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ieee transactions on ultrasonics, ferroelectrics, and frequency control, vol. 52, no. 12, december 2005 2185

Finite-Element Analysis of CapacitiveMicromachined Ultrasonic Transducers

Goksen G. Yaralioglu, Member, IEEE, A. Sanli Ergun, Member, IEEE,and Butrus T. Khuri-Yakub, Fellow, IEEE

Abstract—In this paper, we present the results of finite-element analysis performed to investigate capacitive mi-cromachined ultrasonic transducers (CMUTs). Both three-dimensional (3-D) and 2-D models were developed using acommercially available finite-element modeling (FEM) soft-ware. Depending on the dimensionality of the model, themembranes were constructed using plane or shell elements.The electrostatic gap was modeled using many parallel platetransducers. An axisymmetric model for a single membranewas built; the electrical input impedance of the device thenwas calculated in vacuum to investigate series and paral-lel resonant frequencies, where the input impedance hasa minimum and a maximum, respectively. A method fordecomposing the membrane capacitance into parasitic andactive parts was demonstrated, and it was shown that theparallel resonant frequency shifted down with increased bi-ased voltage. Calculations then were performed for immer-sion transducers. Acoustic wave propagation was simulatedin the immersion medium, using appropriate elements ina 3-D model. Absorbing boundaries were implemented toavoid the reflections at the end of the medium mesh. Onerow of an array element, modeled with appropriate bound-ary conditions, was used to calculate the output pressure.The results were compared with a simpler model: a singlemembrane in immersion, with symmetry boundary condi-tions on the sidewalls that cause the calculations to reflectthe properties of an infinitely large array. A 2-D model thenwas developed to demonstrate the effect of membrane di-mensions on the output pressure and bandwidth. Our cal-culations revealed that the small signal transmit pressurewas inversely proportional to the square root of gap height.We also compared FEM results with analytical and exper-imental results.

I. Introduction

Invented at Stanford University [1]–[3], capacitive mi-cromachined ultrasound transducers (CMUTs) have be-

come very popular over the last decade for use in med-ical imaging [4], [5]. They easily can compete with theirpiezoelectric counterparts in terms of bandwidth, dynamicrange, and sensitivity [6]. Moreover, ease of fabrication forcomplex device geometries (such as two-dimensional (2-D)arrays) has made CMUTs an even more attractive alter-native to piezoelectric transducers. Recently, images ob-tained from a 2-D CMUT array have been demonstrated[7]. The increased interest in the CMUT technology andapplications has accelerated modeling efforts for these de-vices. Initially, an equivalent circuit method was used to

Manuscript received September 13, 2004; accepted May 12, 2005.The authors are with the Ginzton Laboratory, Stanford University,

Stanford, CA 94305 (e-mail: [email protected]).

predict output pressure and bandwidth [3], [8]. In its sim-plest form, the equivalent circuit has two ports. The elec-trical port contains the clamped capacitance of the trans-ducer; the mechanical port is composed of the mechani-cal impedance of the membrane and the negative springsoftening capacitance (caused by electromechanical inter-action). In immersion, the mechanical side is terminatedby the radiation impedance of the medium. However, thisapproach is not very accurate as it does not include thehydrodynamic mass loading of the immersion medium [9].Lohfink et al. [10] recently demonstrated a method for ac-curate modeling of fluid loading. By defining an equivalentparallel plate capacitor for a CMUT membrane at eachbias voltage, turns ratio was calculated precisely. However,the equivalent circuit approach still lacks the modeling ofboth cross talk between the membranes and the effect ofhigher order resonances. By using finite-element model-ing (FEM) intensively, more accurate calculations can bemade to evaluate the performance of the CMUT devices.Commercial and custom-developed FEM codes have beenused for both static and dynamic analysis.

Static FEM models have been used to accurately cal-culate collapse voltage and device capacitance [11]–[13].Bozkurt et al. [11] developed a FEM model in which theelectrode coverage was optimized for optimum bandwidth.Recently, Bayram et al. [14] calculated average mem-brane displacement for different operation regimes. Dy-namic FEM models also have been developed and used toestimate the output pressure and membrane displacementby performing time-domain analysis [15]. For the evalua-tion of cross talk, more intensive models were introducedto demonstrate propagating waves along the water-waferinterface [16]. Additionally, harmonic analysis was used toinvestigate the cross talk between the membranes in a 2-D model in which elements are assumed to be composedof infinitely long membranes [17]. In a similar geometry, asingle membrane was used to model propagating modes ina periodic membrane structure [18] by assigning appropri-ate phase difference at the boundaries.

