameter SiO2 Si GO Young’s modulus (GPa) 70 130 25 Poisson’s ratio 0.17 0.28 0.28 Density (kg/m3) 2200 2329 200 Relative permittivity (dielectric constant) 4.2 - 100 Table 2 The geometrical characteristics of the GO/CMUT cell [ 31 – 34 ]. Parameters Value Top bi-layer plate radius (µm) 28 Si plate thickness (µm) 2.2 GO plate thickness (nm) 200 Insulation layer thickness (nm) 250 Cavity depth (nm) 250 3. Results and Discussion The integration of GO layer onto CMUT introduces both static and dynamic effects that significantly influence device performance. 3.1. Static Effect of GO/CMUT The GO film influences the static behaviors of CMUT as it increases its overall capacitance. GO material has a high dielectric constant, which enhances the CMUT's capacitance, improving static displacement sensitivity, and hence leads to better electromechanical coupling. GO layer can also alter the top movable plate’s stiffness depending on their thickness and deposition method. A thin GO layer may slightly increase stiffness, affecting the static deflection under DC bias. 3.1.1. Pull-in Analysis To analyze the additional influence of the GO layer on the pull-in behavior of the CMUT, the pull-in voltage of the uncoated CMUT top electrode was initially evaluated using Eq. (10) with the replacement of the eqivalent stiffness parameter \(\:\left({k}_{eq}=\frac{192\:\pi\:\:{D}_{o}}{{a}^{2}}\right)\) with the modified stiffness parameter \(\:{k}_{bi}\) . This preliminary analysis focused exclusively on the mechanical contribution of the top plate's equivalent spring system, disregarding any additional layers. Therefore, a systematic parameter analysis was performed by sweeping the DC bias potential from 0 to 160 V. Figure 5 demonstrates the relationship between the applied DC voltage and the static displacement at the center of the top movable plate. The intersection of the stable (blue solid curve) and unstable (red solid curve) equilibrium solutions reveals a saddle-node bifurcation of the bare CMUT, indicating the onset of static pull-in. The lumped-parameter model predicted a pull-in threshold of 143 V, occurring when the top resonating plate's central displacement reached 83 nm. The introduction of a 200 nm GO layer significantly alters the electromechanical response of the CMUT structure. The methodological approach to investigating this change involved using Eq. (10) with the introduction of \(\:{k}_{bi}\) . A notable shift in the electromechanical response of the CMUT assembly is observed in the pull-in phenomenon in Fig. 5 . While the uncoated top plate exhibited a pull-in voltage of 143 V at a center deflection of 83 nm, the GO-coated CMUT demonstrated an increased pull-in threshold of 147.5 V for the same displacement. This enhancement in pull-in voltage can be attributed to the GO layer’s mechanical stiffening effect, which modifies the structural compliance of the movable plate. The additional stiffness introduced by the GO film resists electrostatic attraction more effectively, thereby delaying the onset of pull-in limit point instability. Furthermore, the stable (blue dotted curve) and unstable (red dotted curve) equilibrium branches of the GO-integrated system shift slightly compared to the bare CMUT, indicating a modified bifurcation behavior. These findings suggest that GO functionalization not only improves the device’s electromechanical stability but also extends its operational range before nonlinear collapse occurs. The C-V characteristics in Fig. 6 , derived from electrostatic-mechanical equilibrium, highlight the impact of GO functionalization. While both devices start from a 0.872 fF zero-bias capacitance, the GO/CMUT (red curve) shows an attenuated response, reaching a maximum capacitance of only 1.20 fF compared to 1.35 fF for the conventional device (green curve). This suppression confirms the increased effective stiffness imparted by the GO layer. 3.1.2. Eigenvalue Analysis For the GO-integrated CMUT, the fundamental resonant frequency \(\:\left({f}_{bi}\right)\) was derived through an eigenvalue extraction procedure applied to the system's governing equation of motion (Eq. (1)). This formulation explicitly accounted for the influence of electrostatic forces inherent in the device's operation. The computational process involved employing Eq. (12) to compute the system's eigenvalues across a range of direct current (DC) bias voltages. These eigenvalues were subsequently converted into their corresponding natural frequencies using the relationship defined in Eq. (13). The initial baseline angular resonance frequency \(\:{\omega\:}_{o}\) of the non-functionalized CMUT was established by applying this analytical framework to its equivalent mass \(\:{\:m}_{eq}\) and stiffness \(\:{k}_{eq}\) parameters. This same methodology was then used to determine the eigenvalue \(\:{\omega\:}_{bi}\) for the GO-integrated device, enabling a direct comparison of the frequency response \(\:{f}_{bi}\) as a function of DC bias voltage. As illustrated in Fig. 7 , the natural frequency characteristics are contrasted for the CMUT both with (denoted by a solid black curve) and without (denoted by a dotted red curve) the GO layer. The data reveals a pronounced spring-hardening behavior attributable to the GO integration. This phenomenon, indicated by the shift in the frequency curve, signifies a displacement-dependent augmentation of the system's effective stiffness, which consequently elevates its resonant frequency. It is observed that although elevating the DC bias voltage produces a consistent reduction in resonant frequency for both configurations, the GO-coated CMUT perpetually operates at a higher resonant frequency than its unmodified counterpart throughout the entire range of applied bias voltages. 3.1.3. Mass Sensitivity It is critical to analyze the mass sensitivity and resonant characteristics of the sensor after functionalizing GO onto the top plate surface of the CMUT to determine the sensor’s feasibility for gas sensing applications. This analysis focuses exclusively on the GO layer to investigate its effect on the core operational characteristics of the CMUT. The GO layer is configured with a predetermined thickness, considerably thinner than the top plate, to simulate practical operational scenarios. The mass of the GO sensing layer was determined to be 0.9852 ng through incorporating the GO parameters documented in Table 1 and Table 2 into Eq. (2). This mass represents approximately 0.78% of the mass of the CMUT's Si top plate. Consequently, the sensor's sensitivity ( S bi ), quantified as the ratio of frequency variation to mass change, can be mathematically represented as [ 35 ]: $$\:\begin{array}{c}{S}_{bi}=\frac{\varDelta\:f}{\varDelta\:m}\#\left(15\right)\end{array}$$ where \(\:\varDelta\:f\) denotes Frequency shift due to mass change ( \(\:\varDelta\:m\) ). Eq. (15) demonstrates that enhanced mass sensitivity necessitates both elevated resonance frequencies and reduced effective mass, conditions that are optimally satisfied through minimization of the sensor's radial dimensions. To analytically characterize the mass loading sensitivity resulting from the proportional mass contribution of the GO sensing layer to the mass of the top Si plate of CMUT, incremental mass variations were introduced to the mass of GO film ( \(\:{m}_{s}\) ). The mass loading ratio was defined as discrete integer increments ranging from 0 to 8 times the \(\:{m}_{s}\) . Figure 8 illustrates the correlation between mass loading sensitivity and resonance frequency shift across different bias voltages ranging from 0 to 80 V with an incremental increase of 5 V. The analysis revealed that mass loading sensitivity scales proportionally with both frequency shift and DC bias voltage, achieving maximum sensitivity of 24.19 Hz/pg at 286.255 Hz under 80 V DC. This is an indication that the spring effect of GO film is the most predominant parameter, leading to an overall increase in resonance frequency shift and consequently enhanced mass loading sensitivity. 3.2. Dynamic Effect of GO/CMUT The incorporation of a GO coating significantly alters the operational dynamics of CMUTs through modifications to both inertial and elastic properties. The GO/CMUT resonant characteristics exhibit a dependence on GO thickness, where enhanced mass loading from thicker films tends to decrease the fundamental resonance frequency, whereas thin, high-stiffness coatings can elevate it. Additionally, GO's intrinsic viscoelastic properties contribute to energy dissipation, effectively lowering the system's \(\:{Q}_{F}\) while expanding its operational bandwidth. Such bandwidth enhancement proves advantageous for broadband acoustic applications, though it may compromise sensitivity in frequency-specific implementations. 3.2.1. Time-Varying Dynamic Response The time-varying transient response of the CMUT was mathematically investigated with particular emphasis on characterizing the GO layer influence through: $$\:\begin{array}{c}\ddot{w}\left(t\right)+\frac{{\omega\:}_{bi}}{{Q}_{F}}\dot{w}\left(t\right)+{\omega\:}_{bi}^{2}\:w\left(t\right)=\frac{{\epsilon\:}_{o\:}\text{}{A}_{m}\text{}{V}_{DC}^{2}}{{2\:m}_{bi}{\text{(}{d}_{o}\text{-\:w(t))}}^{\text{2}}}\:\#\left(16\right)\end{array}$$ An initial examination was performed on bare CMUT, with Fig. 9 (a) presenting the transient response evaluated for 2 µs derived from Eq. (16) in the absence of GO film under 40 V DC bias voltage. It is worth noting that the evaluation for bare CMUT was accounted by the mass of the top plate of the CMUT ( \(\:{m}_{m}\) ) and the angular resonance frequency ( \(\:{\omega\:}_{o}\) ), given by [ 23 ]: $$\:\begin{array}{c}{\omega\:}_{o}=\frac{2.95\:{t}_{m}}{{a}^{2}}\sqrt{\frac{{E}_{m}\:}{{\rho\:}_{m}\:\left(1-{\nu\:}_{m}^{2}\right)}}\#\left(17\right)\end{array}$$ and the corresponding center resonance frequency is then given by: $$\:\begin{array}{c}{f}_{o}=\frac{{\omega\:}_{o}\:}{2\pi\:}\#\left(18\right)\end{array}$$ The results demonstrate a characteristic underdamped oscillatory regime with a predetermined \(\:{Q}_{F}=10\) , where elastic restoring forces substantially outweighed damping effects throughout the vibrational decay process. Key metrics extracted include a peak displacement overshoot of 6.59 nm and stabilization to a steady-state amplitude of 3.56 nm within the 2 µs. These rapid settling dynamics directly contribute to enhanced device precision and operational stability [ 36 ]. Comparative analysis of the GO-functionalized system as shown in Fig. 9 (b), reveals significant modifications to the transient profile. The introduction of GO induces measurable stiffening effects, manifesting as both decreased overshoot magnitude at peak displacement overshoot of 5.66 nm and declined steady-state condition at 3.05 nm. Notably, while the temporal stabilization threshold (< 2 µs) remains comparable to the bare CMUT, the modified mechanical impedance alters the energy dissipation trajectory, reflecting the composite top movable plate's adjusted dynamic properties. Subsequently, the CMUT's steady-state behavior was analyzed both in the presence and absence of the GO layer to examine its impact on harmonic oscillation dynamics, as described by the governing equation: $$\:\begin{array}{c}\ddot{w}\left(t\right)+\frac{{\omega\:}_{bi}}{{Q}_{F}}\dot{w}\left(t\right)+{\omega\:}_{bi}^{2}\:w\left(t\right)=\frac{{\epsilon\:}_{o\:}\text{}{A}_{m}{\left({V}_{DC}+{V}_{AC}\:sin\:\left(2\pi\:{f}_{ex}t\right)\right)}^{2}\text{}}{{2\:m}_{bi}{\text{(}{d}_{o}\text{-\:w(t))}}^{\text{2}}}\:\#\left(19\right)\end{array}$$ The CMUT was first characterized in its bare state by substituting the bi-layer mass ( \(\:{m}_{bi}\) ) with the mass of the Si top plate ( \(\:{m}_{m}\) ) and replacing the bi-layer resonance frequency ( \(\:{\omega\:}_{bi}\) ) with the intrinsic resonance frequency of the unmodified CMUT ( \(\:{\omega\:}_{o}\) ) in Eq. (19). The investigation utilized a quality factor of \(\:{Q}_{F}=10\) , with excitation frequency swept near its fundamental resonance frequency ( \(\:{f}_{ex}={f}_{o}=10\:\text{M}\text{H}\text{z}\) ). The electrical biasing conditions consisted of a 40 V DC offset superimposed with a 10 V AC driving signal. Thus, the results as seen in Fig. 10 (a) revealed a rapid stabilization period, transitioning from transient oscillations to steady-state operation in less than 1 µs, accompanied by peak amplitude around the value of 6.49 nm. The response demonstrated a signal-to-noise ratio (SNR) significantly corresponds to the specified DC and AC excitation parameters. Under the same conditions, a comparative examination of the GO-integrated CMUT system as shown in Fig. 10 (b), analyzed via Eq. (19). The results demonstrated substantial alterations in the equilibrium vibrational characteristics such that the GO layer induces a measurable reduction in stabilized oscillation magnitude at around 5.46 nm, primarily attributable to the dominant stiffness characteristics of GO which effectively suppress vibrational displacement amplitudes at steady-state conditions. 3.2.2. Frequency Characteristics due to Varying AC Voltage To characterize the frequency response of the CMUT, the mid-point displacement amplitude \(\:\left(w\right)\) of the oscillating plate was evaluated both before and after GO layer functionalization. The LTI numerical method was implemented to solve the fundamental dynamical Eq. (19) for the uncoated configuration and for the GO-integrated system. From the stabilized vibrational regime, critical response characteristics were derived through comprehensive processing of the time-history data. Figure 11 (a) and (b) depict the frequency response characteristics of the CMUT device under both unmodified and GO-functionalized conditions, demonstrating the comparative variations in central oscillating plate deflection ( \(\:w\) ). These results were derived for a \(\:{Q}_{F}=10\) , under a constant DC bias voltage of 40 V and an AC actuation voltage incrementally increased from 5 V to 40 V in steps of 5 V. Notably, the selected DC bias voltage represented approximately 30% of the critical pull-in voltage threshold for all assessment configurations. The frequency sweep spans from 100 Hz to 30 MHz, with a resolution of 0.1 MHz. For the bare CMUT, The findings reveal that as the excitation frequency rises, the system's response amplifies until it reaches a second-order superharmonic resonance at \(\:{f}_{ex}=\frac{1}{2}{f}_{o}\:at\:5.055\:MHz\) . Beyond this point, the response continues to grow until it attains the primary resonance at \(\:{f}_{ex}={f}_{o}\:at\:10.134\:MHz\) t, which aligns with the 40 V DC condition identified during the eigenfrequency analysis. It is noteworthy that the response at \(\:{f}_{o}\) exhibits linear behavior under the applied forcing conditions. The double peaks observed in Fig. 11 (a) and (b) arise due to the multi-frequency excitation nature of the electrostatic force. This behavior can be attributed to the quadratic nonlinearity inherent in the electrostatic force, which activates mechanisms that generate responses at both twice and half the excitation frequency within the system. This can be clarified as the following: $$\:{F}_{es}\propto\:{V\left(t\right)}^{2}={\left({V}_{DC}+{V}_{AC}\:sin\:\left(2\pi\:{f}_{ex}t\right)\right)}^{2}$$ $$\:\begin{array}{c}={{V}_{DC}}^{2}+{2V}_{DC}{V}_{AC}{sin}\left(2\pi\:{f}_{ex}t\right)+{{V}_{AC}}^{2}{sin\:}^{2}\left(2\pi\:{f}_{ex}t\right)\end{array}$$ $$\:={{V}_{DC}}^{2}+{2V}_{DC}{V}_{AC}\:sin\:\left(2\pi\:{f}_{ex}t\right)+\frac{1}{2}{{V}_{AC}}^{2}-\frac{1}{2}{{V}_{AC}}^{2}cos\left(4\pi\:{f}_{ex}t\right)$$ $$\:\begin{array}{c}=\left({{V}_{DC}}^{2}+\frac{1}{2}{{V}_{AC}}^{2}\right)+{2V}_{DC}{V}_{AC}{sin}\left(2\pi\:{f}_{ex}t\right)-\frac{1}{2}{{V}_{AC}}^{2}cos\left(4\pi\:{f}_{ex}t\right)\#\left(20\right)\end{array}$$ Equation (20) indicates that the electrostatic force comprises three distinct components: a static term proportional to \(\:\left({{V}_{DC}}^{2}+\frac{1}{2}{{V}_{AC}}^{2}\right)\) , a lower harmonic term at frequency \(\:{f}_{ex}\) proportional to \(\:\left({2V}_{DC}{V}_{AC}\right)\) , and a higher harmonic term at frequency \(\:{2f}_{ex}\) proportional to \(\:\left(\frac{1}{2}{{V}_{AC}}^{2}\right)\) . When the excitation frequency is doubled, the primary resonance of the system is activated, which occurs when the CMUT is driven at half the primary resonance frequency ( \(\:{f}_{ex}=\frac{1}{2}{f}_{o}\) ). In contrast, when the excitation frequency is halved, the primary resonance is initiated, taking place when the CMUT is driven at the primary resonance frequency itself ( \(\:{f}_{ex}={f}_{o}\) ). It is noteworthy that the superharmonic resonance is positively correlated with the quality factor and inversely related to the damping factor. The amplitude of the superharmonic peak grows as the damping factor diminishes, a relationship governed by the equation \(\:\xi\:=\frac{1}{2\:{Q}_{F}}\) , following a trend consistent with the fundamental frequency curve. The integration of GO layer onto the CMUT’s top plate substantially modified its resonant behavior under identical excitation conditions. As evidenced in Fig. 11 (b), the functionalization process induced a systematic increase in resonance frequencies, with the superharmonic resonance shifting to 5.155 MHz and the fundamental mode transitioning to 10.376 MHz. This frequency elevation stems primarily from the GO layer's pronounced elastic properties, which simultaneously reduced oscillation amplitudes relative to the uncoated CMUT (Fig. 11 (a)), demonstrating the material's stiffness-dominated influence on the system's dynamic response. 