An Experimental and Numerical Study of Crosstalk Effects in PMUT Arrays

This study provides a comprehensive characterization of a 4 $\times $ 4 piezoelectric micromachined ultrasonic transducer (PMUT) array suited for in-air applications. Crosstalk can be viewed as a lack of isolation among the transducers, with possible reasons, including the electrical, acoustic, and structural coupling among the transducers. This study focuses on acoustic coupling as the primary cause of crosstalk, since the influence of electrical and structural coupling on crosstalk is shown to be negligible by comparing experimental tests and numerical analysis conducted in vacuum and in air. It is expected that acoustic coupling among diaphragms occurs due to air-connected cavities. Consequently, several scenarios are investigated in which the cavities are either partially or completely acoustically disconnected. The computed response spectra of the PMUTs demonstrate that the presence of secondary peaks, which indicated the presence of crosstalk, is diminished when the PMUTs are decoupled partially or completely as compared with an open cavity. The conducted acoustic tests also demonstrated that crosstalk impacts the intensity and directivity of the acoustic wave produced by the array.

In the design and development of MEMS devices such as PMUT, Capacitive Micromachined Ultrasound Transducers (CMUTs) [23], and Piezoelectric micro-speakers [24], [25], the crosstalk among elements is a critical factor that negatively impacts their performance.Crosstalk, if not properly controlled, can damage the beamforming capacity of the array, reducing the imaging resolution [26] and, in severe cases, can cause reliability issues [27].For the purpose of enhancing the performance of PMUT arrays, precise experimental characterization and numerical simulation of the array are essential [28].Recent research has focused on developing a high-density PMUT array with minimal levels of crosstalk.Xu et al. [29] proposed a design for an array with a high fill factor that mitigates acoustic interference among adjacent transducers.The proposed design increased the sensitivity of the array in comparison with conventional designs of arrays of the same size.Yang et al. [30] introduced a novel PMUT design based on the silicon on insulator (SOI)-bonded technique with a high fill factor and reduced crosstalk.Crosstalk-induced vibrations in diaphragms adjacent to an excited one are minimized by three times compared with conventional designs.
Pirouz et al. [31] investigated crosstalk using an efficient finite-element method (FEM) simulation deployed on a cloud high-performance computing (HPC) platform for an array with over 5000 transducers.Their findings demonstrated that increasing the pitch among transducers significantly reduces crosstalk levels.The frequency-domain analysis also revealed that the PMUTs are activated at frequencies that differ from their own central frequency.As a result, crosstalk influences the frequency response of the transducer, forcing it to deviate from its ideal performance.Weekers et al. [32] and Billen et al. [33] also detected the additional resonance mode in the array that does not correlate to any flexural mode of the PMUTs.It is understood that acoustic crosstalk through the fluid is responsible for the observed phenomenon, since the distance between displacement peaks at this resonance corresponds to one wavelength in the acoustic domain.
Different strategies are implemented to control and eliminate the undesirable presence and effect of crosstalk on the performance of PMUT arrays.A phase-shift excitation approach has been adopted to shrink the blind detection zone by shortening the intensity of crosstalk and ring-down vibrations [34], [35].An array design with a dampening backing structure (B-PMUT) was proposed and demonstrated by Wang et al. [36].The results indicate that the bandwidth of the device with a backing structure is 63% greater than that of the device without a backing structure.The degree of crosstalk between two adjacent transducers fell from −13.4 to −17 dB.Several acoustic isolation approaches, such as the usage of mesa-type structures, ring-shaped pillars made of different materials, or a combination of both, have been implemented to reduce crosstalk [37], [38].
In this study, a 4 × 4 PMUT array designed for inair applications is investigated using numerical simulations and experimental data.This study aims to give an accurate evaluation, modeling, and explanation of the crosstalk phenomenon using FEM models and experiments.Furthermore, the suggested numerical approach can aid in providing a deeper understanding of the behavior of PMUT arrays and a set of design guidelines to mitigate crosstalk.In addition, the influence of residual stresses on the diaphragms is also a crucial component to be considered.Residual stress is an unavoidable aspect of PMUTs microfabrication.To some extent, nonoptimal device performance and variations from model predictions can be attributed to residual stresses in the layered system and the silicon die [39].The performance of a transducer is extremely sensitive to residual stresses, because they alter their operating frequencies and reduce the bandwidth.A precise knowledge of residual stresses can be used as a powerful design and analysis tool to improve the PMUT performance [14], [39], [40], [41].
This article is organized as follows.Section II provides a concise description of the investigated PMUT array.In Section III, experimental testing and numerical simulations are described to quantify and evaluate the influence of residual stresses and the applied voltage bias on the mechanical behavior of a single transducer.In addition, a set of experimental tests supported by numerical simulations are described to investigate the influence of crosstalk on the mechanical and

