Capacitive micromachined ultrasonic transducers (CMUTs) were reported to own high potential in aircoupled ultrasonic applications such as noncontact nondestructive examination and gas flow measurement. The unsealed CMUTs which utilized the squeeze film effect were reported to overcome the narrow output pressure bandwidth of the conventional sealed CMUTs in air operation. This kind of unsealed CMUTs can also be regarded as Helmholtz resonators. In this work, we present the aircoupled unsealed Helmholtz structural CMUTs which utilize both the squeeze film effect and the Helmholtz resonant effect to enhance the output pressure bandwidth. Based on the mechanism of vibration coupling between membrane and air pistons in membrane holes, we propose an analytical model to aid the design process of this kind of CMUTs. We also use finite element method (FEM) to investigate this kind of CMUTs for our analytical model validation. The FEM results show that the significant bandwidth enhancement can be achieved when the Helmholtz resonant frequency is designed close to the fundamental resonant frequency of the CMUT membrane. Compared with the conventional sealed CMUT cell, the 4hole unsealed Helmholtz structural CMUT cell improves both the 3dB fractional bandwidth and SPLbandwidth product around 35 times. Furthermore, it is found that, with more holes under the same hole area ratio or with a smaller ratio of the cavity height to the viscous boundary layer thickness, the Helmholtz resonant effect becomes weaker and thus the output pressure bandwidth decreases.
Capacitive micromachined ultrasonic transducers (CMUTs) were invented as an alternative to conventional piezoelectric ultrasonic transducers in early 1990s. Compared with the piezoelectric ultrasonic transducers, CMUTs have many advantages such as they are suitable for batch production and can be easily integrated with modern electronics. Conventional piezoelectric ultrasonic transducers typically require matching layers to compensate the acoustic impedance mismatch between piezoelectric materials and surrounding gaseous or liquid media. CMUTs feature a better impedance matching with gaseous and liquid media, and therefore they can transmit/receive ultrasound more efficiently into gas and liquid without the matching layer [
In many ultrasonic applications, the transducers were used in immersion [
For aircoupled ultrasonic applications, transducers with wide bandwidth have sharp ultrasonic pulses and short decay time of the vibration amplitude, thus providing high resolution and a small “dead zone” where objects cannot be detected [
In this paper, we present an aircoupled unsealed CMUT cell with Helmholtz resonant cavity, which utilizes both the squeeze film effect and Helmholtz resonant effect to enhance the output pressure bandwidth of the device. For convenience, we hereafter call it as Helmholtz structural CMUT. We propose an analytical model to explain the vibration coupling working mechanism of this kind of CMUT and aid the design process of this kind of CMUT. In order to validate this analytical model and provide deeper investigation on this kind of Helmholtz structural CMUT, the FEM model is built with COMSOL Multiphysics version 5.2a (COMSOL Inc., Stockholm, Sweden) to simulate three cases of 4hole CMUTs with different hole radii.
Figure
(a) A 4hole unsealed circular CMUT cell with Helmholtz resonator structure. (b) Crosssectional diagram of this cell along dashed line of (a).
As Figure
Simplified model of Helmholtz structural CMUT cell.
Then referencing from [
When the DC and AC bias voltages are superimposed, the top plate vibrates as a harmonic oscillator and drives the equivalent air piston to vibrate due to the elastic property of air cushion in the cavity. We assume that the deflected shape of the top plate can be neglected and the uniform displacement of the top plate is denoted as
Uncoupled Helmholtz resonator and uncoupled top plate: firstly, if we set
Coupled Helmholtz resonator and top plate: now, we focus on the coupled vibration system, assuming in harmonic oscillation state, and we solve (
From (
As shown in (
For an aircoupled unsealed Helmholtz structural CMUT cell, when calculating
The squeeze film effect includes squeeze film stiffening effect and squeeze film damping effect. The proportion of the stiffening effect and damping effect is determined by the squeeze number [
For the plate, although there are other damping mechanisms, the squeeze film damping is the most noticeable once it is presented in MEMS devices [
For the equivalent air piston, not only the acoustic radiation damping but also the viscous damping needs to be considered. When the air pistons vibrate, due to the viscosity of the air, they are damped by the viscous force at the side walls of the holes. From [
After getting
In order to investigate the aircoupled unsealed Helmholtz structural CMUTs and validate the analytical model, we have developed the finite element method (FEM) model to simulate the device. Unsealed CMUTs would induce squeeze film effect; usually the squeeze film effect in MEMS devices is described by nonlinear compressible Reynolds equation, which is simplified from the continuity equation and the NavierStokes equation when the modified Reynolds number satisfies [
The commercial software COMSOL Multiphysics version 5.2a is used to build our FEM model. The Electromechanical (
We use four symmetric distributed holes here to avoid the damage of the center part of membrane which makes main contribution to membrane’s vibration. Table
Material properties used in COMSOL (
Parameter  Value/Preset function in COMSOL 