3.2.3. Frequency Characteristics due to Varying DC Voltage Subsequently, frequency characteristics due to varying DC bias voltage and fixed AC actuation voltage were investigated for two configurations. The first configuration analyzed unfunctionalized CMUT performance across a DC bias range of 0-117.5 V with a constant 10 V AC excitation, while maintaining a quality factor of \(\:{Q}_{F}=10\) . The second configuration evaluated the modified system incorporating GO functional layer on the CMUT’s top plate, assessed under an extended DC bias range of 0-121.5 V with identical AC driving conditions (10 V) and equivalent quality factor \(\:{Q}_{F}=10\) . Initially, the CMUT was analyzed both in its bare form and with an integrated GO layer under zero DC bias conditions. As illustrated in Fig. 12 (a), the computational findings reveal that without an applied DC voltage, the excitation frequency exhibited a twofold increase. Specifically, the primary resonance peak occurred at 5.055 MHz for the uncoated CMUT, whereas the GO-coated CMUT, as illustrated in Fig. 12 (b) displayed a shifted resonance at 5.155 MHz, with both configurations demonstrating attenuated vibrational amplitudes. Under this operating condition, the system exhibits a primarily linear response due to the dominance of linear stiffness effects. As the DC bias voltage is incrementally increased, as shown in Fig. 13 , the response exhibited a bifurcation of the resonant behavior into second-order superharmonic and distinct primary modes. This phenomenon stems from the quadratic nonlinear characteristics of electrostatic forces, which produce dual-frequency excitation effects. Consequently, the system manifests simultaneous vibrational responses at both frequency-doubled and frequency-halved components relative to the input signal. It is evident that higher DC bias voltages caused a decrease in the fundamental resonance frequency, primarily due to the predominant softening effect on the effective stiffness induced by the electrostatic force, thereby constraining the operational bandwidth. Specifically, the series resonance frequency fell from 10.11 MHz to 9.32 MHz for the bare CMUT (Fig. 13 (a)) and fell from 10.34 MHz to 9.55 MHz for the functionalized GO/CMUT (Fig. 13 (b)) with increasing applied DC bias voltage. Though, the progressive elevation of DC bias voltage enhanced the system's sensitivity, evidenced by amplified resonance amplitude in both scenarios. These observations collectively highlight the critical role of DC bias on the vibrational characteristics of both device configurations. It is important to highlight that when the DC voltage exceeds 110 V with presence of 10 V AC in the both configurations, the system begins to exhibit bifurcation instabilities where the dynamic pull-in takes place. For AC + DC actuation, dynamic pull-in occurs at lower voltages than static pull-in due to resonant energy buildup. Hence, optimal AC actuation voltage exists when operated at 10–30% of DC bias voltage to avoid dynamic pull-in phenomenon. Figure 14 illustrates the system's stability characteristics, showing the cyclic fold bifurcation thresholds (CF 1 and CF 2 ) for the fundamental vibration mode. For the unmodified CMUT configuration, shown in Fig. 14 (a), these stability limits occurred at VDC = 117.5 V with VAC = 10 V, while the GO-integrated device, shown in Fig. 14 (b) exhibited similar behavior at slightly higher bias conditions (VDC = 121.5 V, and VAC = 10 V). Under these specific operating parameters, the system maintains a single stable equilibrium state, effectively suppressing the linear jump phenomena associated with bifurcation instabilities. The observed stability enhancement in the GO-modified device suggests improved dynamic performance at elevated bias voltages compared to the bare CMUT structure. When the driving frequency is swept under maximum combined DC and AC voltage excitation, the system first encounters a second-order superharmonic resonance at 4.17 MHz for the unmodified CMUT and 4.36 MHz for the GO-modified device. This resonant behavior persists until reaching the first stability threshold (CF 1 ) at 7.34 MHz and 7.53 MHz for the respective configurations, where the low-amplitude solution branch becomes unstable. At this critical juncture, where stable and unstable solution manifolds coalesce, the system undergoes a sudden jump to the second stability threshold (CF 2 ), occurring at 7.41 MHz for the bare CMUT and 7.60 MHz for its GO-functionalized counterpart. At the CF 1 point, a Floquet multiplier crosses the unit circle at + 1 [ 37 ], indicating the intersection of the lower stable solution branch with an unstable solution branch, represented by a dashed red line. Operating the system near CF 1 is highly advantageous, as the sudden jump at this point establishes a region of remarkable sensitivity, which is particularly useful for mass detection applications. This implies that increasing the amplitude of the motion can amplify linear or nonlinear effects, thereby generating bifurcation points in the frequency domain. In each of the three scenarios, the incorporation of GO layer led to a decrease in the resonance amplitudes of the movable plate while simultaneously increasing the resonance frequency. This upward shift in resonance frequency is attributed to the enhanced effective stiffness of the top plate induced by the GO layer. The magnitude of the frequency shift is largely determined by the mechanical properties and geometric parameters of the GO layer. Within the present study, the influence of the GO film’s elastic behavior outweighs the effect of added mass, resulting in a net rise in the resonance frequency. The quality factor ( \(\:{Q}_{F}\) ) also plays a crucial role in determining the frequency response characteristics of the CMUT, both with and without GO layer as shown in Fig. 15 . The quality factor directly affects the bandwidth ( BW ) of the system, as mathematically expressed in \(\:{Q}_{F}=\frac{{f}_{o}}{\varDelta\:f}=\frac{{f}_{o}}{BW}\) . A high \(\:{Q}_{F}\) corresponds to a narrower BW and a sharper resonance peak, improving sensitivity near resonance but limiting the operational frequency range. Conversely, a lower \(\:{Q}_{F}\) yields an expanded BW , facilitating operation across a wider spectrum at the cost of diminished peak sensitivity. Table 3 provides a comparative analysis of how \(\:{Q}_{F}\) influences the BW in configurations both excluding and incorporating the GO layer under conditions of 40 V DC and 30 V AC. Table 3 Influence of the quality factor on frequency bandwidth with and without the GO layer under conditions of 40 V DC and 30 V AC. Quality factor ( \(\:{\varvec{Q}}_{\varvec{F}}\) ) BW w/o GO layer (MHz) BW with GO layer (MHz) 5 2.026 2.075 10 1.013 1.037 15 0.675 0.691 20 0.506 0.518 4. Conclusion This work presents a comprehensive analytical framework for evaluating the static and dynamic performance of GO-integrated CMUTs. The GO layer’s mechanical stiffening effect significantly enhances device stability, delaying pull-in instability and increasing resonance frequencies. Static modeling confirms a 3.1% rise in pull-in voltage, while dynamic analyses reveal improved mass sensitivity and frequency response due to GO’s elastic properties. The system exhibits linear behavior under combined DC/AC excitation, with distinct superharmonic resonances and bifurcation instability points that can be leveraged for sensing applications. Furthermore, GO’s viscoelasticity optimizes the quality factor, balancing bandwidth and sensitivity. These findings demonstrate the viability of GO/CMUTs as ultrasensitive gas sensors, with tunable performance through geometric and material optimization. Future work will focus on experimental validation and integration with multi-analyte detection systems. ORCID Tirad Owais - https://orcid.org/0000-0002-9301-1707 Declarations Conflicts of Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgement This study is supported via funding from Prince Sattam bin Abdulaziz University project number (PSAU/2025/R/1446). The authors also are grateful to KFUPM, IRC-HTCM (Project # INHE-2309). References Sumner JB (1923) The detection of pentose, formaldehyde and methyl alcohol. J Am Chem Soc 45(10):2378–2380 Hajjaj AZ, Jaber N, Ilyas S, Alfosail FK, Younis MI (2020) Linear and nonlinear dynamics of micro and nano-resonators: Review of recent advances. Int J Non-Linear Mech 119:103328 Kessler Y, Liberzon A, Krylov S (2020) Flow velocity gradient sensing using a single curved bistable microbeam. J Microelectromech Syst 29(5):1020–1025 Lichtenegger A, Gesperger J, Niederleithner M, Ginner L, Woehrer A, Drexler W, Salas M (2020) Ex-vivo Alzheimer’s disease brain tissue investigation: a multiscale approach using 1060-nm swept source optical coherence tomography for a direct correlation to histology. Neurophotonics 7(3):035004–035004 Hajjaj AZ, Jaber N, Alcheikh N, Younis MI (2019) A resonant gas sensor based on multimode excitation of a buckled microbeam. IEEE Sens J 20(4):1778–1785 Thränhardt M, Eccardt PC, Mooshofer H, Hauptmann P, Degertekin L (2009) October). A resonant CMUT sensor for fluid applications. SENSORS, 2009 IEEE. IEEE, pp 878–883 Boubenia, R., Le Moal, P., Bourbon, G., Ramasso, E., & Joseph, E. (2021). CMUT-based sensor for acoustic emission application: Experimental and theoretical contributions to sensitivity optimization. Sensors, 21(6), 2042. Park, S., Yoon, I., Lee, S., Kim, H., Seo, J. W., Chung, Y., … Lee, H. J. (2018).CMUT-based resonant gas sensor array for VOC detection with low operating voltage.Sensors and Actuators B: Chemical, 273 , 1556–1563 Park, K. K., Lee, H., Kupnik, M., Oralkan, Ö., Ramseyer, J. P., Lang, H. P., … Khuri-Yakub,B. T. (2011). Capacitive micromachined ultrasonic transducer (CMUT) as a chemical sensor for DMMP detection. Sensors and Actuators B: Chemical, 160 (1), 1120–1127 Pernu T, Saarilahti J, Sillanpää T (2025) Real-time frequency matching circuit for a high sensitivity cMUT-based gas flow sensor. Sens Actuators A: Phys 383:116252 Park, K. K., Lee, H. J., Yaralioglu, G. G., Ergun, A. S., Oralkan, Ö., Kupnik, M.,… Gimzewski, J. K. (2007). Capacitive micromachined ultrasonic transducers for chemical detection in nitrogen. Applied Physics Letters , 91 (9) Zhao, L., Zhao, Y., Xia, Y., Li, Z., Li, J., Zhang, J., … Jiang, Z. (2018). A novel CMUT-based resonant biochemical sensor using electrospinning technology. IEEE Transactions on Industrial Electronics , 66 (9), 7356–7365 Nathani MU, Nazemi H, Love C, Babu Lopez Y, Swaminathan S, Emadi A (2020) Capacitive based micromachined resonators for low level mass detection. Micromachines 12(1):13 Owais T, Khater M, Al-Qahtani H (2024) Graphene-Based MEMS Devices for Gas Sensing Applications: A Review. Micro Nanostruct, 207954 Schedin F, Geim AK, Morozov SV, Hill EW, Blake P, Katsnelson MI, Novoselov KS (2007) Detection of individual gas molecules adsorbed on graphene. Nat Mater 6(9):652–655 Hill EW, Vijayaragahvan A, Novoselov K (2011) Graphene sensors. IEEE Sens J 11(12):3161–3170 Zhu Y, Murali S, Cai W, Li X, Suk JW, Potts JR, Ruoff RS (2010) Graphene and graphene oxide: synthesis, properties, and applications. Adv Mater 22(35):3906–3924 He Q, Wu S, Yin Z, Zhang H (2012) Graphene-based electronic sensors. Chem Sci 3(6):1764–1772 Liu Y, Dong X, Chen P (2012) Biological and chemical sensors based on graphene materials. Chem Soc Rev 41(6):2283–2307 Georgakilas, V., Otyepka, M., Bourlinos, A. B., Chandra, V., Kim, N., Kemp, K. C.,… Kim, K. S. (2012). Functionalization of graphene: covalent and non-covalent approaches,derivatives and applications. Chemical reviews , 112 (11), 6156–6214 Bottari, G., Herranz, M. Á., Wibmer, L., Volland, M., Rodríguez-Pérez, L., Guldi,D. M., … Torres, T. (2017). Chemical functionalization and characterization of graphene-based materials. Chemical Society Reviews , 46 (15), 4464–4500 Hsu TR (2008) MEMS and microsystems: design, manufacture, and nanoscale engineering. Wiley Fan, X., Smith, A. D., Forsberg, F., Wagner, S., Schröder, S., Akbari, S. S. A.,… Niklaus, F. (2020). Manufacture and characterization of graphene membranes with suspended silicon proof masses for MEMS and NEMS applications. Microsystems & Nanoengineering , 6 (1), 17 Wygant IO, Kupnik M, Khuri-Yakub BT (2008), November Analytically calculating membrane displacement and the equivalent circuit model of a circular CMUT cell. In 2008 IEEE ultrasonics symposium (pp. 2111–2114). IEEE Nazemi H, Emadi A (2019), October A new advanced analytical model for bi-layer circular CMUT-based gas sensors. In 2019 IEEE SENSORS (pp. 1–4). IEEE Brenner K, Ergun AS, Firouzi K, Rasmussen MF, Stedman Q (2019) & Khuri–Yakub, B. Advances in capacitive micromachined ultrasonic transducers. Micromachines, 10 (2), 152 De Boer MP, Michalske TA (1999) Accurate method for determining adhesion of cantilever beams. J Appl Phys 86(2):817–827 Van Spengen WM, Puers R, Mertens R, De Wolf I (2004) A comprehensive model to predict the charging and reliability of capacitive RF MEMS switches. J Micromech Microeng 14(4):514 Loh OY, Espinosa HD (2012) Nanoelectromechanical contact switches. Nat Nanotechnol 7(5):283–295 Ergun AS, Yaralioglu GG, Khuri-Yakub BT (2003) Capacitive micromachined ultrasonic transducers: Theory and technology. J Aerospace Eng 16(2):76–84 Owais T (2020) Capacitive Micromachined Ultrasound Transducers for Non-Destructive Testing Applications (Master's thesis, University of Waterloo) Cao C, Daly M, Singh CV, Sun Y, Filleter T (2015) High strength measurement of monolayer graphene oxide. Carbon 81:497–504 Bayer T, Bishop SR, Perry NH, Sasaki K, Lyth SM (2016) Tunable mixed ionic/electronic conductivity and permittivity of graphene oxide paper for electrochemical energy conversion. ACS Appl Mater Interfaces 8(18):11466–11475 Wan J, Jiang JW, Park HS (2017) Negative Poisson's ratio in graphene oxide. Nanoscale 9(11):4007–4012 Mølgaard MJ, Laustsen M, Jakobsen MH, Andresen TL, Thomsen EV (2018) Combined colorimetric and gravimetric CMUT sensor for detection of benzyl methyl ketone. Sens Actuators B 275:483–489 Ahmadi N, Fathalilou M, Rezazadeh G (2024) Neo-Hookean modeling of nonlinear coupled behavior in circular plates supported by micro-pillars. Sci Rep 14(1):25428 Nayfeh AH, Ibrahim RA (2001) Nonlinear interactions: analytical, computational, and experimental methods. Appl Mech Rev 54(4):B60–B61 Additional Declarations The authors declare no competing interests. Cite Share Download PDF Status: Posted Version 2 posted You are reading this latest preprint version Show more versions Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. We do this by developing innovative software and high quality services for the global research community. Our growing team is made up of researchers and industry professionals working together to solve the most critical problems facing scientific publishing. Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-7179088","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":492343240,"identity":"41ed85d7-119d-42c1-9800-06760cdc123c","order_by":0,"name":"Tirad Owais","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAABFElEQVRIie3PMUvEMBTA8YRAusT9HYL3FV4pHDep38OlRejUTsJRELyUQrpUXLv5GURwTgnoEpxvuEE5uMnhxNEO9nTyIHKjQ/5LSODHeyHE5/uPBWX5skE4ohUzpPh5w7+JMCZsi2nEanVO7F4E0vRQ2CKpGxvtR8anGcKBAnrdZh/YqeUZCaoHIP2xk4TaIo4UsFGb38edWudSPM6ADls6SdlgHCrg4UDMuzK5hGwCVDI3qQTqRIE4WWQr3W3J+G0yLDZ3/4XzRGoLQBvL4m8CYiDcOAkKZqgsAGmtItTP61yJ9GKaqCf3lNvX+rPHq3lZsRXo2TK/CczdYtNfuqfo33dN+PaInWCYIneJz+fz+Xb7Ahh1XymxYyluAAAAAElFTkSuQmCC","orcid":"","institution":"King Fahd University of Petroleum and Minerals","correspondingAuthor":true,"prefix":"","firstName":"Tirad","middleName":"","lastName":"Owais","suffix":""},{"id":492343241,"identity":"08a0ae93-3f22-4c54-989d-da41216179de","order_by":1,"name":"Mahmoud Khater","email":"","orcid":"","institution":"King Fahd University of Petroleum and Minerals","correspondingAuthor":false,"prefix":"","firstName":"Mahmoud","middleName":"","lastName":"Khater","suffix":""},{"id":492343242,"identity":"60466926-cfd9-482e-acf9-0c3a57661d07","order_by":2,"name":"Hussain Al-Qahtani","email":"","orcid":"","institution":"King Fahd University of Petroleum and Minerals","correspondingAuthor":false,"prefix":"","firstName":"Hussain","middleName":"","lastName":"Al-Qahtani","suffix":""}],"badges":[],"createdAt":"2025-07-21 15:38:16","currentVersionCode":2,"declarations":{"humanSubjects":false,"vertebrateSubjects":false,"conflictsOfInterestStatement":false,"humanSubjectEthicalGuidelines":false,"humanSubjectConsent":false,"humanSubjectClinicalTrial":false,"humanSubjectCaseReport":false,"vertebrateSubjectEthicalGuidelines":false},"doi":"10.21203/rs.3.rs-7179088/v2","doiUrl":"https://doi.org/10.21203/rs.3.rs-7179088/v2","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":93713277,"identity":"886ec582-d812-407f-9b5a-6a134080f458","added_by":"auto","created_at":"2025-10-16 18:48:08","extension":"docx","order_by":0,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":2398082,"visible":true,"origin":"","legend":"","description":"","filename":"StaticandDynamicAnalysesofElectrostaticallyActuatedMEMSGOCMUTGasSensor.docx","url":"https://assets-eu.researchsquare.com/files/rs-7179088/v2/a4b51064f13abaa4e7ab6c86.docx"},{"id":93713276,"identity":"25c88c75-15e1-4eab-9b9d-ab27227567dd","added_by":"auto","created_at":"2025-10-16 18:48:08","extension":"json","order_by":1,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":342,"visible":true,"origin":"","legend":"","description":"","filename":"rs7179088.json","url":"https://assets-eu.researchsquare.com/files/rs-7179088/v2/770d44e74a739613429a4c59.json"},{"id":93714052,"identity":"8a51e346-8145-4986-9609-1761f3bce9c4","added_by":"auto","created_at":"2025-10-16 18:56:08","extension":"xml","order_by":2,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":119062,"visible":true,"origin":"","legend":"","description":"","filename":"rs71790882enriched.xml","url":"https://assets-eu.researchsquare.com/files/rs-7179088/v2/eba965940c76053ef8901818.xml"},{"id":93714869,"identity":"04492a14-4aaf-4699-b4e0-09567f33cefd","added_by":"auto","created_at":"2025-10-16 19:12:08","extension":"png","order_by":3,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":86862,"visible":true,"origin":"","legend":"","description":"","filename":"floatimage1.png","url":"https://assets-eu.researchsquare.com/files/rs-7179088/v2/ae5e2f54e277d78235418a09.png"},{"id":93958994,"identity":"e4d98d40-9698-4453-af74-437e76354b39","added_by":"auto","created_at":"2025-10-20 16:48:30","extension":"png","order_by":4,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":506167,"visible":true,"origin":"","legend":"","description":"","filename":"floatimage10.png","url":"https://assets-eu.researchsquare.com/files/rs-7179088/v2/4dfb7ba253c2fdbf2d56b83b.png"},{"id":93958409,"identity":"08ed3e90-411c-4b27-9952-47129b3596cb","added_by":"auto","created_at":"2025-10-20 16:39:25","extension":"png","order_by":5,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":363989,"visible":true,"origin":"","legend":"","description":"","filename":"floatimage11.png","url":"https://assets-eu.researchsquare.com/files/rs-7179088/v2/60f197b598caa76370731762.png"},{"id":93714986,"identity":"a5bb45e6-5a83-46e3-949c-612a925ceab5","added_by":"auto","created_at":"2025-10-16 19:20:08","extension":"png","order_by":6,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":133425,"visible":true,"origin":"","legend":"","description":"","filename":"floatimage12.png","url":"https://assets-eu.researchsquare.com/files/rs-7179088/v2/57621f6bfd07253d6a08f357.png"},{"id":93714347,"identity":"3d4b66dc-becc-4ee4-a3bd-f204d9afc7e3","added_by":"auto","created_at":"2025-10-16 19:04:08","extension":"png","order_by":7,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":232815,"visible":true,"origin":"","legend":"","description":"","filename":"floatimage13.png","url":"https://assets-eu.researchsquare.com/files/rs-7179088/v2/60b2ba8399516f9350a954bb.png"},{"id":93714057,"identity":"fcf65884-b4f0-40d2-a904-5c2f5fb5f69f","added_by":"auto","created_at":"2025-10-16 18:56:08","extension":"png","order_by":8,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":158696,"visible":true,"origin":"","legend":"","description":"","filename":"floatimage14.png","url":"https://assets-eu.researchsquare.com/files/rs-7179088/v2/48ae2ccff5b65a8b60865b99.png"},{"id":93713283,"identity":"fcea0a04-bfba-4ef9-9933-2e15da26db14","added_by":"auto","created_at":"2025-10-16 18:48:08","extension":"png","order_by":9,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":199798,"visible":true,"origin":"","legend":"","description":"","filename":"floatimage15.png","url":"https://assets-eu.researchsquare.com/files/rs-7179088/v2/e99aeb878663f91ea27ad16f.png"},{"id":93714871,"identity":"9380f22f-0fe9-4bdd-ab64-f3d18bc39959","added_by":"auto","created_at":"2025-10-16 19:12:08","extension":"png","order_by":10,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":150702,"visible":true,"origin":"","legend":"","description":"","filename":"floatimage2.png","url":"https://assets-eu.researchsquare.com/files/rs-7179088/v2/3ef87fed300d11c5de3fa579.png"},{"id":93713287,"identity":"1e5825e0-3fb8-4c08-988e-e02f71ea5786","added_by":"auto","created_at":"2025-10-16 18:48:08","extension":"png","order_by":11,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":41300,"visible":true,"origin":"","legend":"","description":"","filename":"floatimage3.png","url":"https://assets-eu.researchsquare.com/files/rs-7179088/v2/8f89b6d50007a4de5839bd63.png"},{"id":93714351,"identity":"d6b4f527-b79b-41f7-a5be-b6c8a08aca7e","added_by":"auto","created_at":"2025-10-16 19:04:08","extension":"png","order_by":12,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":39382,"visible":true,"origin":"","legend":"","description":"","filename":"floatimage4.png","url":"https://assets-eu.researchsquare.com/files/rs-7179088/v2/8338418b388983a5038ee2a7.png"},{"id":93714355,"identity":"e95524ed-51e6-424c-97ac-bbd1df18a1c0","added_by":"auto","created_at":"2025-10-16 19:04:08","extension":"png","order_by":13,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":82767,"visible":true,"origin":"","legend":"","description":"","filename":"floatimage5.png","url":"https://assets-eu.researchsquare.com/files/rs-7179088/v2/0975fdcaf0a63a50e2a3eb3d.png"},{"id":93713306,"identity":"6862385a-da53-4467-890a-babf89a8f3ca","added_by":"auto","created_at":"2025-10-16 18:48:09","extension":"png","order_by":14,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":26479,"visible":true,"origin":"","legend":"","description":"","filename":"floatimage6.png","url":"https://assets-eu.researchsquare.com/files/rs-7179088/v2/8882c16dca757e67131af198.png"},{"id":93714066,"identity":"4cbb3e05-5a5e-4ffe-a458-5ffb30bc5fec","added_by":"auto","created_at":"2025-10-16 18:56:08","extension":"png","order_by":15,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":27944,"visible":true,"origin":"","legend":"","description":"","filename":"floatimage7.png","url":"https://assets-eu.researchsquare.com/files/rs-7179088/v2/96702075d058f48ba5130428.png"},{"id":93714356,"identity":"d73543ac-edc0-4527-96b2-61acd2e8bf51","added_by":"auto","created_at":"2025-10-16 19:04:08","extension":"png","order_by":16,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":91471,"visible":true,"origin":"","legend":"","description":"","filename":"floatimage8.png","url":"https://assets-eu.researchsquare.com/files/rs-7179088/v2/a699be82eb928c8f109c6f33.png"},{"id":93713307,"identity":"f4d5eddd-1d8f-47de-a7c4-45e7a9846c4e","added_by":"auto","created_at":"2025-10-16 18:48:09","extension":"png","order_by":17,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":170605,"visible":true,"origin":"","legend":"","description":"","filename":"floatimage9.png","url":"https://assets-eu.researchsquare.com/files/rs-7179088/v2/d14f09b7b11364a3bcd4c505.png"},{"id":93713305,"identity":"9afc4231-d27c-41e5-acb7-e963b547b9de","added_by":"auto","created_at":"2025-10-16 18:48:09","extension":"png","order_by":18,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":17511,"visible":true,"origin":"","legend":"","description":"","filename":"Onlinefloatimage1.png","url":"https://assets-eu.researchsquare.com/files/rs-7179088/v2/50a4e823a2b6cb99808044f2.png"},{"id":93714062,"identity":"e6ebbc91-4f7f-49fc-83b7-9a195c9b7498","added_by":"auto","created_at":"2025-10-16 18:56:08","extension":"png","order_by":19,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":93114,"visible":true,"origin":"","legend":"","description":"","filename":"Onlinefloatimage10.png","url":"https://assets-eu.researchsquare.com/files/rs-7179088/v2/265914f9ecabca833da7640d.png"},{"id":93714988,"identity":"ac88c50d-4896-4930-b83a-9eaec413ff25","added_by":"auto","created_at":"2025-10-16 19:20:08","extension":"png","order_by":20,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":75604,"visible":true,"origin":"","legend":"","description":"","filename":"Onlinefloatimage11.png","url":"https://assets-eu.researchsquare.com/files/rs-7179088/v2/92df62148fd1794c444072f9.png"},{"id":93714358,"identity":"e4296536-c870-4697-ac3f-af78e9afa43f","added_by":"auto","created_at":"2025-10-16 19:04:08","extension":"png","order_by":21,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":36425,"visible":true,"origin":"","legend":"","description":"","filename":"Onlinefloatimage12.png","url":"https://assets-eu.researchsquare.com/files/rs-7179088/v2/068a479a63042998ccf319f5.png"},{"id":93714874,"identity":"b91d9137-f857-4b18-b870-b7628c620d49","added_by":"auto","created_at":"2025-10-16 19:12:08","extension":"png","order_by":22,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":48959,"visible":true,"origin":"","legend":"","description":"","filename":"Onlinefloatimage13.png","url":"https://assets-eu.researchsquare.com/files/rs-7179088/v2/f3960efed2fc059b8bd8b355.png"},{"id":93713291,"identity":"6837b07a-d918-4ed0-a05c-e972e8df1d21","added_by":"auto","created_at":"2025-10-16 18:48:08","extension":"png","order_by":23,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":44608,"visible":true,"origin":"","legend":"","description":"","filename":"Onlinefloatimage14.png","url":"https://assets-eu.researchsquare.com/files/rs-7179088/v2/89dbc0cb39e74f738014fe35.png"},{"id":93713295,"identity":"084a3ec2-c6fa-44d5-b3c0-1626553c8304","added_by":"auto","created_at":"2025-10-16 18:48:08","extension":"png","order_by":24,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":45739,"visible":true,"origin":"","legend":"","description":"","filename":"Onlinefloatimage15.png","url":"https://assets-eu.researchsquare.com/files/rs-7179088/v2/a9fb5e88ca727d65d31f9f1c.png"},{"id":93714360,"identity":"88452eba-9b7c-42ea-b00f-d6d9398bcc6f","added_by":"auto","created_at":"2025-10-16 19:04:09","extension":"png","order_by":25,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":20433,"visible":true,"origin":"","legend":"","description":"","filename":"Onlinefloatimage2.png","url":"https://assets-eu.researchsquare.com/files/rs-7179088/v2/5b32d1296a4380cbc6522b6b.png"},{"id":93713302,"identity":"2384f829-05b3-4f16-9957-b5e4c9eba478","added_by":"auto","created_at":"2025-10-16 18:48:09","extension":"png","order_by":26,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":11326,"visible":true,"origin":"","legend":"","description":"","filename":"Onlinefloatimage3.png","url":"https://assets-eu.researchsquare.com/files/rs-7179088/v2/b7b26533466bcc1bd45471a6.png"},{"id":93714875,"identity":"418bff29-3637-4b25-94bc-9fe656615093","added_by":"auto","created_at":"2025-10-16 19:12:09","extension":"png","order_by":27,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":6707,"visible":true,"origin":"","legend":"","description":"","filename":"Onlinefloatimage4.png","url":"https://assets-eu.researchsquare.com/files/rs-7179088/v2/d21ec91b1a16ff0af611fcb2.png"},{"id":93713303,"identity":"3e9c1def-fd04-450c-9a2a-d5774ef617ac","added_by":"auto","created_at":"2025-10-16 18:48:09","extension":"png","order_by":28,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":25323,"visible":true,"origin":"","legend":"","description":"","filename":"Onlinefloatimage5.png","url":"https://assets-eu.researchsquare.com/files/rs-7179088/v2/8c281b5d2f75fb48f16b8727.png"},{"id":93713300,"identity":"290912d7-2c8c-42ca-b1ae-868e440c35f5","added_by":"auto","created_at":"2025-10-16 18:48:09","extension":"png","order_by":29,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":11055,"visible":true,"origin":"","legend":"","description":"","filename":"Onlinefloatimage6.png","url":"https://assets-eu.researchsquare.com/files/rs-7179088/v2/584b98dca1db09abdb294a51.png"},{"id":93714070,"identity":"f747ef16-6984-44b8-ad94-7030acd0bed5","added_by":"auto","created_at":"2025-10-16 18:56:09","extension":"png","order_by":30,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":11342,"visible":true,"origin":"","legend":"","description":"","filename":"Onlinefloatimage7.png","url":"https://assets-eu.researchsquare.com/files/rs-7179088/v2/c687729068d32752a24f9f1f.png"},{"id":93713304,"identity":"4519c42a-c169-477b-a0cb-3cf32d7e9ac5","added_by":"auto","created_at":"2025-10-16 18:48:09","extension":"png","order_by":31,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":31188,"visible":true,"origin":"","legend":"","description":"","filename":"Onlinefloatimage8.png","url":"https://assets-eu.researchsquare.com/files/rs-7179088/v2/f6be65e0905a12fa7067f4e7.png"},{"id":93713301,"identity":"f2a86931-5bbb-4b53-ac7c-34142119e48d","added_by":"auto","created_at":"2025-10-16 18:48:09","extension":"png","order_by":32,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":49306,"visible":true,"origin":"","legend":"","description":"","filename":"Onlinefloatimage9.png","url":"https://assets-eu.researchsquare.com/files/rs-7179088/v2/c00a787c5b1cde048028b1f6.png"},{"id":93713308,"identity":"50b53c3b-b203-4093-990b-5da478163497","added_by":"auto","created_at":"2025-10-16 