Design of Piezoelectric Micromachined Ultrasonic Transducers Using High-Order Mode With High Performance and High Frequency

This work proposes the piezoelectric micromachined ultrasonic transducer (pMUT) design using high-order mode. Analytical models are established and used to estimate the performance of pMUT in <inline-formula> <tex-math notation="LaTeX">${n} ^{\text {th}}$ </tex-math></inline-formula>-order axisymmetric mode. To prove the concept, a comprehensive analysis is conducted on the <inline-formula> <tex-math notation="LaTeX">$3^{\text {rd}}$ </tex-math></inline-formula>-order pMUT by finite element method (FEM). The analytical models give guidance for the design of electrode configuration and geometric dimensions, which are verified by FEM. With optimized electrode configuration and thickness, the proposed pMUT design shows extraordinary performance improvement in transmitting and round-trip sensitivity. Approximately <inline-formula> <tex-math notation="LaTeX">$10.2\times $ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$4.12\times $ </tex-math></inline-formula> improvements in transmitting sensitivity and round-trip sensitivity have been achieved compared to the traditional <inline-formula> <tex-math notation="LaTeX">$1^{\text {st}}$ </tex-math></inline-formula>-order pMUT in the same radius, while there is an <inline-formula> <tex-math notation="LaTeX">$8.6\times $ </tex-math></inline-formula> improvement of the receiving voltage in the pulse-echo analysis. The high frequency, round-trip sensitivity, and directivity features of the proposed high-order pMUT design shown in FEM make it very promising for forming a high-frequency large-scale pMUT array.


I. INTRODUCTION
P IEZOELECTRIC micromachined ultrasonic transducers (pMUTs) ultrasonic transducers have been used in the application of fingerprint recognition, object detection, and particularly medical imaging [1], [2].Conventional ultrasonic transducers based on bulk piezoelectric ceramic have poor acoustic coupling to air or liquid, and it is expensive to machine them into a two-dimensional (2D) transducer array for three-dimensional (3D) imaging.Instead, the micromachined ultrasonic transducer has a low acoustic impedance for good coupling to air/liquids.Furthermore, pMUTs have the advantages of small element size, low power consumption, low cost, and easy integration with supporting electronics [3].
The pMUTs with both high transmission and receiving sensitivity are needed for a high signal-to-noise ratio (SNR) detection.The designs with bimorph layers [4], curved shape [5], and Helmholtz resonant cavity [6] are proposed to improve the transmitting performance.However, the bimorph design leads to complex interconnection, and design with a curved shape, Helmholtz tube, or cavity requires additional processes.Furthermore, these works focus on the transmitting performance and neglect the receiving performance.
High-frequency (≥ 10 MHz) pMUT is an attractive option to substitute bulk piezoelectric transducer for arrays without expensive dicing process [2].Higher frequency represents higher spatial resolution but higher attenuation in the medium, which would be able to apply in applications A pMUT device with an unimorph structure including stack layers of electrodes, piezoelectric material, and supporting material could be approximated as a uniform thin plate with clamped boundary.A multi-electrodes actuated circular plate consisting of multi-layers with clamped boundaries will be analyzed in the following, the cross-section view and the top view of the plate are shown as Figs. 1 (a) and 1(b), respectively.Where h i and z i are the height of the top plane and midplane of i-th layer, respectively; r ′ j and r ′′ j are the inner and outer radius for j-th electrode.For the homogeneous clamped circular plate with a radius of a, the deflection function w can be expressed in the following form [9]: A mn ψ mn (r) cos(mθ) sin(ωt) (1a) where m refers to the number of nodal diameters and n represents the number of nodal circles including the boundary circle.For the sake of simplicity, the mode shape with m nodal diameters and n nodal circles is referred to as (m, n).r = r/a is the normalized radial coordinate, and θ is the angle coordinate.A mn is the displacement amplitude of corresponding mode shape, and ψ mn is the eigenfunction of the mode shape profile.J m and I m are the Bessel function and the modified Bessel function of the first kind.The mode shape constant λ mn is determined by boundaries, and its value is given by [9].
In the previous study, the first axisymmetric mode (0, 1) of pMUT is considered to possess the maximum volumetric displacement and velocity, thus producing the highest possible acoustic pressure [7].However, the resonant frequency of the third axisymmetric mode (0, 3) is more than 10 times of mode (0, 1), resulting in a higher acoustic pressure, since the output pressure of pMUT is proportional to ω 2 .In the following discussion, the non-axisymmetric modes (m ̸ = 0) are neglected to simplify the derivation.
To excite the modes effectively, the gap of electrodes should be placed and aligned with the nodal circle of volumetric stress.Solve equation of motion [9] with a boundary of ∇ 2 w| r=1 = 0, the nodal circle radius can be obtained from the root of the equation: where ν is the Poisson ratio.Table 1 summaries the roots of (2) for the first five axisymmetric modes, when ν = 0.3.For instance, the nodal circle of mode (0, 1) is located at r = 0.677.This result is consistent with the fact that most of the conventional pMUT designs [3], [10] optimize the electrode coverage to 70% for maximum displacement.The slight difference is due to the adoption of the deflection function approximation (1 − r 2 ) 2 used in these studies.For mode (0, 3) with ν = 0.3, the nodal circle is located at r = {0.256,0.591, 0.893}.The solution of the stress nodal circle gives guidance for the optimal electrode design.
The natural frequencies of the circular plate in a vacuum are expressed as: where D is the flexure rigidity of the plate and µ is the mass per area.The rigidity D and mass per area µ of the stack layers are calculated as [11].Assume that the mode shape is the same in vacuum and water.For the application with one side of pMUT contact with liquid, the natural frequencies can be evaluated by [12]: where β 0n is the added virtual mass incremental factor, which represents the kinetic energy ratio of the liquid to the plate.
0n is a nondimensional added virtual mass incremental factor and its numerical solution in water is given by [13].