II. DESCRIPTION OF THE PMUT ARRAY
The array, shown in Fig. 1, consists of 16 PMUTs, laid out in a square pattern.Fig. 2 shows that each transducer is composed of multiple layers of various thicknesses and material properties.The material properties of the different layers are listed in Table I.The active lead zirconate titanate (PZT) layer is 2-µm thick.The top and bottom electrodes are made of titanium-tungsten (TiW) and titanium-platinum (TiPt), respectively.At the center of the layered system, the overall thickness is 7.9 µm.The structural layer consists of poly-Si with a thickness of 4 µm.The PZT active layer, with a nominal radius of 260 µm, is coaxial with the transducer cavity, which has a nominal radius of 372.5 µm.The diameter of the cavities is a significant design parameter that determines the resonance frequency of a transducer [14], [43].Due to the microimperfections of the fabrication process, the diameter of each transducer may vary slightly from its intended value.Consequently, the center frequency of each PMUT can vary slightly around its design value.The silicon die, which contains the 16 PMUTs, is square with the side dimensions of 6950 µm and a thickness of 410 µm.The silicon die adheres through an epoxy glue on a test measurements adapter (TMA) board that is connected to an expansion board with 16 pairs of pins.Each pair of pins is intended to electrically drive each transducer independently.The edges of the square-shaped silicon die are attached to a square hole in the TMA board, leaving the transducers on the back side of the die acoustically connected; see Fig. 1.