18:48:09","extension":"xml","order_by":33,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":118223,"visible":true,"origin":"","legend":"","description":"","filename":"rs71790882structuring.xml","url":"https://assets-eu.researchsquare.com/files/rs-7179088/v2/f704ae32c82cf1997538136f.xml"},{"id":93713309,"identity":"d23a72ac-4604-4e9e-a9c9-4339c09fcf1b","added_by":"auto","created_at":"2025-10-16 18:48:09","extension":"html","order_by":34,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":133342,"visible":true,"origin":"","legend":"","description":"","filename":"earlyproof.html","url":"https://assets-eu.researchsquare.com/files/rs-7179088/v2/3373c63c342cb9fca8aa9217.html"},{"id":93960759,"identity":"8a0833ea-cd16-4741-9cf6-545d9fea9d6f","added_by":"auto","created_at":"2025-10-20 16:57:13","extension":"pdf","order_by":0,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":659582,"visible":true,"origin":"","legend":"","description":"","filename":"manuscript.pdf","url":"https://assets-eu.researchsquare.com/files/rs-7179088/v2/60f2319b-7f11-45eb-9914-46b65b4db13b.pdf"}],"financialInterests":"The authors declare no competing interests.","formattedTitle":"Static and Dynamic Analyses of Electrostatically Actuated MEMS GO/CMUT Gas Sensor","fulltext":[{"header":"1. Introduction","content":"\u003cp\u003eExtensive quantities of organic vapors are generated within the industrial sector. It is crucial to monitor their production due to their toxicity, rapid evaporation, and potential for easy dissemination into the surrounding environment. As a result, the initial devices designed to identify such organic vapors emerged during the early 1920s, attributed to James Sumner's efforts in detecting methanol and formaldehyde [\u003cspan citationid=\"CR1\" class=\"CitationRef\"\u003e1\u003c/span\u003e]. Since that time, the need for dependable and easy-to-use gas detection solutions has increased, particularly for the detection of dangerous and toxic gases. Consequently, there has been a surge of interest in developing advanced microelectromechanical systems (MEMS) and nanoscale gas sensors. MEMS gas sensors have demonstrated exceptional reliability, minimal power consumption, seamless integration capabilities, and affordability in detecting minimal quantities of various gases in parts-per-million (ppm), positioning them as leading candidates for highly sensitive sensing applications [\u003cspan citationid=\"CR2\" class=\"CitationRef\"\u003e2\u003c/span\u003e].\u003c/p\u003e\u003cp\u003eExplicitly, resonant MEMS sensors have achieved significant success due to their extremely compact size and exceptional sensitivity, which enable them to be utilized for gas sensing applications [\u003cspan additionalcitationids=\"CR4\" citationid=\"CR3\" class=\"CitationRef\"\u003e3\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR5\" class=\"CitationRef\"\u003e5\u003c/span\u003e]. These sensors operate based on how specific gas affects the resonance frequencies of the resonators in various ways.\u003c/p\u003e\u003cp\u003eIn the last decade, capacitive micromachined ultrasonic transducers (CMUTs) MEMS-based resonant sensors have been thoroughly researched for various fields of sensing, utilizing gravimetric detection [\u003cspan additionalcitationids=\"CR7 CR8 CR9\" citationid=\"CR6\" class=\"CitationRef\"\u003e6\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR10\" class=\"CitationRef\"\u003e10\u003c/span\u003e]. When a sensing film is applied, a CMUT sensor can capture a specific analyte, which is detected by observing the shift in capacitance or in resonance frequency due to the added mass as illustrated in Fig.\u0026nbsp;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003e. In the sensing field, CMUTs exhibit advantageous characteristics, including high mass sensitivity and reliability [\u003cspan additionalcitationids=\"CR12\" citationid=\"CR11\" class=\"CitationRef\"\u003e11\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR13\" class=\"CitationRef\"\u003e13\u003c/span\u003e]. Additionally, CMUTs advantageously possess low detection limits, straightforward functionalization, and compact array designs suitable for multi-analyte detection.\u003c/p\u003e\u003cp\u003e\u003c/p\u003e\u003cp\u003eThe detection capability of a resonant mass-sensitive sensor depends critically on both the sensor mechanism and the functional sensing layer. Consequently, careful consideration must be given to the choice of sensing material. Optimal materials should exhibit several key characteristics: significant mass sensitivity, fast response and recovery kinetics, excellent reversibility, and sustained stability under operational conditions. Currently, polymeric compounds and metal oxides dominate practical applications, though novel alternative materials continue to be actively investigated in research settings.\u003c/p\u003e\u003cp\u003eResearchers have sought in the last decade to come up with effective sensing materials that respond well to gas molecules. Consequently, graphene-based materials were discovered, showing remarkable abilities for gas sensing applications [\u003cspan citationid=\"CR14\" class=\"CitationRef\"\u003e14\u003c/span\u003e]. Graphene, widely recognized for its remarkable mechanical and electrical properties, has demonstrated exceptional sensitivity to external stimuli. In fact, it has the capability to detect even individual molecules within diluted gas samples, as initially demonstrated by the team led by Novoselov in 2007 [\u003cspan citationid=\"CR15\" class=\"CitationRef\"\u003e15\u003c/span\u003e]. Graphene material is a thin layer composed of a lattice of carbon atoms arranged in a honeycomb-like pattern and serving as the fundamental unit for carbon-based substances, has garnered significant attention due to its immense potential for diverse applications. Graphene-based materials have lately garnered significant interest as contributors to gas sensing applications, primarily because of the ultra-thin, two-dimensional nature of graphene sheets and their exceptional properties [\u003cspan additionalcitationids=\"CR17 CR18\" citationid=\"CR16\" class=\"CitationRef\"\u003e16\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR19\" class=\"CitationRef\"\u003e19\u003c/span\u003e]. Numerous applications of graphene depend on graphene oxides (GO), which possess active functional groups (additives) on their surfaces. The functional groups grant them additional benefits, including excellent dispersibility and straightforward functionalization [\u003cspan citationid=\"CR20\" class=\"CitationRef\"\u003e20\u003c/span\u003e, \u003cspan citationid=\"CR21\" class=\"CitationRef\"\u003e21\u003c/span\u003e].\u003c/p\u003e\u003cp\u003eHerein, our goal involves presenting a bi-layer lumped-mass analytical model, employing GO as a typical sensing layer. We aim to evaluate the static and dynamic characteristics of the integrated GO/CMUT to capture the gas surface interactions, electrical behavior, and mechanical responses of the integrated MEMS movable gas sensors. Static and dynamic modeling of GO-integrated CMUTs is crucial for advancing gas sensing applications. Static modeling helps optimize GO\u0026rsquo;s gas adsorption properties, ensuring high sensitivity and selectivity by analyzing steady-state changes in electrical parameters upon gas exposure. Meanwhile, dynamic modeling captures real-time response kinetics, enabling fast detection and recovery by assessing transient behaviors, resonance shifts, and GO-CMUT interactions under varying conditions. Without accurate analytical modeling, GO/CMUT-based gas sensors risk sluggish response, poor stability, or unreliable operation, underscoring the necessity of comprehensive static and dynamic modeling for developing high-precision, and durable sensing system. The developed analytical approach maintains validity for any gas-sensing material combination where adsorption phenomena occur.\u003c/p\u003e"},{"header":"2. Analytical Bi-Layer Lumped-Mass Model","content":"\u003cp\u003eDue to the intricate nature of analytical modeling for GO/CMUT, a bi-layer lumped-mass model is employed to simplify the analysis of CMUT by representing a spatially distributed system with discrete elements. The bi-layer lumped-mass system generally consists of an effective mass, an effective spring, and an effective damper, as depicted in Fig.\u0026nbsp;\u003cspan refid=\"Fig2\" class=\"InternalRef\"\u003e2\u003c/span\u003e.\u003c/p\u003e\u003cp\u003e\u003c/p\u003e\u003cp\u003eThis model is commonly used to estimate the behavior of complex systems by consolidating various physical, mechanical, electrical, and other properties into these discrete components, thereby reducing overall complexity. The accuracy and relevance of the results hinge on how effectively the physics of the individual elements are represented. Figure\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e3\u003c/span\u003e illustrates the CMUT featuring a deformed top bi-layer plate.\u003c/p\u003e\u003cp\u003e\u003c/p\u003e\u003cp\u003eThe top bi-layer plate of the CMUT is modelled in a circle shape, as described in Fig.\u0026nbsp;\u003cspan refid=\"Fig4\" class=\"InternalRef\"\u003e4\u003c/span\u003e, for the reason that a circular top plate experiences the least stress at its edges compared to other shapes [\u003cspan citationid=\"CR22\" class=\"CitationRef\"\u003e22\u003c/span\u003e].\u003c/p\u003e\u003cp\u003e\u003c/p\u003e\u003cp\u003eUnder the assumption of a conductive top movable plate and no external load, the governing equation of motion for the lumped-mass system is given by:\u003cdiv id=\"Equa\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equa\" name=\"EquationSource\"\u003e\n$$\\:\\begin{array}{c}{m}_{bi}\\ddot{w}+{b}_{bi}\\dot{w}+{k}_{bi}w=\\:{F}_{es}\\#\\left(1\\right)\\end{array}$$\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e\u003cp\u003ewhere \u003cem\u003em\u003c/em\u003e\u003csub\u003e\u003cem\u003ebi\u003c/em\u003e\u003c/sub\u003e represents the modified effective mass accounting for the additional mass contribution from the GO coating can be determined by combining the mass of the GO film with that of the CMUT's top plate, \u003cem\u003eb\u003c/em\u003e\u003csub\u003e\u003cem\u003ebi\u003c/em\u003e\u003c/sub\u003e represents the effective damping, \u003cem\u003ek\u003c/em\u003e\u003csub\u003e\u003cem\u003ebi\u003c/em\u003e\u003c/sub\u003e represents the effective stiffness, \u003cem\u003eF\u003c/em\u003e\u003csub\u003e\u003cem\u003ee\u003c/em\u003es\u003c/sub\u003e represents the attractive or electrostatic force generated at applied DC voltage, and w represents the bi-layer plate displacement. The top bi-layer plate's effective mass (\u003cem\u003em\u003c/em\u003e\u003csub\u003e\u003cem\u003ebi\u003c/em\u003e\u003c/sub\u003e) can be expressed as [\u003cspan citationid=\"CR23\" class=\"CitationRef\"\u003e23\u003c/span\u003e]:\u003cdiv id=\"Equb\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equb\" name=\"EquationSource\"\u003e\n$$\\:\\begin{array}{c}{m}_{bi}=1.84\\:\\left({m}_{m}+{m}_{s}\\right)=1.84\\:\\left({\\rho\\:}_{m}\\:{A}_{m}\\:{t}_{m}+{\\rho\\:}_{s}\\:{A}_{s}\\:{t}_{s}\\right)\\#\\left(2\\right)\\end{array}$$\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e\u003cp\u003eThis formulation accounts for the mechanical properties of both constituent layers: the silicon (Si) structural top plate of the CMUT and the GO sensing element. The Si component is characterized by its mass density (\u003cem\u003eρ\u003c/em\u003e\u003csub\u003e\u003cem\u003em\u003c/em\u003e\u003c/sub\u003e), cross-sectional area (\u003cem\u003eA\u003c/em\u003e\u003csub\u003e\u003cem\u003em\u003c/em\u003e\u003c/sub\u003e\u0026thinsp;\u003cem\u003e=\u0026thinsp;πa\u003c/em\u003e\u003csup\u003e\u003cem\u003e2\u003c/em\u003e\u003c/sup\u003e), and vertical dimension (\u003cem\u003et\u003c/em\u003e\u003csub\u003e\u003cem\u003em\u003c/em\u003e\u003c/sub\u003e), while the GO layer is similarly defined by its density (\u003cem\u003eρ\u003c/em\u003e\u003csub\u003e\u003cem\u003es\u003c/em\u003e\u003c/sub\u003e), active area (\u003cem\u003eA\u003c/em\u003e\u003csub\u003e\u003cem\u003es\u003c/em\u003e\u003c/sub\u003e\u003cem\u003e= A\u003c/em\u003e\u003csub\u003e\u003cem\u003em\u003c/em\u003e\u003c/sub\u003e \u003cem\u003e=A\u003c/em\u003e), and thickness profile (\u003cem\u003et\u003c/em\u003e\u003csub\u003e\u003cem\u003es\u003c/em\u003e\u003c/sub\u003e). The expression of the linear effective stiffness of the equivalent spring force is given by [\u003cspan citationid=\"CR23\" class=\"CitationRef\"\u003e23\u003c/span\u003e]:\u003cdiv id=\"Equc\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equc\" name=\"EquationSource\"\u003e\n$$\\:\\begin{array}{c}{k}_{bi}=\\frac{192\\:\\pi\\:\\:{D}_{bi}}{{a}^{2}}\\#\\left(3\\right)\\end{array}$$\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e\u003cp\u003ewhere \u003cem\u003ea\u003c/em\u003e represents the top bi-layer plate radius. The ultrathin GO coating combined with the top Si surface layer of the CMUT forms a composite top plate structure, which adds mechanical rigidity to the top layer of the CMUT, altering its static deflection profile. The flexural rigidity (\u003cem\u003eD\u003c/em\u003e\u003csub\u003e\u003cem\u003eo\u003c/em\u003e\u003c/sub\u003e) of the uncoated circular top plate of the CMUT is given by [\u003cspan citationid=\"CR24\" class=\"CitationRef\"\u003e24\u003c/span\u003e]:\u003cdiv id=\"Equd\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equd\" name=\"EquationSource\"\u003e\n$$\\:\\begin{array}{c}{D}_{o}=\\frac{{E}_{m}\\:{{t}_{m}}^{3}}{12\\left(1-{{\\nu\\:}_{m}}^{2}\\right)}\\#\\left(4\\right)\\end{array}$$\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e\u003cp\u003eHowever, when the sensing GO layer is introduced on top of the circular top plate of the CMUT, the bi-layer flexural rigidity (\u003cem\u003eD\u003c/em\u003e\u003csub\u003e\u003cem\u003ebi\u003c/em\u003e\u003c/sub\u003e) can be calculated using the following parameters [\u003cspan citationid=\"CR25\" class=\"CitationRef\"\u003e25\u003c/span\u003e]:\u003cdiv id=\"Eque\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Eque\" name=\"EquationSource\"\u003e\n$$\\:\\begin{array}{c}{D}_{bi}=\\frac{\\varLambda\\:\\lambda\\:-{\\chi\\:}^{2}}{\\varLambda\\:}\\#\\left(5\\right)\\end{array}$$\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e\u003cp\u003ewhere,\u003cdiv id=\"Equf\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equf\" name=\"EquationSource\"\u003e\n$$\\:\\begin{array}{c}\\varLambda\\:=\\frac{{E}_{m}\\:{t}_{m}}{1-{{\\nu\\:}_{m}}^{2}}+\\frac{{E}_{s}\\:{t}_{s}}{1-{{\\nu\\:}_{s}}^{2}}\\#\\left(6\\right)\\end{array}$$\u003c/div\u003e\u003c/div\u003e\u003cdiv id=\"Equg\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equg\" name=\"EquationSource\"\u003e\n$$\\:\\begin{array}{c}\\lambda\\:=\\frac{{E}_{m}\\:{t}_{m}^{3}}{3\\left(1-{{\\nu\\:}_{m}}^{2}\\right)}+\\frac{{E}_{s}\\left({\\left({t}_{m}+{t}_{s}\\right)}^{3}-{t}_{m}^{3}\\right)}{3\\left(1-{{\\nu\\:}_{s}}^{2}\\right)}\\#\\left(7\\right)\\end{array}$$\u003c/div\u003e\u003c/div\u003e\u003cdiv id=\"Equh\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equh\" name=\"EquationSource\"\u003e\n$$\\:\\begin{array}{c}\\chi\\:=\\frac{{E}_{m}\\:{t}_{m}^{2}}{2\\left(1-{{\\nu\\:}_{m}}^{2}\\right)}+\\frac{{E}_{s}\\left({\\left({t}_{m}+{t}_{s}\\right)}^{2}-{t}_{m}^{2}\\right)}{2\\left(1-{{\\nu\\:}_{s}}^{2}\\right)}\\#\\left(8\\right)\\end{array}$$\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e\u003cp\u003ewhere \u003cem\u003eE\u003c/em\u003e\u003csub\u003e\u003cem\u003em\u003c/em\u003e\u003c/sub\u003e, \u003cem\u003eν\u003c/em\u003e\u003csub\u003e\u003cem\u003em\u003c/em\u003e\u003c/sub\u003e, \u003cem\u003eE\u003c/em\u003e\u003csub\u003e\u003cem\u003es\u003c/em\u003e\u003c/sub\u003e and \u003cem\u003eν\u003c/em\u003e\u003csub\u003e\u003cem\u003es\u003c/em\u003e\u003c/sub\u003e represent the Young\u0026rsquo;s modulus and Poisson\u0026rsquo;s ratio of the GO sensing layer material and top Si layer of the CMUT, respectively. The electrostatic force input \u003cem\u003eF\u003c/em\u003e\u003csub\u003e\u003cem\u003ees\u003c/em\u003e\u003c/sub\u003e used to drive the bi-layer parallel plate capacitor can be obtained by applying Castigliano's Theory, which involves taking the derivative of the potential energy \u003cem\u003eU\u003c/em\u003e with respect to the top plate deflection \u003cem\u003ew\u003c/em\u003e as:\u003cdiv id=\"Equi\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equi\" name=\"EquationSource\"\u003e\n$$\\:\\begin{array}{c}{F}_{es}\\text{=}\\text{}\\text{}\\frac{\\partial\\:U}{\\partial\\:w}=\\frac{{\\epsilon\\:}_{o\\:}\\text{}A\\text{}{V}^{\\text{2}}}{\\text{2}{\\text{(}{d}_{o}\\text{-\\:w)}}^{\\text{2}}}\\#\\left(9\\right)\\end{array}$$\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e\u003cp\u003ewhere \u003cem\u003ed\u003c/em\u003e\u003csub\u003e\u003cem\u003eo\u003c/em\u003e\u003c/sub\u003e denotes the gap height and \u003cem\u003eε\u003c/em\u003e\u003csub\u003e\u003cem\u003eo\u003c/em\u003e\u003c/sub\u003e denotes the vacuum permittivity. The value of \u003cem\u003eε\u003c/em\u003e\u003csub\u003e\u003cem\u003eo\u003c/em\u003e\u003c/sub\u003e is equal to 8.8542\u0026times;10\u003csup\u003e\u0026minus;\u0026thinsp;12\u003c/sup\u003e F/m.