B. TRANSMITTING SENSITIVITY
For transmission, the sound pressure output in far-field at a distance l is revised from [14]: where R 0 = ka 2 /2 is the Rayleigh distance; k = ω/c 0 is the wave number in loaded medium.The surface pressure is P 0 = ρ 0 c 0 ωu av .ρ 0 and c 0 are the density and acoustic speed of the medium.The amplitude directivity factor D(θ) is defined as the pressure at angle θ relative to that at θ = 0.The average displacement amplitude u av of mode (0, n) is given by: where α 0n represents the effective factor of mode (0, n) to a uniform piston disk with the same radius.The α 0n for the first five axisymmetric modes are {0.3291,-0.1309, 0.0736, -0.0486, 0.03506}.The effective factors for even-order axisymmetric modes are negative, indicating the acoustic wave is emitted in a negative direction, resulting in the valleys of the transmitting response, while odd-order modes emit in a positive direction and produce peaks.
The displacement amplitude A 0n should be Q times of static displacement, i.e., A 0n = A 0 Q 0n , where Q 0n is the corresponding quality factor of mode (0, n); A 0 is the static displacement and is given by [15] and [16]:  where A s is the static displacement sensitivity (in a unit of m/V).M p is the bending moment produced by the piezoelectric layer.z p is the distance between the mid-plane of the piezoelectric layer and the neutral axis.Equation (9) shows that the piezoelectric coefficient involved in the generation of bending moment includes both e 31 and e 33 .I D and I p in (7) are integrals related to the elastic energy and coupled energy between the electric domain and mechanic domain [17].Neglecting the Poisson ratio ν related terms, I D is a constant for specific mode shape.The value of I D can be obtained by numerical integration.I p is an integral related to the electrode configuration.The maximum value of I p can be obtained by alternately applying voltages of opposite polarities separated by zeros of the integrand.With the ideal full coverage optimal electrode (r ′′ j = r ′ j+1 ), the maximum value of I p is: Table .2 lists the non-dimensional parameters which are necessary for the calculation of sound pressure for the first five axisymmetric modes (m = 0) in the circular clamped plate immersed in water.
Combine ( 5)-( 9), the relationship between output pressure and the key parameters can be obtained: where α 0n and λ 0n are the mode shape relevant constants and are positively related to n, which help the high-order modes achieve higher ω 0n,L and S Tx .I p /I D is determined by the electrode design, the maximum value I p,max /I D of which corresponds to the optimal electrode configuration.1/(1 + β 0n ) represents the descent of resonant frequency of the plate when it is in contact with water compared with that when it is in the vacuum.The |e 31,f | term shows that the piezoelectric material with high |e 31,f | are preferred.z p /µ is related to the thicknesses of stack layers.
When the material properties of pMUT and acoustic medium are given, the normalized transmitting sensitivity for n th -order mode can be obtained.For instance, neglect difference of |e 31,f |, z p and Q, given a = 50 µm, µ = 8.4 mg/m 2 , ρ 0 = 1000 kg/m 3 , the normalized transmitting sensitivity for 1st, 3rd, 5th modes are {1, 8.2, 18.2}, respectively.This result illustrates the remarkable improvement of emission performance for high-order mode compared with fundamental mode.
The Q-factor is difficult to parse with a single formula.There are multiple sources of loss: where 1/Q rad is due to the energy radiated to the loaded medium; Q rad ∝ 1/ v/(ωa 2 ) [18]; v is kinematic viscosity, for water, v = 1.14 × 10 −6 m 2 s −1 .1/Q support is the support loss, and Q support ∝ (a/t) 3 [19].1/Q TED is the thermoelastic damping, and 1/Q other represents the other intrinsic losses.