III. CROSSTALK EFFECT ANALYSIS A. Single-PMUT Analysis
This section outlines experimental testing and numerical simulations intended to investigate a single PMUT from the array.First, it is explored how the voltage bias and residual stresses affect its initial configuration and frequency response.Due to the presence of residual stresses, the diaphragms and the die are distorted and not flat.Several analytical solutions are proposed for calculating the residual stresses in the layered system, which can ultimately facilitate their design [14], [39], [40].The transducers are extremely sensitive to residual stresses, as they influence not only their initial configuration but also their central frequency, the device bandwidth, and the overall mechanical and acoustic dynamic behavior [44], [45], [46].We present here numerical simulations that account for the presence of residual stresses as one of the primary inputs to be addressed.The level of stress experienced by each layer in the transducer and the silicon die is determined experimentally and used as an input to the numerical model.
A single diaphragm static deformation and its variation as a function of an applied dc voltage bias are explored first.A 3-D model of a single diaphragm, from the 4 × 4 array, is simulated.Fig. 3 depicts a cross-sectional view of the intricate geometry of the layered system discussed in this work.As depicted in Fig. 3, one of the design strategies utilized in the fabrication of the transducer is the removal of specific layers in certain regions.As one moves away from the diaphragm center, it can be observed that in some regions, all layers are eliminated with the exception of poly crystalline silicon.This technique is commonly referred to as mesa structures, in which certain portions of the structure are removed to impede the propagation of waves through the material, reducing the levels of crosstalk due to the structural coupling among the diaphragms [37].The silicon die bottom side is fixed.The 3-D model, shown in Fig. 4, is discretized using wedge elements belonging to the quadratic family.The top surface of the PMUT is discretized with triangular elements, and the resulting mesh is swept throughout the thickness.The optimal mesh is determined using an iterative convergent procedure that selects the largest mesh size that maintains accurate results.
In the numerical models, residual stresses are handled as an additional input.The various materials are subjected to prestresses that vary in magnitude and sign.Even though the net residual stress can be either tensile or compressive, it is better to have tensile state stress, as PMUTs with high net compressive stress can have reliability problems related to buckling and a sudden drop in resonance frequency [45], [47].At the PZT layer, an electromechanically coupled problem is solved using a linear constitutive model of the piezoelectric material represented in a stress-charge form.To investigate the effect of the imposed dc bias voltage, on the initial deformation of the diaphragms, a stationary study is carried out for dc voltages ranging from 0 to 40 V with 5-V increments.The dc voltage is applied on the top electrode, while the bottom one is at ground.The model adopted accounts for the geometrical nonlinearities.When considered, the relationship between displacement and strain is complex and nonlinear, in contrast to the assumption of small displacements (u) and strain (ε).In the case where geometrical nonlinearities are accounted for, the strain is computed with the deformation gradient tensor (F) [48]  where I represents the identity tensor.The Green-Lagrange strain tensor (E), a suitable measure of strain for large deformations, is then calculated as follows: The geometric nonlinearities affect not only the strain-displacement relationship but also the governing equations (equilibrium equations), requiring the use of iterative nonlinear solution techniques.The nonlinear static solution to the problem is sought by means of the Newton-Raphson scheme.The initial deformed configuration and its variation with the imposed voltage bias of a single PMUT are investigated experimentally using the Polytec MSA 500 in white-light interferometry mode, which allows structures to be experimentally profiled in 3-D.
Figs. 5 and 6 depict the variation of the deformed initial configuration at varying voltage bias, computed relative to a flat diaphragm, obtained from the numerical simulations and experimental tests, respectively.It can be observed that the deformation increases with the increase of the applied voltage amplitude.The deformation of the layered system grows at a slower pace for higher voltages, depicted by the dotted lines in Figs. 5 and 6, showing an increase in the PMUT stiffness, i.e., hardening effect.One limitation of whitelight interferometry is the presence of materials that may  be transparent or semitransparent to light.The passivation layers of the PMUT are, in fact, invisible to the white light used in the interferometer, potentially leading to an incorrect interpretation of the results.In addition, the experimental test pertains to measurements associated with the upper surface of the PMUT topography, while the simulation results are computed on the polysilicon layer.In order to ensure a valid and accurate comparison of the outcomes, it is necessary to account for the thicknesses of the transparent layers within the PMUT.Fig. 7 illustrates a comparison between the maximum deflection of the diaphragms obtained from the simulation and the experimental results, as reported in Table II.
In both vacuum and air, the effect of the voltage bias on the frequency response of the single transducer is examined.For in-vacuum analysis, the model used to compute the initial deformation, shown in Fig. 4, is used.For in-air analysis, the model depicted in Fig. 8 is adopted.The air domain is reproduced in the model by a hemisphere surrounding the PMUT.The hemisphere has a radius of 8 × λ 0 , where λ 0 is the reference wavelength of an acoustic wave traveling in the air.For an acoustic wave traveling in air with a velocity of c a = 342.9m/s at a frequency of f 0 = 100 kHz, this gives a radius of 27.432 mm.The size of the acoustic domain has been chosen to be sufficiently large in order to enable the investigation of both the near-field and far-field behavior of the array.The validity of the far-field approximation is verified when considering regions that are located beyond the complex near-field domain and at a significant distance  from the emission source.When analyzing a circular vibrating transducer, it is valid to use this approximation for points that are located at distances greater than the Rayleigh distance.The Rayleigh distance, denoted as D R , can be calculated using the formula D R = πd 2 /4λ = S/λ = 127.12µm, where d represents the diameter of the PMUT (745 µm) and S represents the area of the PMUT (0.436 mm 2 ).The size of the top acoustic domain, defined by the radius of the hemisphere, exceeds, by far, the value of the computed D R .This proves that the computational acoustic domain fully encompasses the complex near field and extends to include the far-field.