\u003c/p\u003e\u003cp\u003eThe pull-in voltage of the GO/CMUT is a critical parameter that defines the voltage at which the electrostatic force \u003cem\u003eF\u003c/em\u003e\u003csub\u003e\u003cem\u003ees\u003c/em\u003e\u003c/sub\u003e acting on the top plate overcomes the mechanical restoring force \u003cem\u003eF\u003c/em\u003e\u003csub\u003e\u003cem\u003es\u003c/em\u003e\u003c/sub\u003e\u003cem\u003e=k\u003c/em\u003e\u003csub\u003e\u003cem\u003eeq\u003c/em\u003e\u003c/sub\u003e \u003cem\u003ew\u003c/em\u003e, causing the top plate to collapse into the gap and become in contact with the substrate due to pull-in limit point instability condition. The pull-in voltage investigation focuses on the electromechanical response of the deformable electrode under gradually increasing electrostatic potentials, where the applied DC voltage induces electrostatic attraction, causing structural deformation. The maximum deflection occurs at the center of the top resonating plate, reaching 1/3 of the gap height do before the system transitions into an unstable regime, marked by pronounced linear or nonlinear dynamics [\u003cspan citationid=\"CR26\" class=\"CitationRef\"\u003e26\u003c/span\u003e]. Understanding this phenomenon is essential for optimizing the design and performance of GO/CMUTs in gas sensing applications. In electrostatically actuated MEMS, pull-in limit point instability is a distinct feature that holds significant consequences. It frequently results in device failure, encompassing issues like stiction (adhesion), electrostatic discharge, and dielectric charging [\u003cspan additionalcitationids=\"CR28\" citationid=\"CR27\" class=\"CitationRef\"\u003e27\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR29\" class=\"CitationRef\"\u003e29\u003c/span\u003e]. The pull-in voltage can be determined by [\u003cspan citationid=\"CR30\" class=\"CitationRef\"\u003e30\u003c/span\u003e]:\u003cdiv id=\"Equj\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equj\" name=\"EquationSource\"\u003e\n$$\\:\\begin{array}{c}{V}_{pi}=\\sqrt{\\frac{\\text{8}{k}_{bi}{{d}_{o}}^{\\text{3}}\\:}{27\\:{\\epsilon\\:}_{o\\:}\\text{}A}}\\#\\left(10\\right)\\end{array}$$\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e\u003cp\u003eThe Capacitance-Voltage (C-V) characteristic, essential for defining the CMUT operational range and pull-in voltage, was modeled by solving the electromechanical force equilibrium. The plate displacement \u003cem\u003ew\u003c/em\u003e(\u003cem\u003eV\u003c/em\u003e\u003csub\u003e\u003cem\u003eDC\u003c/em\u003e\u003c/sub\u003e) was obtained by numerically integrating the resulting ordinary differential equation from 0 to \u003cem\u003eV\u003c/em\u003e\u003csub\u003e\u003cem\u003epi\u003c/em\u003e\u003c/sub\u003e. Capacitance was calculated as:\u003cdiv id=\"Equk\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equk\" name=\"EquationSource\"\u003e\n$$\\:\\begin{array}{c}C({V}_{DC}\\text{)}\\text{}\\text{=}\\frac{{\\epsilon\\:}_{o\\:}\\text{}{A}_{m}\\text{}}{{d}_{o}\\text{-\\:w(}{V}_{DC}\\text{)}\\:}\\#\\left(11\\right)\\end{array}$$\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e\u003cp\u003eThis approach efficiently generates the full C-V curve, which reveals a maximum stable capacitance of \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{C}_{pi}=1.5{C}_{o}\\)\u003c/span\u003e\u003c/span\u003e. The derivative of this curve, d\u003cem\u003eC/\u003c/em\u003ed\u003cem\u003eV\u003c/em\u003e, provides the small-signal sensitivity, which peaks at the optimal bias point for maximizing electromechanical coupling while ensuring stability.\u003c/p\u003e\u003cp\u003eIn the context of GO/CMUTs, the eigenvalue analysis involves solving the system's equation of motion that is described in Eq.\u0026nbsp;(1) to find the natural frequencies \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\left({f}_{bi}\\right)\\)\u003c/span\u003e\u003c/span\u003e. This is crucial for understanding the device's resonance behavior. Herein, as we developed the system based on the lumped-mass model, only the linear center resonance frequency at which the system tends to oscillate in the absence of damping and driving forces is investigated as a function of the varying static DC voltage. Thus, the eigenvalue as a function of DC bias voltage of the integrated GO/CMUT system, accounting for changes in both effective mass and stiffness can be obtained by taking the Taylor series of Eq.\u0026nbsp;(1), which yields to:\u003cdiv id=\"Equl\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equl\" name=\"EquationSource\"\u003e\n$$\\:\\begin{array}{c}{\\omega\\:}_{bi}=\\:\\pm\\:\\sqrt{\\frac{\\left({k}_{bi}-{\\epsilon\\:}_{o\\:}\\text{}A\\text{}{V}_{DC}^{2}\\right)}{{\\:m}_{bi}\\:{d}_{o}^{3}}}\\#\\left(12\\right)\\end{array}$$\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e\u003cp\u003eand the corresponding shift in the linear natural frequency can be obtained by:\u003cdiv id=\"Equm\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equm\" name=\"EquationSource\"\u003e\n$$\\:\\begin{array}{c}{f}_{bi}=\\:\\frac{{\\omega\\:}_{bi}}{2\\pi\\:}\\#\\left(13\\right)\\end{array}$$\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e\u003cp\u003eThe balance between the modified mass and modified stiffness effects in the resonance frequency is quantified by the dimensionless parameter:\u003cdiv id=\"Equn\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equn\" name=\"EquationSource\"\u003e\n$$\\:\\begin{array}{c}\\phi\\:=\\frac{{k}_{bi}\\:{t}_{s}^{2}\\:}{{E}_{m}\\:{t}_{m}^{2}}\\#\\left(14\\right)\\end{array}$$\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e\u003cp\u003eFor dimensionless parameter values φ\u0026thinsp;\u0026lt;\u0026thinsp;1, the system exhibits stiffness-dominated behavior (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{k}_{bi}\\)\u003c/span\u003e\u003c/span\u003e-driven), resulting in an increased resonant frequency (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{f}_{bi}\\)\u003c/span\u003e\u003c/span\u003e\u0026uarr;); conversely, when φ\u0026thinsp;\u0026gt;\u0026thinsp;1, mass-dominated effects (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{m}_{bi}\\)\u003c/span\u003e\u003c/span\u003e-driven) prevail, leading to a decreased resonant frequency (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{f}_{bi}\\)\u003c/span\u003e\u003c/span\u003e\u0026darr;). Unlike Si and SiO\u003csub\u003e2\u003c/sub\u003e, whose properties have been thoroughly characterized [\u003cspan citationid=\"CR31\" class=\"CitationRef\"\u003e31\u003c/span\u003e], GO demonstrates considerable variation in its mechanical and dielectric behavior depending on synthesis conditions. This inherent variability necessitates careful compilation of GO parameters from different references [\u003cspan additionalcitationids=\"CR33\" citationid=\"CR32\" class=\"CitationRef\"\u003e32\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR34\" class=\"CitationRef\"\u003e34\u003c/span\u003e]. Thus, Table\u0026nbsp;\u003cspan refid=\"Tab1\" class=\"InternalRef\"\u003e1\u003c/span\u003e and Table\u0026nbsp;\u003cspan refid=\"Tab2\" class=\"InternalRef\"\u003e2\u003c/span\u003e provide information on the parameters and properties of the selected materials that are utilized for GO/CMUT analytical modeling.\u003c/p\u003e\u003cp\u003e\u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab1\" border=\"1\"\u003e\u003ccaption language=\"En\"\u003e\u003cdiv class=\"CaptionNumber\"\u003eTable 1\u003c/div\u003e\u003cdiv class=\"CaptionContent\"\u003e\u003cp\u003eMaterial characteristics of SiO2, \u0026lt;\u0026thinsp;100\u0026thinsp;\u0026gt;\u0026thinsp;Si and GO [\u003cspan additionalcitationids=\"CR32 CR33\" citationid=\"CR31\" class=\"CitationRef\"\u003e31\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR34\" class=\"CitationRef\"\u003e34\u003c/span\u003e].\u003c/p\u003e\u003c/div\u003e\u003c/caption\u003e\u003ccolgroup cols=\"4\"\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e\u003cthead\u003e\u003ctr\u003e\u003cth align=\"left\" colname=\"c1\"\u003e\u003cp\u003eParameter\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c2\"\u003e\u003cp\u003eSiO2\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c3\"\u003e\u003cp\u003e\u0026lt;\u0026thinsp;100\u0026thinsp;\u0026gt;\u0026thinsp;Si\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c4\"\u003e\u003cp\u003eGO\u003c/p\u003e\u003c/th\u003e\u003c/tr\u003e\u003c/thead\u003e\u003ctbody\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eYoung\u0026rsquo;s modulus (GPa)\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e\u003cp\u003e70\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c3\"\u003e\u003cp\u003e130\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c4\"\u003e\u003cp\u003e25\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003ePoisson\u0026rsquo;s ratio\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e\u003cp\u003e0.17\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c3\"\u003e\u003cp\u003e0.28\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c4\"\u003e\u003cp\u003e0.28\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eDensity (kg/m3)\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e\u003cp\u003e2200\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c3\"\u003e\u003cp\u003e2329\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c4\"\u003e\u003cp\u003e200\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eRelative permittivity (dielectric constant)\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e\u003cp\u003e4.2\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c3\"\u003e\u003cp\u003e-\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c4\"\u003e\u003cp\u003e100\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003c/tbody\u003e\u003c/colgroup\u003e\u003c/table\u003e\u003c/div\u003e\u003c/p\u003e\u003cp\u003e\u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab2\" border=\"1\"\u003e\u003ccaption language=\"En\"\u003e\u003cdiv class=\"CaptionNumber\"\u003eTable 2\u003c/div\u003e\u003cdiv class=\"CaptionContent\"\u003e\u003cp\u003eThe geometrical characteristics of the GO/CMUT cell [\u003cspan additionalcitationids=\"CR32 CR33\" citationid=\"CR31\" class=\"CitationRef\"\u003e31\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR34\" class=\"CitationRef\"\u003e34\u003c/span\u003e].\u003c/p\u003e\u003c/div\u003e\u003c/caption\u003e\u003ccolgroup cols=\"2\"\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e\u003cthead\u003e\u003ctr\u003e\u003cth align=\"left\" colname=\"c1\"\u003e\u003cp\u003eParameters\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c2\"\u003e\u003cp\u003eValue\u003c/p\u003e\u003c/th\u003e\u003c/tr\u003e\u003c/thead\u003e\u003ctbody\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eTop bi-layer plate radius (\u0026micro;m)\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e\u003cp\u003e28\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eSi plate thickness (\u0026micro;m)\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e\u003cp\u003e2.2\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eGO plate thickness (nm)\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e\u003cp\u003e200\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eInsulation layer thickness (nm)\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e\u003cp\u003e250\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eCavity depth (nm)\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e\u003cp\u003e250\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003c/tbody\u003e\u003c/colgroup\u003e\u003c/table\u003e\u003c/div\u003e\u003c/p\u003e"},{"header":"3. Results and Discussion","content":"\u003cp\u003eThe integration of GO layer onto CMUT introduces both static and dynamic effects that significantly influence device performance.\u003c/p\u003e\u003cdiv id=\"Sec4\" class=\"Section2\"\u003e\u003ch2\u003e3.1. Static Effect of GO/CMUT\u003c/h2\u003e\u003cp\u003eThe GO film influences the static behaviors of CMUT as it increases its overall capacitance. GO material has a high dielectric constant, which enhances the CMUT's capacitance, improving static displacement sensitivity, and hence leads to better electromechanical coupling. GO layer can also alter the top movable plate\u0026rsquo;s stiffness depending on their thickness and deposition method. A thin GO layer may slightly increase stiffness, affecting the static deflection under DC bias.\u003c/p\u003e\u003cdiv id=\"Sec5\" class=\"Section3\"\u003e\u003ch2\u003e3.1.1. Pull-in Analysis\u003c/h2\u003e\u003cp\u003eTo analyze the additional influence of the GO layer on the pull-in behavior of the CMUT, the pull-in voltage of the uncoated CMUT top electrode was initially evaluated using Eq.\u0026nbsp;(10) with the replacement of the eqivalent stiffness parameter \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\left({k}_{eq}=\\frac{192\\:\\pi\\:\\:{D}_{o}}{{a}^{2}}\\right)\\)\u003c/span\u003e\u003c/span\u003e with the modified stiffness parameter \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{k}_{bi}\\)\u003c/span\u003e\u003c/span\u003e. This preliminary analysis focused exclusively on the mechanical contribution of the top plate's equivalent spring system, disregarding any additional layers. Therefore, a systematic parameter analysis was performed by sweeping the DC bias potential from 0 to 160 V. Figure\u0026nbsp;\u003cspan refid=\"Fig5\" class=\"InternalRef\"\u003e5\u003c/span\u003e demonstrates the relationship between the applied DC voltage and the static displacement at the center of the top movable plate. The intersection of the stable (blue solid curve) and unstable (red solid curve) equilibrium solutions reveals a saddle-node bifurcation of the bare CMUT, indicating the onset of static pull-in. The lumped-parameter model predicted a pull-in threshold of 143 V, occurring when the top resonating plate's central displacement reached 83 nm.\u003c/p\u003e\u003cp\u003eThe introduction of a 200 nm GO layer significantly alters the electromechanical response of the CMUT structure. The methodological approach to investigating this change involved using Eq.