C. RECEIVING SENSITIVITY
The elastic energy U D and coupled energy U p of thin plate can be associated with displacement [11]: where u 2 0,rx is the displacement amplitude in the reception.Neglecting the loses and assuming that the work U s of sound waves act on the film is converted completely to the elastic energy U D , and the piezoelectric energy U p is thoroughly converted to electrical energy U e , the following equations can be derived: where S is the cavity area of pMUT; C 0 is the device static capacitance, and u 2 0,rx is the displacement amplitude in reception.Then the pMUT receiving sensitivity can be derived as: where A s is the static displacement sensitivity; S is the cavity area of pMUT, C 0 is the device static capacitance.Thus, the devices with high displacement sensitivity are preferred in the receiving.With n increases, the effective displacement factor α 0n decreases, leading to the disadvantage of highorder modes (n > 1) in receiving sensitivity.

D. SENSITIVITY-BANDWIDTH PRODUCT
The sensitivity-bandwidth (SBW) product is given by: where S x is sensitivity, x can be replaced with Tx, Rx, and RT, representing the transmitting, receiving, and roundtrip performance, respectively.Particularly, FBW is -6 dB fractional bandwidth for round-trip sensitivity.
The round-trip sensitivity is defined as S RT = S Tx •S Rx .The round-trip sensitivity bandwidth product SBW RT is adopted as the optimization criterion in this work rather than a single parameter of displacement [20], sound pressure level [21] and electromechanical coupling coefficient [17], since it represents the global transmitting-receiving sensitivity and bandwidth in loaded condition, which consider the process from excitation to reception.Besides, the sensitivity and bandwidth (FBW ≈ 1/Q) are mutually compromising factors, thus making SBW a widely applicable and relatively stabilized coefficient [22].Combine (11), ( 15) and ( 16), with the same cavity radius, the dependency of SBW RT with resonant frequency f 0 and displacement sensitivity A s are: The round-trip performance is fundamentally determined by the product of resonant frequency and displacement sensitivity.