The implemented geometric and mesh size parameters for the acoustic domain of the numerical models utilized in this study are listed in Table III.Both the top and bottom sides of the air domain encompass the near-field region in relation to the Rayleigh distance.The transducer cavity is also filled with air.An absorbing boundary condition (ABC) is applied to the hemisphere external surface and the bottom surface of the air domain filling the cavity.The nonreflecting first-order ABC applied to the external boundary of the hemisphere, denoted by S ABC , is expressed as follows [20]: where p is the acoustic pressure, c denotes the acoustic wave velocity in air, and n denotes the unit outward normal vector to the fluid domain.This BC is introduced to simulate an infinite domain with respect to the reference wavelength λ 0 of the acoustic wave traveling in the air by permitting nearly negligible incident wave reflection.Utilizing ABC in the far-field region, where the propagating wavefront is nearly spherical, can be considered equivalent to a perfectly matching layer (PML).The selection of ABC over PML was based on the fact that PML requires domain assignment, whereas ABC is exclusively assigned to a boundary.This approach eliminates the necessity of establishing an additional supplementary domain exclusively for the acoustic domain in order to implement a PML, consequently minimizing the computational burden.A hard-wall BC is imposed on the lateral surfaces of the air domain on the cavity side, as well as on the portion of the hemisphere domain flat surface that is not shared with the wet boundaries, where the acoustic domain comes into contact with the structural domain, of the layered system.In the air domain, the Helmholtz equation for the pressure acoustic field is solved.The acoustic-structure interaction is governed by the continuity of normal stresses T • n, where T represents the stress tensor, and normal acceleration ü • n at wet boundaries, i.e., the relation between the fluid load exerted on the structure and the structural acceleration experienced by the fluid.The coupling condition can be expressed as follows: −Tn = pn on S I .
Equation ( 4) represents the fluid momentum balance equation projected along the unit outward normal vector n of the interface surface S I .The air domain is totally meshed with tetrahedral elements belonging to the quadratic family, with a mesh size of λ 0 /2, which equals 3.43 mm.This mesh size is chosen to ensure proper sampling and representation of the acoustic wave in the simulation, thereby enhancing the accuracy and reliability of the results [4], [20].For the acoustic field close to the transducer, the mesh size is decreased to conform to the more refined mesh size implemented on the PMUT.
A laser Doppler vibrometric (LDV) test using Polytec MSA 500 is conducted to obtain the first eigenfrequencies and the corresponding eigenmodes of the transducer.The frequency range of the test is between 80 and 500 kHz.Fig. 9 depicts the experimental arrangement utilized for the LDV testing.A transparent glass plate is used to close the vacuum chamber from its top side.The transparent surface is invisible to the light emitted by Polytec, thereby enabling the measurement of PMUT vibrations.A chirp signal, generated by Polytec, with an amplitude of 0.1 V ac and an offset of 0.1 V dc is used to excite a single transducer.The chirp analysis enables the assessment of the performance of PMUTs across a wide range of frequencies.This is achieved by applying a low-amplitude signal with a frequency that varies, over time, within the desired frequency range of interest.Fig. 10 illustrates the in-air experimental test findings of the response spectrum of PMUT P3, highlighting the first six eigenfrequencies.The transversal displacement reported is the mean of the displacements measured at preselected scanning points generated on the transducer using Polytec.Additional in-air and in-vacuum vibrometric tests are conducted on PMUT P6 in the frequency range 80-160 kHz to examine the effect of the voltage bias on the transducers frequency response.In this test, a sinusoidal signal with an amplitude of 0.1 V ac is paired with the dc voltages of 4, 10, and 20 V. The response spectra, presented in Fig. 11, show that the central frequencies of the PMUT have shifted upward under higher applied voltage bias levels for both vacuum and air analyses.In-vacuum eigenfrequencies are 97, 104.9, and 117.5 kHz for the voltage biases of 4, 10, and 20 V, respectively, while in-air values are 107.4,110.9, and 126.8 kHz, respectively.Therefore, the applied static voltage has a stiffening effect.The measured eigenmodes in vacuum correspond to purely mechanical ones.Furthermore, the results of the in-air measurements highlight the presence of secondary peaks in the response spectrum.Secondary peaks are mainly attributable to crosstalk among PMUTs, and for the tested transducer, they are positioned at the central frequency of the adjacent PMUTs in the array.Comparing experimental results, the absence of secondary peaks in the frequency response P6 in the in-vacuum experiment implies that crosstalk is primarily transmitted through the acoustic domain and not through the shared solid domains.It is worth noting that the central frequency of the PMUTs is higher in air compared with vacuum.The presence of air volume on the back, which contributes to a spring-like effect, is the primary cause for this apparent increase in frequency.The simulation of the increased frequency observed in Fig. 11, resulting from the additional spring-like effect, can be achieved by incorporating a larger air domain in the simulation, taking into account the presence of rigid walls in the cylinder.However, it is important to note that this approach would lead to a significant increase in the computational cost.