\u0026nbsp;(10) with the introduction of \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{k}_{bi}\\)\u003c/span\u003e\u003c/span\u003e. A notable shift in the electromechanical response of the CMUT assembly is observed in the pull-in phenomenon in Fig.\u0026nbsp;\u003cspan refid=\"Fig5\" class=\"InternalRef\"\u003e5\u003c/span\u003e. While the uncoated top plate exhibited a pull-in voltage of 143 V at a center deflection of 83 nm, the GO-coated CMUT demonstrated an increased pull-in threshold of 147.5 V for the same displacement. This enhancement in pull-in voltage can be attributed to the GO layer\u0026rsquo;s mechanical stiffening effect, which modifies the structural compliance of the movable plate. The additional stiffness introduced by the GO film resists electrostatic attraction more effectively, thereby delaying the onset of pull-in limit point instability. Furthermore, the stable (blue dotted curve) and unstable (red dotted curve) equilibrium branches of the GO-integrated system shift slightly compared to the bare CMUT, indicating a modified bifurcation behavior. These findings suggest that GO functionalization not only improves the device\u0026rsquo;s electromechanical stability but also extends its operational range before nonlinear collapse occurs.\u003c/p\u003e\u003cp\u003e\u003c/p\u003e\u003cp\u003eThe C-V characteristics in Fig.\u0026nbsp;\u003cspan refid=\"Fig6\" class=\"InternalRef\"\u003e6\u003c/span\u003e, derived from electrostatic-mechanical equilibrium, highlight the impact of GO functionalization. While both devices start from a 0.872 fF zero-bias capacitance, the GO/CMUT (red curve) shows an attenuated response, reaching a maximum capacitance of only 1.20 fF compared to 1.35 fF for the conventional device (green curve). This suppression confirms the increased effective stiffness imparted by the GO layer.\u003c/p\u003e\u003cp\u003e\u003c/p\u003e\u003c/div\u003e\u003cdiv id=\"Sec6\" class=\"Section3\"\u003e\u003ch2\u003e3.1.2. Eigenvalue Analysis\u003c/h2\u003e\u003cp\u003eFor the GO-integrated CMUT, the fundamental resonant frequency \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\left({f}_{bi}\\right)\\)\u003c/span\u003e\u003c/span\u003e was derived through an eigenvalue extraction procedure applied to the system's governing equation of motion (Eq.\u0026nbsp;(1)). This formulation explicitly accounted for the influence of electrostatic forces inherent in the device's operation. The computational process involved employing Eq.\u0026nbsp;(12) to compute the system's eigenvalues across a range of direct current (DC) bias voltages. These eigenvalues were subsequently converted into their corresponding natural frequencies using the relationship defined in Eq.\u0026nbsp;(13). The initial baseline angular resonance frequency \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\omega\\:}_{o}\\)\u003c/span\u003e\u003c/span\u003e of the non-functionalized CMUT was established by applying this analytical framework to its equivalent mass\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\:m}_{eq}\\)\u003c/span\u003e\u003c/span\u003e and stiffness \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{k}_{eq}\\)\u003c/span\u003e\u003c/span\u003e parameters. This same methodology was then used to determine the eigenvalue \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\omega\\:}_{bi}\\)\u003c/span\u003e\u003c/span\u003e for the GO-integrated device, enabling a direct comparison of the frequency response \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{f}_{bi}\\)\u003c/span\u003e\u003c/span\u003e as a function of DC bias voltage.\u003c/p\u003e\u003cp\u003eAs illustrated in Fig.\u0026nbsp;\u003cspan refid=\"Fig7\" class=\"InternalRef\"\u003e7\u003c/span\u003e, the natural frequency characteristics are contrasted for the CMUT both with (denoted by a solid black curve) and without (denoted by a dotted red curve) the GO layer. The data reveals a pronounced spring-hardening behavior attributable to the GO integration. This phenomenon, indicated by the shift in the frequency curve, signifies a displacement-dependent augmentation of the system's effective stiffness, which consequently elevates its resonant frequency. It is observed that although elevating the DC bias voltage produces a consistent reduction in resonant frequency for both configurations, the GO-coated CMUT perpetually operates at a higher resonant frequency than its unmodified counterpart throughout the entire range of applied bias voltages.\u003c/p\u003e\u003cp\u003e\u003c/p\u003e\u003c/div\u003e\u003cdiv id=\"Sec7\" class=\"Section3\"\u003e\u003ch2\u003e3.1.3. Mass Sensitivity\u003c/h2\u003e\u003cp\u003eIt is critical to analyze the mass sensitivity and resonant characteristics of the sensor after functionalizing GO onto the top plate surface of the CMUT to determine the sensor\u0026rsquo;s feasibility for gas sensing applications. This analysis focuses exclusively on the GO layer to investigate its effect on the core operational characteristics of the CMUT. The GO layer is configured with a predetermined thickness, considerably thinner than the top plate, to simulate practical operational scenarios. The mass of the GO sensing layer was determined to be 0.9852 ng through incorporating the GO parameters documented in Table\u0026nbsp;\u003cspan refid=\"Tab1\" class=\"InternalRef\"\u003e1\u003c/span\u003e and Table\u0026nbsp;\u003cspan refid=\"Tab2\" class=\"InternalRef\"\u003e2\u003c/span\u003e into Eq.\u0026nbsp;(2). This mass represents approximately 0.78% of the mass of the CMUT's Si top plate. Consequently, the sensor's sensitivity (\u003cem\u003eS\u003c/em\u003e\u003csub\u003e\u003cem\u003ebi\u003c/em\u003e\u003c/sub\u003e), quantified as the ratio of frequency variation to mass change, can be mathematically represented as [\u003cspan citationid=\"CR35\" class=\"CitationRef\"\u003e35\u003c/span\u003e]:\u003cdiv id=\"Equo\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equo\" name=\"EquationSource\"\u003e\n$$\\:\\begin{array}{c}{S}_{bi}=\\frac{\\varDelta\\:f}{\\varDelta\\:m}\\#\\left(15\\right)\\end{array}$$\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e\u003cp\u003ewhere \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\varDelta\\:f\\)\u003c/span\u003e\u003c/span\u003e denotes Frequency shift due to mass change (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\varDelta\\:m\\)\u003c/span\u003e\u003c/span\u003e). Eq.\u0026nbsp;(15) demonstrates that enhanced mass sensitivity necessitates both elevated resonance frequencies and reduced effective mass, conditions that are optimally satisfied through minimization of the sensor's radial dimensions. To analytically characterize the mass loading sensitivity resulting from the proportional mass contribution of the GO sensing layer to the mass of the top Si plate of CMUT, incremental mass variations were introduced to the mass of GO film (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{m}_{s}\\)\u003c/span\u003e\u003c/span\u003e). The mass loading ratio was defined as discrete integer increments ranging from 0 to 8 times the \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{m}_{s}\\)\u003c/span\u003e\u003c/span\u003e. Figure\u0026nbsp;\u003cspan refid=\"Fig8\" class=\"InternalRef\"\u003e8\u003c/span\u003e illustrates the correlation between mass loading sensitivity and resonance frequency shift across different bias voltages ranging from 0 to 80 V with an incremental increase of 5 V. The analysis revealed that mass loading sensitivity scales proportionally with both frequency shift and DC bias voltage, achieving maximum sensitivity of 24.19 Hz/pg at 286.255 Hz under 80 V DC. This is an indication that the spring effect of GO film is the most predominant parameter, leading to an overall increase in resonance frequency shift and consequently enhanced mass loading sensitivity.\u003c/p\u003e\u003cp\u003e\u003c/p\u003e\u003c/div\u003e\u003c/div\u003e\u003cdiv id=\"Sec8\" class=\"Section2\"\u003e\u003ch2\u003e3.2. Dynamic Effect of GO/CMUT\u003c/h2\u003e\u003cp\u003eThe incorporation of a GO coating significantly alters the operational dynamics of CMUTs through modifications to both inertial and elastic properties. The GO/CMUT resonant characteristics exhibit a dependence on GO thickness, where enhanced mass loading from thicker films tends to decrease the fundamental resonance frequency, whereas thin, high-stiffness coatings can elevate it. Additionally, GO's intrinsic viscoelastic properties contribute to energy dissipation, effectively lowering the system's \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{Q}_{F}\\)\u003c/span\u003e\u003c/span\u003e while expanding its operational bandwidth. Such bandwidth enhancement proves advantageous for broadband acoustic applications, though it may compromise sensitivity in frequency-specific implementations.\u003c/p\u003e\u003cdiv id=\"Sec9\" class=\"Section3\"\u003e\u003ch2\u003e3.2.1. Time-Varying Dynamic Response\u003c/h2\u003e\u003cp\u003eThe time-varying transient response of the CMUT was mathematically investigated with particular emphasis on characterizing the GO layer influence through:\u003cdiv id=\"Equp\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equp\" name=\"EquationSource\"\u003e\n$$\\:\\begin{array}{c}\\ddot{w}\\left(t\\right)+\\frac{{\\omega\\:}_{bi}}{{Q}_{F}}\\dot{w}\\left(t\\right)+{\\omega\\:}_{bi}^{2}\\:w\\left(t\\right)=\\frac{{\\epsilon\\:}_{o\\:}\\text{}{A}_{m}\\text{}{V}_{DC}^{2}}{{2\\:m}_{bi}{\\text{(}{d}_{o}\\text{-\\:w(t))}}^{\\text{2}}}\\:\\#\\left(16\\right)\\end{array}$$\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e\u003cp\u003eAn initial examination was performed on bare CMUT, with Fig.\u0026nbsp;\u003cspan refid=\"Fig9\" class=\"InternalRef\"\u003e9\u003c/span\u003e(a) presenting the transient response evaluated for 2 \u0026micro;s derived from Eq.\u0026nbsp;(16) in the absence of GO film under 40 V DC bias voltage. It is worth noting that the evaluation for bare CMUT was accounted by the mass of the top plate of the CMUT (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{m}_{m}\\)\u003c/span\u003e\u003c/span\u003e) and the angular resonance frequency (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\omega\\:}_{o}\\)\u003c/span\u003e\u003c/span\u003e), given by [\u003cspan citationid=\"CR23\" class=\"CitationRef\"\u003e23\u003c/span\u003e]:\u003cdiv id=\"Equq\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equq\" name=\"EquationSource\"\u003e\n$$\\:\\begin{array}{c}{\\omega\\:}_{o}=\\frac{2.95\\:{t}_{m}}{{a}^{2}}\\sqrt{\\frac{{E}_{m}\\:}{{\\rho\\:}_{m}\\:\\left(1-{\\nu\\:}_{m}^{2}\\right)}}\\#\\left(17\\right)\\end{array}$$\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e\u003cp\u003eand the corresponding center resonance frequency is then given by:\u003cdiv id=\"Equr\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equr\" name=\"EquationSource\"\u003e\n$$\\:\\begin{array}{c}{f}_{o}=\\frac{{\\omega\\:}_{o}\\:}{2\\pi\\:}\\#\\left(18\\right)\\end{array}$$\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e\u003cp\u003eThe results demonstrate a characteristic underdamped oscillatory regime with a predetermined \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{Q}_{F}=10\\)\u003c/span\u003e\u003c/span\u003e, where elastic restoring forces substantially outweighed damping effects throughout the vibrational decay process. Key metrics extracted include a peak displacement overshoot of 6.59 nm and stabilization to a steady-state amplitude of 3.56 nm within the 2 \u0026micro;s. These rapid settling dynamics directly contribute to enhanced device precision and operational stability [\u003cspan citationid=\"CR36\" class=\"CitationRef\"\u003e36\u003c/span\u003e].\u003c/p\u003e\u003cp\u003eComparative analysis of the GO-functionalized system as shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig9\" class=\"InternalRef\"\u003e9\u003c/span\u003e(b), reveals significant modifications to the transient profile. The introduction of GO induces measurable stiffening effects, manifesting as both decreased overshoot magnitude at peak displacement overshoot of 5.66 nm and declined steady-state condition at 3.05 nm. Notably, while the temporal stabilization threshold (\u0026lt;\u0026thinsp;2 \u0026micro;s) remains comparable to the bare CMUT, the modified mechanical impedance alters the energy dissipation trajectory, reflecting the composite top movable plate's adjusted dynamic properties.\u003c/p\u003e\u003cp\u003e\u003c/p\u003e\u003cp\u003eSubsequently, the CMUT's steady-state behavior was analyzed both in the presence and absence of the GO layer to examine its impact on harmonic oscillation dynamics, as described by the governing equation:\u003cdiv id=\"Equs\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equs\" name=\"EquationSource\"\u003e\n$$\\:\\begin{array}{c}\\ddot{w}\\left(t\\right)+\\frac{{\\omega\\:}_{bi}}{{Q}_{F}}\\dot{w}\\left(t\\right)+{\\omega\\:}_{bi}^{2}\\:w\\left(t\\right)=\\frac{{\\epsilon\\:}_{o\\:}\\text{}{A}_{m}{\\left({V}_{DC}+{V}_{AC}\\:sin\\:\\left(2\\pi\\:{f}_{ex}t\\right)\\right)}^{2}\\text{}}{{2\\:m}_{bi}{\\text{(}{d}_{o}\\text{-\\:w(t))}}^{\\text{2}}}\\:\\#\\left(19\\right)\\end{array}$$\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e\u003cp\u003eThe CMUT was first characterized in its bare state by substituting the bi-layer mass (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{m}_{bi}\\)\u003c/span\u003e\u003c/span\u003e) with the mass of the Si top plate (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{m}_{m}\\)\u003c/span\u003e\u003c/span\u003e) and replacing the bi-layer resonance frequency (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\omega\\:}_{bi}\\)\u003c/span\u003e\u003c/span\u003e) with the intrinsic resonance frequency of the unmodified CMUT (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\omega\\:}_{o}\\)\u003c/span\u003e\u003c/span\u003e) in Eq.\u0026nbsp;(19). The investigation utilized a quality factor of \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{Q}_{F}=10\\)\u003c/span\u003e\u003c/span\u003e, with excitation frequency swept near its fundamental resonance frequency (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{f}_{ex}={f}_{o}=10\\:\\text{M}\\text{H}\\text{z}\\)\u003c/span\u003e\u003c/span\u003e). The electrical biasing conditions consisted of a 40 V DC offset superimposed with a 10 V AC driving signal. Thus, the results as seen in Fig.\u0026nbsp;\u003cspan refid=\"Fig10\" class=\"InternalRef\"\u003e10\u003c/span\u003e(a) revealed a rapid stabilization period, transitioning from transient oscillations to steady-state operation in less than 1 \u0026micro;s, accompanied by peak amplitude around the value of 6.49 nm. The response demonstrated a signal-to-noise ratio (SNR) significantly corresponds to the specified DC and AC excitation parameters.\u003c/p\u003e\u003cp\u003eUnder the same conditions, a comparative examination of the GO-integrated CMUT system as shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig10\" class=\"InternalRef\"\u003e10\u003c/span\u003e(b), analyzed via Eq.\u0026nbsp;(19). The results demonstrated substantial alterations in the equilibrium vibrational characteristics such that the GO layer induces a measurable reduction in stabilized oscillation magnitude at around 5.46 nm, primarily attributable to the dominant stiffness characteristics of GO which effectively suppress vibrational displacement amplitudes at steady-state conditions.\u003c/p\u003e\u003cp\u003e\u003c/p\u003e\u003c/div\u003e\u003cdiv id=\"Sec10\" class=\"Section3\"\u003e\u003ch2\u003e3.2.2. Frequency Characteristics due to Varying AC Voltage\u003c/h2\u003e\u003cp\u003eTo characterize the frequency response of the CMUT, the mid-point displacement amplitude \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\left(w\\right)\\)\u003c/span\u003e\u003c/span\u003e of the oscillating plate was evaluated both before and after GO layer functionalization. The LTI numerical method was implemented to solve the fundamental dynamical Eq.