E. DIRECTIVITY
Directivity indicates the spatial distribution of radiant energy.High directivity raises the signal level and decreases the noise level in non-target directions.The radiation pattern product theorem [14] shows that the directivity of the pMUT array is equal to the product of the array factor and the element.Thus, high-directional pMUT elements are conducive to forming high-directional arrays.The normalized directivity of mode (0, n) is given by:

III. OPTIMIZATION
To prove the concept and validate the superiority of the proposed high-order pMUT, mode (0, 3) is selected as an example for the following analysis considering the trade-off between the transmitting and receiving sensitivity.
The electrode design and geometric optimization will be discussed.

A. SETUP OF FEM MODELS
To evaluate the proposed high-order pMUT design, 2D axisymmetric FEM models are set up in COMSOL  Multiphysics.Default material parameters in COMSOL are adopted.Figs. 2 (a)-(c) show the configuration of COMSOL models for simulating transmitting sensitivity, receiving sensitivity, and pulse-echo response, respectively.The geometries are not shown in proportion, in order to illustrate setups.Table 3 summarizes the initial geometry parameters.The sensitivity simulations in the frequency domain give the full band information and optimization guidance which is more stable and efficient, while the pulseecho simulation in the time domain gives the round-trip performance including the transmitting and receiving process, which evaluates the total performance comprehensively and is closer to the practical scenarios of time-of-fly (ToF) applications.Solid mechanics and electrostatics module are used in the pMUT device, and the mechanical damping and dielectric loss are set to 1/1000 and 1/100, respectively.The medium loss due to liquid viscosity is considerable in high-frequency pMUT, thus the thermoviscous acoustic module is adopted in the acoustic propagation medium, and the radiating boundary in the acoustic domain is used to absorb the radiated acoustic waves.
Fig. 2 (a) show the FEM model for transmitting sensitivity.The peripheral boundary of the loaded medium (water) is set as radiating boundary, and the peripheral area of solids is set to the perfectly matched layer (PML).The width and height of the water are set to 500 µm and 2 mm, respectively.The top electrodes are alternately connected to +1 V/ −1 V, while the bottom electrode is connected to the ground.The transmitting sensitivity S Tx is defined as the absolute sound pressure at 2 mm above the center of pMUT divided by the excitation voltage (in the unit of [Pa/V]).
In the receiving model shown as Fig. 2 (b), the water is set to ''Background pressure'' boundary with 1 Pa pressure and ''Wave direction'' along (0, 0, -1).The receiving electrodes are terminated to a 1 M resistor.The receiving sensitivity S Rx is defined as the receiving voltage divided by the ground pressure (in the unit of [V/Pa]).
Fig. 2 (c) shows the schematic of the FEM model simulating pulse-echo response.The width and height of the water are set to 500 µm and 10 mm, respectively.The thermoviscous acoustic domain in the time domain is used.The excitation source is a Gaussian pulse voltage source in series with a 10 k resistor.The Gaussian pulse is exp(−(t − 2T 0 ) 2 /(T 0 /2) 2 ) sin(ω 0 t).T 0 is the period corresponding to the resonant frequency f 0 .

B. MODE SHAPE AND STRESS
Stress distribution guides optimal electrode design.To effectively excite the pMUT, the gap of the electrode circle should be aligned with the nodal point of the stress.The theoretical displacement and stress can be obtained by (1b) and ∇ 2 ψ 0n , respectively.The eigenmode study is employed in the FEM analysis, the stress is calculated from the Gauss point on the neutral axis.Fig. 3 (a) shows the theoretical and FEM simulated results of normalized displacement distribution of mode (0, 1) and mode (0, 3), while Fig. 3 (b) shows the normalized stress distribution.The slight discrepancy of displacement between theory and FEM is due to the difference between the FEM physical boundary and the theoretical clamped boundary.The normalized radius r of the stress nodal circle of FEM for mode (0, 1) and mode (0, 3) are located at r = {0.686}and {0.270, 0.596, 0.928}, respectively.The average error between the theoretical and FEM results is only 2.7%, and the performance variation is less than 2%.Thus, the radius of the theoretical stress nodal circle given in Table 1 is still instructive.The optimal electrode corresponding to FEM results will be adopted in the following analysis.