B. Two-PMUT Analysis
The numerical investigation of the crosstalk effect starts by considering two neighboring PMUTs in the array.An air domain surrounds the top of the two diaphragms, and a rectangular air domain connects their cavities; see Fig. 12.The acoustic domain and structural components of the PMUTs have the same dimensions, BCs, and meshing as those outlined in the study of a single PMUT.Since the occurrence of acoustic crosstalk on the lower side of the PMUT is predominantly attributed to near-field behavior, it is sufficient to construct an acoustic model that is computationally efficient yet capable of accurately explaining the occurrence of secondary peaks in the PMUT's frequency response.As depicted in Fig. 1(b), the array is glued to the TMA board, resulting in the formation of a rectangular air domain on the lower side of the array.This air domain is defined by the inner side walls of the TMA, which extend along its thickness.The thickness of this board is approximately 4 mm, which is used to simulate the lower acoustic domain.This distance exceeds the estimated nearfield distance.Moreover, the characteristics of the acoustic domain play a significant role in determining the level of crosstalk resulting from acoustic coupling.Table IV presents the characteristics of the acoustic domain employed in the simulation.
The radius of the two transducers deviates somewhat from the design value in order to account for production imperfections.The excited PMUT cavity has a radius of 390 µm, whereas the radius of the observed PMUT cavity is 372.5 µm.
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TABLE IV MATERIAL PROPERTIES OF THE ACOUSTIC DOMAIN (AIR) AT A REFERENCE TO A TEMPERATURE AND ABSOLUTE PRESSURE
OF 293.15 [K] AND 1 [atm], RESPECTIVELY Fig. 13.Computed frequency response of the excited "blue line" and observed "red line" PMUTs.
Fig. 14.Two-PMUT model simulation configurations for "full acoustic coupling" in air, PML, hard-wall BC, and "no acoustic coupling" in vacuum.
So, the central frequencies of the two transducers differ slightly, with the excited one having a central frequency of 106 kHz and the observed one having a central frequency of 120 kHz.A frequency sweep analysis between 80 and 140 kHz is performed.A bias voltage of 4 V dc and a small-amplitude harmonic signal of 0.1 V ac are applied to the top electrode on the excited PMUT, while the top electrode of the nonexcited one is set to a floating condition.The bottom electrode of both transducers is set to ground.The computed frequency response of the excited and observed PMUTs is shown in Fig. 13.The appearance of a secondary peak in each of the curves demonstrates the existence of crosstalk.The secondary peak in each PMUT response spectrum is located at the center frequency of the adjacent one.As demonstrated by the experimental data depicted in Fig. 11, crosstalk occurs mostly via acoustic coupling and not mechanical coupling among transducers.It is assumed that acoustic crosstalk occurs mostly via the shared acoustic domain on the cavity side and not through the air domain encircling the array on top.To test this claim and to have a better understanding of the impact of the connection of the cavities between the transducers, numerical simulations are performed with the cavities either fully or partially disconnected as illustrated in Fig. 14.
In addition to the case where cavities are fully acoustically coupled, a second model is generated to create a condition where communication is partially impeded.This is done by simulating the bottom shared air domain with a PML, whose objective is to dampen the propagation of acoustic waves in the domain.Therefore, it closely resembles the properties of a typical damping material.The third model proposed contains cavities, filled with air, that are fully disconnected by assigning a hard-wall BC, preventing PMUT-to-PMUT communication across the bottom air domain.Finally, the two transducers are simulated in vacuum.The response spectra of the four simulated scenarios for both the excited and observed PMUTs are shown in Fig. 15.The response spectra of the excited one indicate that the presence of secondary peaks decreases when the transducers are disconnected partially or completely compared with the "open-cavity" case, i.e., the case of air domain connecting the cavities.In addition, when we proceed from the most disconnected "vacuum" to the most connected "open-cavity" scenario, the vibration amplitude of the excited PMUT falls, while the displacement amplitude of the observed one increases.This is because when cavities are connected, there is a greater transfer of energy, i.e., crosstalk, between the transducers.
The maximum values of the vibration amplitudes computed from the observed PMUT for the "open-cavity," "PML," "hardwall," and "vacuum" conditions are 0.0213, 0.0085, 0.0038, and 0.001 [µm], respectively.The open-cavity case exhibits the highest vibration amplitude, which is caused by crosstalk resulting from structural coupling and acoustic coupling.In the case of the hard wall, where crosstalk is caused by structural and acoustic coupling, only from the top side is a reduction of 82.3% in the vibration amplitude observed in comparison with the prior case.In the case of vacuum, where only structural coupling occurs, crosstalk-induced vibration is reduced by 95.2%.This demonstrates that the structural coupling for the given array design is negligible in comparison with the acoustic coupling in general.
The depth of the cavity can result in varying radiation impedances at the bottom of the PMUT, resulting in varying levels of acoustic pressure transmitted from or reflected back to the device.Since acoustic coupling has been shown to cause more crosstalk than structural coupling, a cavity with a depth that creates a radiation impedance leading to more acoustic emission from the bottom is likely to cause more crosstalk.Dynamic response of the excited and observed PMUTs simulated in air "red line" and vacuum "blue line." Further vibrometric time-domain analysis is performed, using Polytec MSA 500, in both air and vacuum on the two-PMUT model.The transducer is activated using a single-cycle sinusoidal pulse with an amplitude of 8 V pp and a voltage bias of 4 V imposed on the top electrode, while the bottom one is set to ground.In the analysis, both the excited and observed diaphragm ring-down responses are computed and analyzed.
Fig. 16 illustrates the results of a numerical simulation of the dynamic response of two transducers working in air and vacuum.The observed rapid decay in the amplitude of the PMUT vibration in the vacuum is attributed to the implementation of mechanical damping in the simulation.A viscous damping model has been implemented, resulting in an additional to the stresses within the material that is directly proportional to the rate of elastic strain.This contribution, denoted as T v , in the case of geometric nonlinearities being considered, is given as [48] follows: where J = det(F) is the elastic volume ratio and C = F T F is the right Cauchy-Green tensor.η v denotes the shear viscosity, and η d represents the bulk viscosity.They are defined as follows: The values of Young's modulus E and Poisson's ratio ν for each material in the PMUT are given in Table I.The logarithmic decrement technique is used to compute an invacuum quality factor Q of 75, based on the data presented in Fig. 16.The obtained value aligns with the value derived from the bandwidth of the results depicted in Fig. 15.It is worth noting that the vibration amplitude of the observed one is substantially greater in the air than in the vacuum.This demonstrates that an acoustic coupling has a greater effect on crosstalk than a mechanical coupling.
The dynamic behavior of the two-PMUT numerical model is expected to be fairly comparable to the dynamic response of two PMUTs positioned on the array perimeter, as they have the minimum number of neighboring transducers.Therefore, the results of PMUTs P9-P10 are selected for comparison.Due to the fact that, in reality, more transducers in the array would influence the dynamic response of P9-P10 than the two isolated transducers in the numerical simulation, and some differences can be expected.The experimental results obtained from the in-air tests of P9-P10 are compared with the numerical simulation; see Fig. 17.It is evident that the numerical model is able to represent the crosstalk behavior of the transducers to a significant degree.Because the energy lost in activating adjacent transducers is greater in the experiment, the numerical simulation predicts relatively larger displacement amplitudes for both excited and observed PMUTs.It is difficult to establish a direct correlation between the reduction in vibration amplitude and the presence of neighboring PMUTs.Several factors can influence this relationship, such as the design of the array, the alignment of the PMUTs, the material used, and the type of acoustic domain in which the PMUT operates.The ring-down response of PMUT P10 at the array corner is qualitatively captured by the simulated excited one.Based on the numerical and experimental results of the excited PMUT, the quality factor Q, computed through the logarithmic decrement formula, is found to be equal to 38 and 37, respectively.