\u0026nbsp;(19) for the uncoated configuration and for the GO-integrated system. From the stabilized vibrational regime, critical response characteristics were derived through comprehensive processing of the time-history data. Figure\u0026nbsp;\u003cspan refid=\"Fig11\" class=\"InternalRef\"\u003e11\u003c/span\u003e(a) and (b) depict the frequency response characteristics of the CMUT device under both unmodified and GO-functionalized conditions, demonstrating the comparative variations in central oscillating plate deflection (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:w\\)\u003c/span\u003e\u003c/span\u003e). These results were derived for a \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{Q}_{F}=10\\)\u003c/span\u003e\u003c/span\u003e, under a constant DC bias voltage of 40 V and an AC actuation voltage incrementally increased from 5 V to 40 V in steps of 5 V. Notably, the selected DC bias voltage represented approximately 30% of the critical pull-in voltage threshold for all assessment configurations. The frequency sweep spans from 100 Hz to 30 MHz, with a resolution of 0.1 MHz. For the bare CMUT, The findings reveal that as the excitation frequency rises, the system's response amplifies until it reaches a second-order superharmonic resonance at \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{f}_{ex}=\\frac{1}{2}{f}_{o}\\:at\\:5.055\\:MHz\\)\u003c/span\u003e\u003c/span\u003e. Beyond this point, the response continues to grow until it attains the primary resonance at \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{f}_{ex}={f}_{o}\\:at\\:10.134\\:MHz\\)\u003c/span\u003e\u003c/span\u003e t, which aligns with the 40 V DC condition identified during the eigenfrequency analysis. It is noteworthy that the response at \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{f}_{o}\\)\u003c/span\u003e\u003c/span\u003e exhibits linear behavior under the applied forcing conditions.\u003c/p\u003e\u003cp\u003eThe double peaks observed in Fig.\u0026nbsp;\u003cspan refid=\"Fig11\" class=\"InternalRef\"\u003e11\u003c/span\u003e(a) and (b) arise due to the multi-frequency excitation nature of the electrostatic force. This behavior can be attributed to the quadratic nonlinearity inherent in the electrostatic force, which activates mechanisms that generate responses at both twice and half the excitation frequency within the system. This can be clarified as the following:\u003cdiv id=\"Equt\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equt\" name=\"EquationSource\"\u003e\n$$\\:{F}_{es}\\propto\\:{V\\left(t\\right)}^{2}={\\left({V}_{DC}+{V}_{AC}\\:sin\\:\\left(2\\pi\\:{f}_{ex}t\\right)\\right)}^{2}$$\u003c/div\u003e\u003c/div\u003e\u003cdiv id=\"Equu\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equu\" name=\"EquationSource\"\u003e\n$$\\:\\begin{array}{c}={{V}_{DC}}^{2}+{2V}_{DC}{V}_{AC}{sin}\\left(2\\pi\\:{f}_{ex}t\\right)+{{V}_{AC}}^{2}{sin\\:}^{2}\\left(2\\pi\\:{f}_{ex}t\\right)\\end{array}$$\u003c/div\u003e\u003c/div\u003e\u003cdiv id=\"Equv\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equv\" name=\"EquationSource\"\u003e\n$$\\:={{V}_{DC}}^{2}+{2V}_{DC}{V}_{AC}\\:sin\\:\\left(2\\pi\\:{f}_{ex}t\\right)+\\frac{1}{2}{{V}_{AC}}^{2}-\\frac{1}{2}{{V}_{AC}}^{2}cos\\left(4\\pi\\:{f}_{ex}t\\right)$$\u003c/div\u003e\u003c/div\u003e\u003cdiv id=\"Equw\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equw\" name=\"EquationSource\"\u003e\n$$\\:\\begin{array}{c}=\\left({{V}_{DC}}^{2}+\\frac{1}{2}{{V}_{AC}}^{2}\\right)+{2V}_{DC}{V}_{AC}{sin}\\left(2\\pi\\:{f}_{ex}t\\right)-\\frac{1}{2}{{V}_{AC}}^{2}cos\\left(4\\pi\\:{f}_{ex}t\\right)\\#\\left(20\\right)\\end{array}$$\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e\u003cp\u003e\u003c/p\u003e\u003cp\u003eEquation (20) indicates that the electrostatic force comprises three distinct components: a static term proportional to \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\left({{V}_{DC}}^{2}+\\frac{1}{2}{{V}_{AC}}^{2}\\right)\\)\u003c/span\u003e\u003c/span\u003e, a lower harmonic term at frequency \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{f}_{ex}\\)\u003c/span\u003e\u003c/span\u003e proportional to \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\left({2V}_{DC}{V}_{AC}\\right)\\)\u003c/span\u003e\u003c/span\u003e, and a higher harmonic term at frequency \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{2f}_{ex}\\)\u003c/span\u003e\u003c/span\u003e proportional to \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\left(\\frac{1}{2}{{V}_{AC}}^{2}\\right)\\)\u003c/span\u003e\u003c/span\u003e. When the excitation frequency is doubled, the primary resonance of the system is activated, which occurs when the CMUT is driven at half the primary resonance frequency (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{f}_{ex}=\\frac{1}{2}{f}_{o}\\)\u003c/span\u003e\u003c/span\u003e). In contrast, when the excitation frequency is halved, the primary resonance is initiated, taking place when the CMUT is driven at the primary resonance frequency itself (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{f}_{ex}={f}_{o}\\)\u003c/span\u003e\u003c/span\u003e). It is noteworthy that the superharmonic resonance is positively correlated with the quality factor and inversely related to the damping factor. The amplitude of the superharmonic peak grows as the damping factor diminishes, a relationship governed by the equation \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\xi\\:=\\frac{1}{2\\:{Q}_{F}}\\)\u003c/span\u003e\u003c/span\u003e, following a trend consistent with the fundamental frequency curve.\u003c/p\u003e\u003cp\u003eThe integration of GO layer onto the CMUT\u0026rsquo;s top plate substantially modified its resonant behavior under identical excitation conditions. As evidenced in Fig.\u0026nbsp;\u003cspan refid=\"Fig11\" class=\"InternalRef\"\u003e11\u003c/span\u003e(b), the functionalization process induced a systematic increase in resonance frequencies, with the superharmonic resonance shifting to 5.155 MHz and the fundamental mode transitioning to 10.376 MHz. This frequency elevation stems primarily from the GO layer's pronounced elastic properties, which simultaneously reduced oscillation amplitudes relative to the uncoated CMUT (Fig.\u0026nbsp;\u003cspan refid=\"Fig11\" class=\"InternalRef\"\u003e11\u003c/span\u003e(a)), demonstrating the material's stiffness-dominated influence on the system's dynamic response.\u003c/p\u003e\u003c/div\u003e\u003cdiv id=\"Sec11\" class=\"Section3\"\u003e\u003ch2\u003e3.2.3. Frequency Characteristics due to Varying DC Voltage\u003c/h2\u003e\u003cp\u003eSubsequently, frequency characteristics due to varying DC bias voltage and fixed AC actuation voltage were investigated for two configurations. The first configuration analyzed unfunctionalized CMUT performance across a DC bias range of 0-117.5 V with a constant 10 V AC excitation, while maintaining a quality factor of \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{Q}_{F}=10\\)\u003c/span\u003e\u003c/span\u003e. The second configuration evaluated the modified system incorporating GO functional layer on the CMUT\u0026rsquo;s top plate, assessed under an extended DC bias range of 0-121.5 V with identical AC driving conditions (10 V) and equivalent quality factor \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{Q}_{F}=10\\)\u003c/span\u003e\u003c/span\u003e.\u003c/p\u003e\u003cp\u003eInitially, the CMUT was analyzed both in its bare form and with an integrated GO layer under zero DC bias conditions. As illustrated in Fig.\u0026nbsp;\u003cspan refid=\"Fig12\" class=\"InternalRef\"\u003e12\u003c/span\u003e(a), the computational findings reveal that without an applied DC voltage, the excitation frequency exhibited a twofold increase. Specifically, the primary resonance peak occurred at 5.055 MHz for the uncoated CMUT, whereas the GO-coated CMUT, as illustrated in Fig.\u0026nbsp;\u003cspan refid=\"Fig12\" class=\"InternalRef\"\u003e12\u003c/span\u003e(b) displayed a shifted resonance at 5.155 MHz, with both configurations demonstrating attenuated vibrational amplitudes. Under this operating condition, the system exhibits a primarily linear response due to the dominance of linear stiffness effects.\u003c/p\u003e\u003cp\u003e\u003c/p\u003e\u003cp\u003eAs the DC bias voltage is incrementally increased, as shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig13\" class=\"InternalRef\"\u003e13\u003c/span\u003e, the response exhibited a bifurcation of the resonant behavior into second-order superharmonic and distinct primary modes. This phenomenon stems from the quadratic nonlinear characteristics of electrostatic forces, which produce dual-frequency excitation effects. Consequently, the system manifests simultaneous vibrational responses at both frequency-doubled and frequency-halved components relative to the input signal. It is evident that higher DC bias voltages caused a decrease in the fundamental resonance frequency, primarily due to the predominant softening effect on the effective stiffness induced by the electrostatic force, thereby constraining the operational bandwidth. Specifically, the series resonance frequency fell from 10.11 MHz to 9.32 MHz for the bare CMUT (Fig.\u0026nbsp;\u003cspan refid=\"Fig13\" class=\"InternalRef\"\u003e13\u003c/span\u003e(a)) and fell from 10.34 MHz to 9.55 MHz for the functionalized GO/CMUT (Fig.\u0026nbsp;\u003cspan refid=\"Fig13\" class=\"InternalRef\"\u003e13\u003c/span\u003e(b)) with increasing applied DC bias voltage. Though, the progressive elevation of DC bias voltage enhanced the system's sensitivity, evidenced by amplified resonance amplitude in both scenarios. These observations collectively highlight the critical role of DC bias on the vibrational characteristics of both device configurations.\u003c/p\u003e\u003cp\u003e\u003c/p\u003e\u003cp\u003eIt is important to highlight that when the DC voltage exceeds 110 V with presence of 10 V AC in the both configurations, the system begins to exhibit bifurcation instabilities where the dynamic pull-in takes place. For AC\u0026thinsp;+\u0026thinsp;DC actuation, dynamic pull-in occurs at lower voltages than static pull-in due to resonant energy buildup. Hence, optimal AC actuation voltage exists when operated at 10\u0026ndash;30% of DC bias voltage to avoid dynamic pull-in phenomenon. Figure\u0026nbsp;\u003cspan refid=\"Fig14\" class=\"InternalRef\"\u003e14\u003c/span\u003e illustrates the system's stability characteristics, showing the cyclic fold bifurcation thresholds (CF\u003csub\u003e1\u003c/sub\u003e and CF\u003csub\u003e2\u003c/sub\u003e) for the fundamental vibration mode. For the unmodified CMUT configuration, shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig14\" class=\"InternalRef\"\u003e14\u003c/span\u003e(a), these stability limits occurred at VDC\u0026thinsp;=\u0026thinsp;117.5 V with VAC\u0026thinsp;=\u0026thinsp;10 V, while the GO-integrated device, shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig14\" class=\"InternalRef\"\u003e14\u003c/span\u003e(b) exhibited similar behavior at slightly higher bias conditions (VDC\u0026thinsp;=\u0026thinsp;121.5 V, and VAC\u0026thinsp;=\u0026thinsp;10 V). Under these specific operating parameters, the system maintains a single stable equilibrium state, effectively suppressing the linear jump phenomena associated with bifurcation instabilities. The observed stability enhancement in the GO-modified device suggests improved dynamic performance at elevated bias voltages compared to the bare CMUT structure.\u003c/p\u003e\u003cp\u003e\u003c/p\u003e\u003cp\u003eWhen the driving frequency is swept under maximum combined DC and AC voltage excitation, the system first encounters a second-order superharmonic resonance at 4.17 MHz for the unmodified CMUT and 4.36 MHz for the GO-modified device. This resonant behavior persists until reaching the first stability threshold (CF\u003csub\u003e1\u003c/sub\u003e) at 7.34 MHz and 7.53 MHz for the respective configurations, where the low-amplitude solution branch becomes unstable. At this critical juncture, where stable and unstable solution manifolds coalesce, the system undergoes a sudden jump to the second stability threshold (CF\u003csub\u003e2\u003c/sub\u003e), occurring at 7.41 MHz for the bare CMUT and 7.60 MHz for its GO-functionalized counterpart.\u003c/p\u003e\u003cp\u003eAt the CF\u003csub\u003e1\u003c/sub\u003e point, a Floquet multiplier crosses the unit circle at +\u0026thinsp;1 [\u003cspan citationid=\"CR37\" class=\"CitationRef\"\u003e37\u003c/span\u003e], indicating the intersection of the lower stable solution branch with an unstable solution branch, represented by a dashed red line. Operating the system near CF\u003csub\u003e1\u003c/sub\u003e is highly advantageous, as the sudden jump at this point establishes a region of remarkable sensitivity, which is particularly useful for mass detection applications. This implies that increasing the amplitude of the motion can amplify linear or nonlinear effects, thereby generating bifurcation points in the frequency domain.\u003c/p\u003e\u003cp\u003eIn each of the three scenarios, the incorporation of GO layer led to a decrease in the resonance amplitudes of the movable plate while simultaneously increasing the resonance frequency. This upward shift in resonance frequency is attributed to the enhanced effective stiffness of the top plate induced by the GO layer. The magnitude of the frequency shift is largely determined by the mechanical properties and geometric parameters of the GO layer. Within the present study, the influence of the GO film\u0026rsquo;s elastic behavior outweighs the effect of added mass, resulting in a net rise in the resonance frequency.\u003c/p\u003e\u003cp\u003eThe quality factor (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{Q}_{F}\\)\u003c/span\u003e\u003c/span\u003e) also plays a crucial role in determining the frequency response characteristics of the CMUT, both with and without GO layer as shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig15\" class=\"InternalRef\"\u003e15\u003c/span\u003e.\u003c/p\u003e\u003cp\u003e\u003c/p\u003e\u003cp\u003eThe quality factor directly affects the bandwidth (\u003cem\u003eBW\u003c/em\u003e) of the system, as mathematically expressed in \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{Q}_{F}=\\frac{{f}_{o}}{\\varDelta\\:f}=\\frac{{f}_{o}}{BW}\\)\u003c/span\u003e\u003c/span\u003e. A high \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{Q}_{F}\\)\u003c/span\u003e\u003c/span\u003e corresponds to a narrower \u003cem\u003eBW\u003c/em\u003e and a sharper resonance peak, improving sensitivity near resonance but limiting the operational frequency range. Conversely, a lower \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{Q}_{F}\\)\u003c/span\u003e\u003c/span\u003e yields an expanded \u003cem\u003eBW\u003c/em\u003e, facilitating operation across a wider spectrum at the cost of diminished peak sensitivity. Table\u0026nbsp;\u003cspan refid=\"Tab3\" class=\"InternalRef\"\u003e3\u003c/span\u003e provides a comparative analysis of how \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{Q}_{F}\\)\u003c/span\u003e\u003c/span\u003e influences the \u003cem\u003eBW\u003c/em\u003e in configurations both excluding and incorporating the GO layer under conditions of 40 V DC and 30 V AC.\u003c/p\u003e\u003cp\u003e\u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab3\" border=\"1\"\u003e\u003ccaption language=\"En\"\u003e\u003cdiv class=\"CaptionNumber\"\u003eTable 3\u003c/div\u003e\u003cdiv class=\"CaptionContent\"\u003e\u003cp\u003eInfluence of the quality factor on frequency bandwidth with and without the GO layer under conditions of 40 V DC and 30 V AC.