C. THICKNESS
In ToF applications, the SBW RT should be concerned.There are only two dimensions might affect SBW, i.e., radius a and stack layer thicknesses.They determine the resonant frequency f 0 and static displacement sensitivity A s , and further influence the SBWs.Combine (3), ( 4), ( 8) and ( 17), noting that f 0 ∝ 1/a 2 and A s ∝ a 2 , then the SBW RT that determined by the product of them should be independent of the radius a.The location of the neutral axis is dominated by the thickness of the piezoelectric and passive layers (t p and t ox ).Since t p is typically 0.5-1 µm while t ox can vary in a wide range, the effect of t ox on performance will be analyzed.
Figs. 4 (a)-(f) show the FEM simulated frequency responses of sensitivities in different modes vary with t ox .Figs. 4 (a), (c), and (e) show transmitting sensitivity S Tx , receiving sensitivity S Rx and round-trip sensitivity S RT of (0, 1) mode pMUT, respectively.While Figs. 4 (b), (d), and (f) show the corresponding performance of (0, 3) mode pMUT.pMUTs in both modes are excited by their optimize electrode configurations that have been discussed in Section III-B.The maximum S Tx of (0, 3) mode is 12.5 times that of (0, 1) mode, which is a remarkable improvement in transmitting  performance.However, S Rx of (0, 3) mode is lower than (0, 1) mode, and there is a 30% reduction for maximum value.Finally, the S RT of (0, 3) mode is 5.7 times that of (0, 1) mode.The improvement of S Tx and reduction of S Rx are predicted as (11) and (15).Fig. 5 shows the major performance metrics vary with the thickness of oxide t ox .Figs. 5 (a) and (b) show the dependency of resonant frequency f 0 and static displacement sensitivity A s on t ox .The relationship of f 0 , A s with t ox will be discussed first.It can be estimated from ( 4) and ( 8), when t ox is small enough, t ≈ t p , z p ≈ 0, thus A s ≈ 0 and f s obtains a minimum value; when t ox is high enough, t ≈ t ox , the flexural rigidity D ∝ t 3 , z p ≈ t/2, and µ ∝ t ox , resulting in A s ∝ 1/t ox and f 0 ∝ t ox .As shown in Fig. 5 (a) and (b) that when t ox > 2 µm A s is negatively correlated with t ox and f 0 is almost linear with t ox .The clamped plate theory tends to overestimate f 0 of mode (0, 3) when t ox > 2 µm, due to the assumption that a ≫ t made in the theory is no longer valid.It is worth mentioning that Fig. 5 (a) shows that there is an optimal value of t ox for A s around 0.8 µm.
Figs. 5 (c)-(e) show the effect of t ox on the normalized SBWs, which are extracted from the sensitivity frequency responses in Fig. 4. The trend of normalized SBW Tx , SBW Rx and SBW Rx with t ox can be roughly estimated by (11), (15) and (17), respectively.SBWs eliminates the effect of Q-factors, thus Q is neglected in the following qualitative analysis.Equation (11) implies that SBW Tx ∝ f 2 0 A s , which is monotonically increasing with total thickness, which is shown as Fig. 5 (c).Equation (15) shows that SBW Rx ∝ A s α 0n , thus the tendency of SBW Rx in Fig. 5 (d) is similar with A s in Fig. 5 (b).
Finally, SBW RT is proportional to (f 0 A s ) 2 , when t ox is high enough, the SBW RT should tend to be a constant.Fig. 5 (e) shows that at t ox = 1.75 µm the simulated maximum SBW RT of (0, 3) mode is 1.83.For (0, 1) mode, SBW RT tends to be constant when t ox exceeds 4 µm in theory, and the simulation shows that t ox = 4.25 µm corresponds to maximum of SBW RT .The simulated SBW RT of (0, 1) is consistent well with theory, while that of (0, 3) mode declines rapidly and deviates from the theory curve after t ox = 1.75 µm.On one hand, the clamped plate theory tends to overestimate f 0 .On the other hand, the additional losses might come from the support loss (Q support ∝ (a/t) 3 ) and the thermal viscosity loss at a higher frequency.Thus, to obtain the corresponding maximum SBW RT , t ox = 4.25 µm is adopted as the optimal value for (0, 1) mode, while t ox = 1.75 µm adopted for (0, 3) mode in the following analysis.sensitivity S RT of two pMUT designs that optimized for (0, 1) and (0, 3) mode, which are labeled as (0, 1) and (0, 3), respectively.The S Tx is shown in logarithms form, due to the gigantic difference between the two designs.Fig. 6 (a) shows that (0, 3) pMUT demonstrates extraordinary transmitting performance improvement compared to (0, 1) pMUT.The maximum S Tx of (0, 3) pMUT is 10.2× of (0, 1).As shown in Fig. 6 (b), the receiving sensitivity of (0, 3) pMUT, however, is only 0.41× of (0, 1) pMUT.Fig. 6 (c) shows that maximum S RT of (0, 3) pMUT is 4.12× of (0, 1) pMUT.Table 4 summarized the performance comparison of two pMUT designs.The SBW RT of (0, 3) is about 1.85× of (0, 1) pMUT, which is consistent with Fig. 5 (e).