C. Full Array Analysis
Frequency spectra responses are measured both in vacuum and air for the 16 PMUTs.The diaphragms are activated, as described in Sections III-B, with a chirp signal having an amplitude of 0.1 V ac and an offset of 4 V dc .Fig. 18 depicts a comparison between the measured response spectra of PMUT P14 in vacuum and air.In both instances, the transducer is excited with the same amount of energy; however, the amplitude of vibration in the air is less than in a vacuum.This is due to the fact that, in the in-vacuum test, the energy is primarily confined to the excited diaphragm.In contrast, in the in-air case, the decrease in amplitude is attributed to the dissipation of energy into the surrounding air.Part of this energy is responsible for inducing movement in neighboring diaphragms, i.e., crosstalk.
Fig. 19 depicts the measured response spectra of all 16 PMUTs in air.It can be observed that the frequency responses of corner PMUTs P2, 8, 10, and 16, highlighted using thick, continuous lines, are extremely comparable.They appear to be characterized by the fewest secondary peaks, which indicates less crosstalk.This is due to the fact that they have the minimum number of adjacent transducers.Therefore, the level of crosstalk is also directly related to the number of adjacent transducers.
A series of time-dependent tests are performed to study the crosstalk among the transducers of the array operating in air.Using the LDV mode of the Polytec MSA 500, the timedependent displacement of the diaphragms, relative to their initial deformed configuration, is measured experimentally.In the first experiment, PMUT P3 is excited with a single sinusoidal pulse at its central frequency of 113 kHz and an amplitude of 8 V pp centered on a bias of 4 V dc , and the displacement of all transducers over time is measured.Fig. 20 displays the results of activating P3 with the single-pulse input signal.All transducers experience a level of vibration as a of crosstalk.The single pulse activates the system with a broad spectrum of frequencies.As a result, all observed PMUTs tend to vibrate at a frequency, f 1-16 , close to their natural frequency.f 1-16 represents the frequencies, derived from a standard Fourier transform analysis, at which the transducers are vibrating.Excitation of P3 at its central frequency, which is close to the fundamental frequencies of the neighboring transducers, produced a vibration with a beatlike pattern.The numerical analysis of the two-PMUT model captured this behavior as well; see Fig.   the displacement is an additional metric used to quantify the level of crosstalk.It is observed that the closer transducers vibrate with greater displacement, up to 10% of the maximum displacement of the excited one.The differences observed in the responses of the neighboring PMUTs located at the same distance from PMUT 3 are due to two primary factors: the location and the central frequency of the neighboring PMUTs.Moreover, while the response of a PMUT in this scenario is primarily influenced by its relation to the activated PMUT, the interrelationship among all PMUTs in the array will impact the overall performance of each individual PMUT.
In the second experiment, P3 is excited using a tone burst signal with 150 pulses at a constant frequency of 113 kHz.This allows the transducers to reach a steady-state vibration.In this case, the vibration frequency is compelled by the harmonic excitation.As a result, all PMUTs vibrate at 113 kHz, which is the resonance frequency of P3 determined by means of the previously described frequency sweep test through the LDV mode conducted at 4 V dc .The displacement amplitude of all diaphragms has increased compared with the case of a single input pulse; see Fig. 21.The vibration of neighboring PMUTs reached up to 16% of the maximum displacement amplitude of P3.
Furthermore, the influence of crosstalk on the acoustic behavior of the array operating in air must be investigated by characterizing the generation and propagation of acoustic waves emitted from the array.The acoustic test setup consists of an ETA450 ultraoptical microphone mounted on a 3-D moving stage, a waveform generator, and a digital oscilloscope.The microphone is employed to record the acoustic pressure generated by the array.Its frequency bandwidth ranges from 50 kHz to 2 MHz, making it excellent for the acoustic test of the device.As depicted in Fig. 22, the device and microphone are mounted on a stage.The stage enables 3-D tests, since it allows for movement along three axes: the xand y-axes, on which the microphone rests, and the normal to the array z-axis.The experimental data are obtained by activating PMUT P3 once with a single sinusoidal pulse of 8-V pp amplitude centered at a 4-V dc bias and with 150 pulses and then measuring the acoustic wave emitted.The microphone is positioned 2 cm, along the z-axis, from the center of the die.Hence, the microphone can move and measure the acoustic pressure along the x y plane at a 2-cm distance.In accordance with the calculated Rayleigh distance, D R = 127.12µm, this distance falls within the far-field range of the investigated array.A clear signal with a good signal-tonoise ratio was possible at this distance.To better understand the effect of crosstalk on the array performance, the array is analyzed first with open cavities and then with cavities filled with an absorbent filling to minimize the acoustic crosstalk.
Fig. 23 depicts the outcomes of the investigated scenarios.Fig. 23(a) and (c) indicates that the maximum pressure amplitude in the open cavity is P rms = 0.7 Pa for the single-pulse case and P rms = 10 Pa for the 150 pulse case.With 150 pulses as an excitation input, a steady-state vibration is reached, corresponding to the maximum oscillation amplitude of the transducers.Consequently, this results in a higher pressure output.In both cases, the peak pressure intensity regions, highlighted by circles, have been shifted to the left.Therefore, crosstalk also affects the directivity of the acoustic wave created by the interaction of diaphragms.In scenarios where the cavities are filled with absorbent material, the reduction in crosstalk influence is evident.Since fewer diaphragms are interacting and their interaction is at a lower level, lower and more focused pressure amplitudes are produced; see Fig. 23(b) and (d).It is worth mentioning that the microphone used has a directivity of 360 • .Hence, undesirable reflections from the stage are also detected.They are represented by  the high-intensity pressures at the edges of the pressure maps depicted in Fig. 23.
The numerical simulations of the acoustic analysis are carried out by means of the Fast Object-Oriented C++ Ultrasound Simulator (FOCUS) software ultrasonic simulation tool, which calculates the array-generated pressure fields.Fig. 24 displays the array generated by FOCUS as well as the computational acoustic domain grid upon which the pressure field is computed.The grid depicted in Fig. 24(b) is positioned 2 cm away from the array along the z-axis.The acoustic field produced by a single impulse and by 150 pulses at a frequency Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.  of 113 kHz, applied to PMUT P3, is computed.The array depicted in Fig. 24(a) receives as input the time-dependent displacement history of the transducers, which is obtained from vibrometric tests, of the two excitation scenarios, in air.Fig. 25 displays four time instants produced from the simulation of an array with open cavities where crosstalk is anticipated to be greatest.Over time, it is clear that numerous pressure circular maps that are not concentric begin to emerge.Acoustic crosstalk among transducers is responsible for the vibration that produces the circular maps, which emerge first from the crosstalk-induced vibrations of the diaphragms adjacent to P3.On the other hand, Fig. 26 reveals that there are no additional circular pressure maps in the case where cavities are partially isolated by adding the absorbent filling material in the cavities.This is due to the great reduction in crosstalk, which resulted in the minimizing of vibration of nearby transducers.Higher intensities of pressure waves are transmitted when more transducers interact, while the scenario with low crosstalk indicated a more concentrated acoustic pressure.Fig. 27 shows the P rms numerical values generated for all the studied scenarios.As a result of the crosstalk, the singlepulse excitation case with open cavities shows a noncircular pressure map with higher intensities when compared with the case in which the acoustic crosstalk is minimized by filling the cavities.For the situation of 150 pulses, a similar conclusion can be drawn.The discrepancies between numerical and experimental results are caused by the fact that all PMUTs in the simulation have the same radius, which is a limitation of the simulation tool, whereas, in the actual device, the transducers radii vary slightly.Moreover, the experimental tests are influenced by the acoustic noise and reflections coming from the stage.