\u003c/p\u003e\u003c/div\u003e\u003c/caption\u003e\u003ccolgroup cols=\"3\"\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e\u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e\u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e\u003cthead\u003e\u003ctr\u003e\u003cth align=\"left\" colname=\"c1\"\u003e\u003cp\u003eQuality factor (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\varvec{Q}}_{\\varvec{F}}\\)\u003c/span\u003e\u003c/span\u003e)\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c2\"\u003e\u003cp\u003e\u003cem\u003eBW\u003c/em\u003e w/o GO layer (MHz)\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c3\"\u003e\u003cp\u003e\u003cem\u003eBW\u003c/em\u003e with GO layer (MHz)\u003c/p\u003e\u003c/th\u003e\u003c/tr\u003e\u003c/thead\u003e\u003ctbody\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003e5\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e\u003cp\u003e2.026\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e\u003cp\u003e2.075\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003e10\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e\u003cp\u003e1.013\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e\u003cp\u003e1.037\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003e15\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e\u003cp\u003e0.675\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e\u003cp\u003e0.691\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003e20\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e\u003cp\u003e0.506\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e\u003cp\u003e0.518\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003c/tbody\u003e\u003c/colgroup\u003e\u003c/table\u003e\u003c/div\u003e\u003c/p\u003e\u003c/div\u003e\u003c/div\u003e"},{"header":"4. Conclusion","content":"\u003cp\u003eThis work presents a comprehensive analytical framework for evaluating the static and dynamic performance of GO-integrated CMUTs. The GO layer\u0026rsquo;s mechanical stiffening effect significantly enhances device stability, delaying pull-in instability and increasing resonance frequencies. Static modeling confirms a 3.1% rise in pull-in voltage, while dynamic analyses reveal improved mass sensitivity and frequency response due to GO\u0026rsquo;s elastic properties. The system exhibits linear behavior under combined DC/AC excitation, with distinct superharmonic resonances and bifurcation instability points that can be leveraged for sensing applications. Furthermore, GO\u0026rsquo;s viscoelasticity optimizes the quality factor, balancing bandwidth and sensitivity. These findings demonstrate the viability of GO/CMUTs as ultrasensitive gas sensors, with tunable performance through geometric and material optimization. Future work will focus on experimental validation and integration with multi-analyte detection systems.\u003c/p\u003e\u003cp\u003e\u003cb\u003eORCID\u003c/b\u003e\u003c/p\u003e\u003cp\u003eTirad Owais - \u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003ehttps://orcid.org/0000-0002-9301-1707\u003c/span\u003e\u003cspan address=\"https://orcid.org/0000-0002-9301-1707\" targettype=\"URL\" class=\"RefTarget\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e"},{"header":"Declarations","content":"\u003cp\u003e\u003ch2\u003eConflicts of Interest\u003c/h2\u003e\u003cp\u003eThe authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.\u003c/p\u003e\u003c/p\u003e\u003ch2\u003eAcknowledgement\u003c/h2\u003e\u003cp\u003eThis study is supported via funding from Prince Sattam bin Abdulaziz University project number (PSAU/2025/R/1446). The authors also are grateful to KFUPM, IRC-HTCM (Project # INHE-2309).\u003c/p\u003e"},{"header":"References","content":"\u003col\u003e\u003cli\u003e\u003cspan\u003eSumner JB (1923) The detection of pentose, formaldehyde and methyl alcohol. J Am Chem Soc 45(10):2378\u0026ndash;2380\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eHajjaj AZ, Jaber N, Ilyas S, Alfosail FK, Younis MI (2020) Linear and nonlinear dynamics of micro and nano-resonators: Review of recent advances. Int J Non-Linear Mech 119:103328\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eKessler Y, Liberzon A, Krylov S (2020) Flow velocity gradient sensing using a single curved bistable microbeam. J Microelectromech Syst 29(5):1020\u0026ndash;1025\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eLichtenegger A, Gesperger J, Niederleithner M, Ginner L, Woehrer A, Drexler W, Salas M (2020) Ex-vivo Alzheimer\u0026rsquo;s disease brain tissue investigation: a multiscale approach using 1060-nm swept source optical coherence tomography for a direct correlation to histology. Neurophotonics 7(3):035004\u0026ndash;035004\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eHajjaj AZ, Jaber N, Alcheikh N, Younis MI (2019) A resonant gas sensor based on multimode excitation of a buckled microbeam. IEEE Sens J 20(4):1778\u0026ndash;1785\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eThr\u0026auml;nhardt M, Eccardt PC, Mooshofer H, Hauptmann P, Degertekin L (2009) October). A resonant CMUT sensor for fluid applications. SENSORS, 2009 IEEE. IEEE, pp 878\u0026ndash;883\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eBoubenia, R., Le Moal, P., Bourbon, G., Ramasso, E., \u0026amp; Joseph, E. (2021). CMUT-based sensor for acoustic emission application: Experimental and theoretical contributions to sensitivity optimization. Sensors, 21(6), 2042.\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003ePark, S., Yoon, I., Lee, S., Kim, H., Seo, J. W., Chung, Y., \u0026hellip; Lee, H. J. (2018).CMUT-based resonant gas sensor array for VOC detection with low operating voltage.Sensors and Actuators B: \u003cem\u003eChemical, 273\u003c/em\u003e, 1556\u0026ndash;1563\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003ePark, K. K., Lee, H., Kupnik, M., Oralkan, \u0026Ouml;., Ramseyer, J. P., Lang, H. P., \u0026hellip; Khuri-Yakub,B. T. (2011). Capacitive micromachined ultrasonic transducer (CMUT) as a chemical sensor for DMMP detection. \u003cem\u003eSensors and Actuators B: Chemical, 160\u003c/em\u003e(1), 1120\u0026ndash;1127\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003ePernu T, Saarilahti J, Sillanp\u0026auml;\u0026auml; T (2025) Real-time frequency matching circuit for a high sensitivity cMUT-based gas flow sensor. Sens Actuators A: Phys 383:116252\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003ePark, K. K., Lee, H. J., Yaralioglu, G. G., Ergun, A. S., Oralkan, \u0026Ouml;., Kupnik, M.,\u0026hellip; Gimzewski, J. K. (2007). Capacitive micromachined ultrasonic transducers for chemical detection in nitrogen. \u003cem\u003eApplied Physics Letters\u003c/em\u003e, \u003cem\u003e91\u003c/em\u003e(9)\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eZhao, L., Zhao, Y., Xia, Y., Li, Z., Li, J., Zhang, J., \u0026hellip; Jiang, Z. (2018). A novel CMUT-based resonant biochemical sensor using electrospinning technology. \u003cem\u003eIEEE Transactions on Industrial Electronics\u003c/em\u003e, \u003cem\u003e66\u003c/em\u003e(9), 7356\u0026ndash;7365\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eNathani MU, Nazemi H, Love C, Babu Lopez Y, Swaminathan S, Emadi A (2020) Capacitive based micromachined resonators for low level mass detection. Micromachines 12(1):13\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eOwais T, Khater M, Al-Qahtani H (2024) Graphene-Based MEMS Devices for Gas Sensing Applications: A Review. Micro Nanostruct, 207954\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eSchedin F, Geim AK, Morozov SV, Hill EW, Blake P, Katsnelson MI, Novoselov KS (2007) Detection of individual gas molecules adsorbed on graphene. Nat Mater 6(9):652\u0026ndash;655\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eHill EW, Vijayaragahvan A, Novoselov K (2011) Graphene sensors. IEEE Sens J 11(12):3161\u0026ndash;3170\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eZhu Y, Murali S, Cai W, Li X, Suk JW, Potts JR, Ruoff RS (2010) Graphene and graphene oxide: synthesis, properties, and applications. Adv Mater 22(35):3906\u0026ndash;3924\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eHe Q, Wu S, Yin Z, Zhang H (2012) Graphene-based electronic sensors. Chem Sci 3(6):1764\u0026ndash;1772\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eLiu Y, Dong X, Chen P (2012) Biological and chemical sensors based on graphene materials. Chem Soc Rev 41(6):2283\u0026ndash;2307\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eGeorgakilas, V., Otyepka, M., Bourlinos, A. B., Chandra, V., Kim, N., Kemp, K. C.,\u0026hellip; Kim, K. S. (2012). Functionalization of graphene: covalent and non-covalent approaches,derivatives and applications. \u003cem\u003eChemical reviews\u003c/em\u003e, \u003cem\u003e112\u003c/em\u003e(11), 6156\u0026ndash;6214\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eBottari, G., Herranz, M. \u0026Aacute;., Wibmer, L., Volland, M., Rodr\u0026iacute;guez-P\u0026eacute;rez, L., Guldi,D. M., \u0026hellip; Torres, T. (2017). Chemical functionalization and characterization of graphene-based materials. \u003cem\u003eChemical Society Reviews\u003c/em\u003e, \u003cem\u003e46\u003c/em\u003e(15), 4464\u0026ndash;4500\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eHsu TR (2008) MEMS and microsystems: design, manufacture, and nanoscale engineering. Wiley\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eFan, X., Smith, A. D., Forsberg, F., Wagner, S., Schr\u0026ouml;der, S., Akbari, S. S. A.,\u0026hellip; Niklaus, F. (2020). Manufacture and characterization of graphene membranes with suspended silicon proof masses for MEMS and NEMS applications. \u003cem\u003eMicrosystems \u0026amp; Nanoengineering\u003c/em\u003e, \u003cem\u003e6\u003c/em\u003e(1), 17\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eWygant IO, Kupnik M, Khuri-Yakub BT (2008), November Analytically calculating membrane displacement and the equivalent circuit model of a circular CMUT cell. In \u003cem\u003e2008 IEEE ultrasonics symposium\u003c/em\u003e (pp. 2111\u0026ndash;2114). IEEE\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eNazemi H, Emadi A (2019), October A new advanced analytical model for bi-layer circular CMUT-based gas sensors. In \u003cem\u003e2019 IEEE SENSORS\u003c/em\u003e (pp. 1\u0026ndash;4). IEEE\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eBrenner K, Ergun AS, Firouzi K, Rasmussen MF, Stedman Q (2019) \u0026amp; Khuri\u0026ndash;Yakub, B. Advances in capacitive micromachined ultrasonic transducers. \u003cem\u003eMicromachines, 10\u003c/em\u003e(2), 152\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eDe Boer MP, Michalske TA (1999) Accurate method for determining adhesion of cantilever beams. J Appl Phys 86(2):817\u0026ndash;827\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eVan Spengen WM, Puers R, Mertens R, De Wolf I (2004) A comprehensive model to predict the charging and reliability of capacitive RF MEMS switches. J Micromech Microeng 14(4):514\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eLoh OY, Espinosa HD (2012) Nanoelectromechanical contact switches. Nat Nanotechnol 7(5):283\u0026ndash;295\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eErgun AS, Yaralioglu GG, Khuri-Yakub BT (2003) Capacitive micromachined ultrasonic transducers: Theory and technology. J Aerospace Eng 16(2):76\u0026ndash;84\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eOwais T (2020) \u003cem\u003eCapacitive Micromachined Ultrasound Transducers for Non-Destructive Testing Applications\u003c/em\u003e (Master's thesis, University of Waterloo)\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eCao C, Daly M, Singh CV, Sun Y, Filleter T (2015) High strength measurement of monolayer graphene oxide. Carbon 81:497\u0026ndash;504\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eBayer T, Bishop SR, Perry NH, Sasaki K, Lyth SM (2016) Tunable mixed ionic/electronic conductivity and permittivity of graphene oxide paper for electrochemical energy conversion. ACS Appl Mater Interfaces 8(18):11466\u0026ndash;11475\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eWan J, Jiang JW, Park HS (2017) Negative Poisson's ratio in graphene oxide. Nanoscale 9(11):4007\u0026ndash;4012\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eM\u0026oslash;lgaard MJ, Laustsen M, Jakobsen MH, Andresen TL, Thomsen EV (2018) Combined colorimetric and gravimetric CMUT sensor for detection of benzyl methyl ketone. Sens Actuators B 275:483\u0026ndash;489\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eAhmadi N, Fathalilou M, Rezazadeh G (2024) Neo-Hookean modeling of nonlinear coupled behavior in circular plates supported by micro-pillars. Sci Rep 14(1):25428\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eNayfeh AH, Ibrahim RA (2001) Nonlinear interactions: analytical, computational, and experimental methods. Appl Mech Rev 54(4):B60\u0026ndash;B61\u003c/span\u003e\u003c/li\u003e\u003c/ol\u003e"}],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":true,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":true,"hideJournal":true,"highlight":"","institution":"","isAcceptedByJournal":false,"isAuthorSuppliedPdf":false,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":false,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true},"keywords":"MEMS, GO/CMUT, pull-in analysis, mass sensitivity, resonance frequency, bifurcation analysis","lastPublishedDoi":"10.21203/rs.3.rs-7179088/v2","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-7179088/v2","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eThis paper presents a detailed analytical investigation into the static and dynamic performance of a capacitive micromachined ultrasonic transducer (CMUT) functionalized with a graphene oxide (GO) sensing layer for gas detection applications. A bi-layer lumped-mass model is developed to capture the electromechanical response of the hybrid system, explicitly accounting for the mechanical stiffening effect of GO and its consequent influence on resonant characteristics. Static analysis reveals that the GO layer enhances operational stability by increasing the pull-in voltage from 143 V for a bare CMUT to 147.5 V. Dynamic assessments demonstrate that GO functionalization elevates resonance frequencies due to a stiffness-dominated regime, yielding a mass sensitivity of 24.19 Hz/pg at an 80 V DC bias. Frequency response analyses under combined DC/AC excitation highlight the system's linear dynamics, including distinct superharmonic resonances and bifurcation thresholds. The results underscore the role of GO in improving sensitivity, stability, and bandwidth, establishing GO/CMUTs as highly promising platforms for next-generation, high-performance gas sensors.\u003c/p\u003e","manuscriptTitle":"Static and Dynamic Analyses of Electrostatically Actuated MEMS GO/CMUT Gas Sensor","msid":"","msnumber":"","nonDraftVersions":[{"code":2,"date":"2025-10-16 18:48:04","doi":"10.21203/rs.3.rs-7179088/v2","editorialEvents":[{"type":"communityComments","content":0}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true}},{"code":1,"date":"2025-07-29 05:33:42","doi":"10.21203/rs.3.rs-7179088/v1","editorialEvents":[{"type":"communityComments","content":0}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true}}],"origin":"","ownerIdentity":"b23cc2ff-1c9c-4b2e-a7af-5d4a8658746c","owner":[],"postedDate":"October 16th, 2025","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"posted","subjectAreas":[],"tags":[],"updatedAt":"2025-09-21T13:53:34+00:00","versionOfRecord":[],"versionCreatedAt":"2025-10-16 18:48:04","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v2","identity":"rs-7179088","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-7179088","identity":"rs-7179088","version":["v2"]},"buildId":"8U1c8b4HqxoKbykW_rLl7","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}