B. DIRECTIVITY
Fig. 7 shows the sound pressure level (SPL) spatial distribution of (0, 3) mode pMUT while (0, 1) mode as a reference when a = 50 µm and driving voltage is 1 V.The reference sound level in water is 1 µPa.In contact with water (c 0 = 1481 m/s), the f 0 of demonstrated (0, 1) and (0, 3) pMUTs are 3.18 MHz and 18.60 MHz, respectively.Fig. 7 (a) shows that (0, 3) mode exhibits not only higher SPL but also more concentrated field distribution, i.e., higher directivity.According to product theorem [14], the overall array directivity is the product of the element directivity and the array factor.Thus, with the same array form (array factor), higher element directivity results in higher overall array directivity.
The simulated and theoretical results of the normalized radiation pattern for the pressure of (0, 1) and (0, 3) pMUTs are shown in Fig. 7 (b).The (0, 1) pMUT almost produces the omnidirectional pattern, while (0, 3) demonstrates much better directivity.The theoretical half power beam width (HPBW) of (0, 3) is 38.9 • , while simulated HPBW is 40.5 • , producing a directivity of ∼5.0, which implies that with the same radiated power and distance, (0, 3) mode will produce about 5 times the sound pressure of the omnidirectional (0, 1) mode at the maximum radiation direction.
The simulated and theoretical SPL along the z-axis is shown in Fig. 7 (c).The Rayleigh distance R 0 of (0, 3) mode is 98.7 µm, while that of (0, 1) mode is 16.8 µm.In the far field (z > R 0 ), the pressure is nearly inversely proportional to distance.The SPL of (0, 3) mode is 20.38 dB higher than mode (0, 1), which is consistent with the previous 10.4× improvement in S Tx .The simulated SPL of (0, 3) mode is slightly lower (0.3 dB) than the theoretical analysis, which is partially due to higher thermal viscous loss at a higher frequency.