IV. CONCLUSION
The crosstalk among the transducers has a significant impact on the array's performance.This study utilized numerical simulation and experimental tests to investigate how crosstalk influences the mechanical and acoustic behavior of a 4 × 4 PMUT array.Initial studies examined how voltage bias and residual stresses affect the initial configuration and frequency response of the layered system.Both are found to increase, as the applied voltage bias increases.
The crosstalk effect is numerically analyzed using a two-PMUT model with air domains encapsulating their tops and cavities.Acoustic coupling via air-connected cavities is thought to be the primary cause of crosstalk.Numerical simulations with the cavities either fully or partially disconnected are conducted to verify this claim.It is found that when the diaphragms are acoustically disconnected, partially or completely, the computed response spectra show that secondary peaks, which indicate crosstalk, are drastically reduced in amplitude compared with an open cavity.In addition, the dynamic responses of two PMUTs are measured experimentally and compared with the results of a numerical simulation, with which there is a high degree of agreement.
Finally, considering a full array case, all 16 PMUTs frequency spectra responses are measured both in a vacuum and in air.The frequency response spectrum of each PMUT computed from the in-air test reveals the presence of secondary peaks precisely at the center frequencies of the surrounding PMUTs.The corner PMUTs had the fewest secondary peaks, as they have the minimum number of neighboring transducers.In addition, a series of time-dependent analyses are carried out to measure the vibration amplitude of all transducers caused by exciting a single PMUT in the array.It is reported that the diaphragms nearest to the excited PMUT vibrate with a larger displacement, up to 10% of its maximum displacement.Furthermore, acoustic analyses are conducted with open cavities and with cavities filled with an absorbent substance to limit acoustic crosstalk.The results of the tested scenarios revealed a greater and more displaced maximum pressure amplitude from the center in the open-cavity situation.As a result, crosstalk influences both the intensity and the directivity of the acoustic wave emitted by the array.Future work will include a deeper investigation into the causes of crosstalk as well as innovative techniques for reducing and controlling it.