Relative permittivity  1 
Dynamic viscosity 
eta(T[1/K]) 
Heat capacity at constant pressure 
Cp(T[1/K]) 
Density 
rho(pA[1/Pa],T[1/K]) 
Thermal conductivity 
k(T[1/K]) 

0.6 
Typical Wave Speed for PML [m/s]  343 




Young’s modulus 
170 
Poisson ratio 
0.28 
Density 
2329 
Relative permittivity  11.7 




Young’s modulus [GPa]  70 
Poisson ratio  0.17 
Density [kg/m^{3}]  2200 
Relative permittivity  4.2 
A 3D model in COMSOL: (a) a quarter of a CMUT cell and spherical air domain; (b) details of the unsealed Helmholtz structural CMUT cell; (c) meshing details of the vertical section of (b).
In order to make a comparison between the unsealed Helmholtz structural CMUTs and the conventional vacuum sealed CMUTs, we also build an FEM model for the latter. The model building process of the conventional cell is slightly different from the unsealed Helmholtz structural CMUT cell. The solidair interfaces for the conventional cell only contain the top surface of the membrane; thus, the “velocity” and “isothermal” are only added on the membrane’s top surface. The cavity is vacuum; thus, for a conventional cell, we do not take the acoustic field of the cavity into
We take three (Standard, Double, and Half) cases in FEM simulation; Table
Parameters of CMUT cells in our design.
Parameters  Double  Standard  Half 


1000  500  250 

60  30  15 

20  10  5 

110  55  27.5 

40, 80, 120, 160, 200  20, 40, 60, 80, 100  10, 20, 30, 40, 50 

6  3  1.5 

750  375  187.5 

4  4  4 

6720  3360  1680 
For hole radius, we use (
Firstly, the Double case is considered to show the vibration coupling working mechanism of the aircoupled unsealed Helmholtz structural CMUTs.
The membrane and air piston’s motions are mainly on vertical direction; thus, in COMSOL, we take the amplitude of the membrane’s velocity
Figure
FEM results of the Double case: (a) magnitude of complex amplitude of air piston velocity; (b) magnitude of complex amplitude of membrane velocity.
The FEM complex amplitude of total volume velocity
FEM results of the Double case: magnitude of complex amplitude of total volume velocity.
Figure
FEM results of the Double case: (a)
The CMUTs from the Double case to the Half case have different working frequencies; thus, at the point with same distance from the CMUT cells, the acoustic attenuation differs among cases. In order to fairly compare the output pressure from the Double case to the Half case, the SPL is taken at the top surfaces of membrane and hole. We first take the area average sound pressure of the top surfaces of membrane and hole then calculate the SPL. Figure
FEM results of the Double case: sound pressure level (SPL) over the membrane and hole’s top surface (dB).
Secondly, for the Standard case in Table
3dB fractional bandwidth and SPLbandwidth product of the Standard case as shown in Figure
Hole radius [ 





no hole 


92.3  138.5  176.5  211.3  244.7 

Separation between 
75.2  29  9  43.8  77.2  
3dB fractional bandwidth  4.8%  11.6%  35.7%  7.3%  3.8%  0.9% 
SPLbandwidth product  1.22  2.88  8.01  1.51  0.77  0.23 
FEM results of the Standard case: sound pressure level (SPL) over the membrane and holes (dB).
As shown in Table
Last, for the Half case in Table
3dB fractional bandwidth and SPLbandwidth product of the Half case as shown in Figure
Hole radius [ 