C. PULSE-ECHO RESPONSE
To compare the device on the same time scale, the height of the water is set to 10 mm, then the total round-trip time is about 13.5 µs.The radius of the (0, 3) and (0, 1) are both 50 µm to compare their efficiency with the same area.The detail of the pulse-echo simulation setting has been shown in Section III-A.
The proposed (0, 3) pMUT exhibits extraordinary roundtrip performance compared with (0, 1) pMUT.The surface pressure (P 0 ) at the center of pMUTs is shown in Fig. 8 (a).Two pMUT designs are driven by the voltage shown in Fig. 8 (c), which is the product of the Gaussian function and the sine function.The (0, 3) pMUT generates a surface pressure of 10.6 kPa, which is 2.15× of (0, 1) pMUT (4.96 kPa).The echo pressure is reflected by the hard boundary and then returns to the surface of pMUTs are shown in Fig. 8 (b).After reflected by the hard boundary, the echo pressure of (0, 3) pMUT (62.0 Pa) is 10.5× of (0, 1) pMUT (5.88 Pa), which is consistent with the improvement of S Tx (10.2×).
The receiving voltage of pMUTs is shown in 8 (d).In the reception, the maximum receiving voltage of (0, 3) pMUT is 6.53 µV, which is 8.6× of (0, 1) pMUT (0.76 µV).The ratio of receiving voltage between the two designs is higher than the ratio of S RT (4.12×), due to the mismatch between terminal resistance and the pMUT reactance.The capacitance of (0, 1) and (0, 3) pMUT are 0.368 and 0.140 pF, which correspond to the reactance of 136 and 61 k , leads to an output voltage ratio of 2.05.Thus, the maximum receiving voltage ratio is about 4.12 × 2.05≈8.5.

V. CONCLUSION AND DISCUSSION
This work proposed the pMUT designs with high-order axisymmetric mode.To prove the concept, 3 rd -order mode is used for the analysis, while traditional 1 st -order axisymmetric mode as a reference.Analytical models for n th -order

FIGURE 1 .
FIGURE 1.(a) Cross-section view and (b) top view of the multi-electrodes actuated circular plate consist of multi-layers with clamped boundary.

FIGURE 2 .
FIGURE 2. The configuration of 2D axisymmetric FEM models of pMUT for (a) transmitting sensitivity (b) receiving sensitivity and (c) pulse-echo response.

FIGURE 3 .
FIGURE 3. (a) Normalized displacement and (b) Normalized stress distribution of mode (0, 1) and mode (0, 3) along radial.The continuous curves are theoretical results, while the dots represent FEM simulated results.The insets at the corner show the mock-up views of the displacement and stress distribution in FEM models while the substrate is hidden.

FIGURE 4 .
FIGURE 4. The FEM simulated frequency response of S x in different mode vary with oxide layer thickness t ox .The curves increase from purple to red, corresponding to t ox increase from 0.25 to 5 µm with a step of 0.25 µm.(a) S Tx , (c) S Rx and (e) S RT of (0, 1) mode pMUT; (b) S Tx , (d) S Rx and (f) S RT of (0, 3) mode pMUT.

FIGURE 5 .
FIGURE 5.The simulated and theoretical performance metrics vary with the thickness of the oxide layer.(a) Resonant frequency f 0 , (b) Static displacement A s , (c) Normalized transmitting sensitivity bandwidth SBW Tx , (d) Normalized Receiving sensitivity bandwidth SBW Rx , and (e) Normalized round-trip sensitivity bandwidth SBW RT .The SBWs are normalized to the maximum value of mode (0, 1).

Figs. 6
Figs. 6 (a)-(c) show the frequency response of transmitting sensitivity S Tx , receiving sensitivity S Tx and round-trip

FIGURE 6 .
FIGURE 6.The simulated transmitting, receiving, and round-trip performance with different pMUT designs in the same radius a = 50 µm.(a) Transmitting sensitivity; (b) Receiving sensitivity; and (c) Round-trip sensitivity of the two pMUTs design that optimized for (0, 1) and (0, 3) mode.

FIGURE 8 .
FIGURE 8.The simulated pulse-echo response of (0, 1) and (0, 3) mode pMUT designs with the same radius.(a) The transmitting surface pressure and (b) The receiving echo pressure at the center of the pMUTs surface.(c) The driving voltage.(d) The receiving voltage generated by the echo.