Fig. 1 .
Fig. 1. 4 × 4 PMUT array.(a) Backside view of the silicon die with epoxy adhesive on the edges.(b) TMA to which the silicon die is adhered.(c) Expansion board containing the TMA and the PMUT array with the adopted numbering system for the PMUTs.

Fig. 2 .
Fig. 2. Profile view of the multilayered system of each PMUT.

Fig. 3 .
Fig. 3. Cross-sectional view of a single-PMUT model, showing the detailed layered structure.

Fig. 4 .
Fig. 4. Details of the 3-D FE model for a single PMUT.

Fig. 5 .Fig. 6 .
Fig. 5. Numerical simulation results of the displacement of the PMUT, computed along the diameter of the bottom surface of the polysilicon layer, and its variation with the applied dc voltage.Inset: deformation of single diaphragm.

Fig. 7 .
Fig. 7. Comparison between numerically simulated and measured PMUT maximum deflection with varying V dc .

Fig. 9 .
Fig. 9. Experimental setup for the LDV analysis.The TMA board containing the array is mounted on cylinder support.(a) In vacuum measurement setup.(b) In air measurement setup.

Fig. 11 .
Fig. 11.LDV test of PMUT P6.The response spectrum shows secondary peaks that are only visible in air due to crosstalk.

Fig. 15 .
Fig. 15.Response spectra of the excited "ext" and observed "obs" PMUTs simulated at four different degrees of acoustic interaction through the cavities.

Fig. 16 .
Fig.16.Dynamic response of the excited and observed PMUTs simulated in air "red line" and vacuum "blue line."

Fig. 17 .
Fig. 17.Comparison between the experimental "blue line" and numerical "red line" results of the dynamic response of excited (P10) and observed (P9) PMUTs operating in air.

17
. The amplitude of

Fig. 20 .
Fig. 20.Dynamic response of the PMUTs due to the excitation of PMUT 3 "highlighted in red" with a single-pulse input signal.

Fig. 21 .
Fig. 21.Dynamic response of the PMUTs due to the excitation of PMUT 3 "highlighted in red" with 150 pulses.

Fig. 22 .
Fig. 22. Stage on which the microphone and array device are positioned for acoustic test.

Fig. 23 .
Fig. 23.Acoustic pressure P rms , generated by exciting PMUT 3 with single and 150 pulses, measured by the microphone at the plane located at a distance of 2 cm from the array.The region of maximum acoustic pressure measured from the array vibrations is highlighted by dashed circles.The red cross indicates the corresponding location of PMUT 3. (a) Single pulse with no absorbent filling in the cavities.(b) Single pulse with an absorbent filling in the cavities.(c) One hundred fifty pulses with no absorbent filling in the cavities.(d) One hundred fifty pulses with an absorbent filling in the cavities.(e) Zoomed-in perspective of the area in the scenario of 150 pulses with no absorbent filling in the cavities.(f) Zoomed-in perspective of the area in the scenario of 150 pulses with an absorbent filling in the cavities.

Fig. 24 .
Fig. 24.Geometrical representation of the PMUT array and the acoustic domain grid generated on FOCUS.(a) Array geometry simulated on FOCUS.(b) Acoustic domain grid on the xy plane 2 cm away from the array.

Fig. 26 .
Fig. 26.Time history of the acoustic wave generated by exciting PMUT 3 with a single pulse, with an absorbent filling material in the cavities to minimize acoustic crosstalk.(a) t = 58.50µs.(b) t = 60.30µs.(c) t = 64.80µs.(d) t = 69.30µs.

TABLE I MATERIAL
[42]ERTIES OF THE DIFFERENT LAYERS OF THE PMUT[42]

TABLE II ESTIMATED
ERROR BETWEEN THE MEASURED d exp AND SIMULATED d sim VALUES FOR THE DEFLECTION OF THE DIAPHRAGMS WITH VARYING DC VOLTAGE INPUT

TABLE III LIST
OF THE IMPLEMENTED GEOMETRIC AND MESH SIZE PARAMETERS FOR THE ACOUSTIC DOMAIN FOR THE 1-PMUT (1-P) AND 2-PMUT (2-P) MODELS