no hole 


184.6  277.1  353.1  422.6  489.6 

Separation between 
150.3  57.8  18.2  87.7  154.7  
3dB fractional bandwidth  5.4%  10.9%  24.0%  14.7%  6.5%  1.3% 
SPLbandwidth product  2.67  5.29  11.06  6.08  2.62  0.67 
FEM results of the Half case: sound pressure level (SPL) over the membrane and holes (dB).
As shown above in Tables
In order to validate our analytical model, we make a comparison between analytical results and FEM results from all cases. For the analytical results, we first use (
Comparison of analytical and FEM results of SPL at a point above the center of the membrane.
One point we would like to mention, in (
In order to evaluate the similarity between the analytical SPL results and the FEM SPL results, we calculate the normalized mean square error (NMSE) between our analytical results and FEM results [
The NMSE values vary between negative infinity (not fit) and 1 (perfect fit). In each subgraph in Figure
Normalized mean square error (NMSE) values between analytical results and FEM results in Figure



NMSE  0.9173  0.9347 





NMSE  0.9355  0.9513 





NMSE  0.9523  0.9689 
The deviations between analytical results and FEM results could be caused by the simplifications of the analytical model such as negligence of the membrane’s deformation and assumption as a plate with uniform displacement. In (
However, this analytical model gives the explanation of the most significant vibration coupling mechanism of the aircoupled unsealed Helmholtz structural CMUTs and can be used to predict the characteristics such as
The flow chart of using analytical model to aid the design process of Helmholtz structural CMUTs.
In this section, we make a comparison of the FEM results of SPLs from the Double case to the Half case and give an explanation of the decreased phenomenon of the two resonant peaks. As we can see from Figure
When sound propagates in structures with small geometric dimensions, the interaction between air and solid boundaries gives rise to the viscous boundary layer and the thermal boundary layer. In these two boundary layers, the loss is much greater than in the free field which is far away from the solid boundaries. The thicknesses of these two boundary layers are calculated as [
For viscous boundary layer, as Figure
Ratio of cavity height to 2
Double case  Standard case  Half case  


3.94  2.79  1.97 
Illustration of viscous boundary layer and flow in cavity and holes.
We have also simulated other two cases whose dimensions are 10 times and 0.1 times of the Standard case in Table
Here we check the bandwidth and SPLbandwidth product enhancement of the unsealed Helmholtz structural CMUTs is indeed affected by the Helmholtz resonator, although the vibration coupling feature of two resonant peaks is not obvious in Figures
FEM results of the Standard case: SPL comparison of two cells with same hole area ratio,
We also simulate another 68hole cell which has the same hole area ratio as the cell (
FEM results of the Half case: SPL comparison of two cells with same hole area ratio,
From discussion above, when the
From Tables
FEM results of the Standard case: sound pressure level (SPL) over the membrane and holes (dB), hole radius changes from 40
In this paper, we present the aircoupled unsealed Helmholtz structural CMUTs which utilize both Helmholtz resonant effect and squeeze film effect to enhance the device’s output pressure bandwidth performance in transmit mode. We have developed an analytical model to explain the vibration coupling between the vibration of membrane and the vibration of air pistons in the holes, which is the basic working mechanism of the Helmholtz structural CMUTs. In order to validate the analytical model, we use FEM model to simulate three cases of the 4hole unsealed Helmholtz structural CMUT cells. From the FEM results, for the Double case, the two resonant peaks in the frequency response clearly show the vibration coupling working mechanism of the Helmholtz structural CMUT cell. For an unsealed Helmholtz structural CMUT cell with various hole radii, compared with the conventional sealed one, the improvements of 3dB fractional SPL bandwidth and SPLbandwidth product are around 35 times in the Standard case and 17 times in the Half case, which are achieved with the hole radius which let
The data used to support the findings of this study are included within the article.
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Small part of this initial work was presented (but not published) as a poster at MDBSBHE 2017: The 11th IEEEEMBS International Summer School and Symposium on Medical Devices and Biosensors. This work was supported by the Science and Technology Development Fund of Macau (FDCT) under 093/2015/A3, 088/2016/A2, the University of Macau under Grants MYRG201800146AMSV, MYRG201600157AMSV, MYRG201500178AMSV, the Nature Science Research Project of Lingnan Normal University under Grant ZL1901, and Science and Technology Research Project of Zhanjiang City under Grant 2018B01003.