Previously, we demonstrated output pressure calcula-tion using 3-D FEM models [9]. In the simplest FEM con-struction, we modeled a single cell of a circular membranearray that was distributed uniformly in lateral directionsover a 2-D rectangular grid. Appropriate boundary condi-tions were applied on the sidewalls of the model, so thatthe single membrane was replicated in the lateral direc-tions. This model assumes that all the cells in the arrayoperate with the same phase. More extensive models in-

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2186 ieee transactions on ultrasonics, ferroelectrics, and frequency control, vol. 52, no. 12, december 2005

Fig. 1. Axisymmetric membrane and the electrostatic elements. Iand J are the bottom and the top nodes of one of the electrostaticelements, respectively. The top electrode is assumed to be on thebottom surface of the membrane. The same membrane geometryhas been used in all of the calculations throughout the paper unlessotherwise noted.

clude a 3-D model of an array element in which the elementhas a finite width in one dimension. In this paper, we willfirst model the membrane with an electrostatic gap, whichwill be included in the model through transducer elements.We will then show 3-D models for the CMUT transducerin immersion, and compare them with a 2-D axisymmetricmodel. Using the 2-D model, we will demonstrate the ef-fect of membrane dimensions and present design steps fora given bandwidth and output pressure. The FEM resultsthen will be validated by comparing them with analyticaland experimental results.

II. FEM Calculations

The 2-D and 3-D models considered in this study wereconstructed using a commercially available FEM package(ANSYS8.0, ANSYS Inc., Canonsburg, PA). In 2-D mod-els, the membrane was built using axisymmetric plane ele-ments; in 3-D models, shell elements were used. The elec-trical ports were included in the model using electrostaticelements (TRANS126). The electrostatic membrane firstwas modeled; the model then was extended for immersiondevices in which fluid-solid interaction was performed ac-curately by ANSYS. For immersion devices, the absorbingboundaries at the end of the liquid mesh also were imple-mented using available elements (FLUID30, FLUID130).The scope of this paper is limited to the harmonic analysisin which the alternating current (AC) signals are assumedto be small relative to the direct current (DC) bias, andin which a linearized model for the transducer was used.

A. Modeling of the Membrane with Electrical Ports

Fig. 1 shows the membrane and the electrostatic ele-ments used in the model. The thickness and the radius ofthe circular membrane were 1.3 µm and 15 µm, respec-tively. The gap was 0.2 µm, and the top electrode was onthe bottom surface of the membrane. The membrane wascomposed of plane elements (PLANE42) that are axisym-metric to the symmetry line of the circular membrane. The

rim of the membrane, where it is connected to the posts,was assumed to be clamped. By using electromechanicaltransducer elements (TRANS126), which are basically par-allel plate capacitors, the electrostatic forces applied be-tween the top and the bottom electrodes were included inthe model. The elements apply opposite electrostatic at-traction forces to the nodes to which they are attached. Inthe model, the top nodes of these elements were attachedto the section of the membrane at which the electrode islocated; the bottom nodes were simply clamped. The areaof the ith component is given by:

Ai = 2πri∆r, (1)

where ri is the radius of the center of the element, and∆r is the mesh size. This approach divides the electro-static gap into many segments; within each segment, theelectrostatic pressure is modeled by a small parallel platecapacitor. Therefore, the electrostatic field is assumed tobe constant within each capacitor, and the fringing fieldsat the rim of the electrode were disregarded. If the num-ber of segments is sufficiently high (approximately 25 overthe electrode region), the membrane’s static deflection canbe calculated very accurately. Using both ANSYS as de-scribed in [12], [13] and the above approach, we comparedthe deflection profiles computed. When the top and bot-tom electrodes were equal in size, both calculations for thedisplacement profile were in agreement within 1%.

The described model can be used for static, harmonic,and transient analysis. In harmonic analysis, the small sig-nal equivalent of the transducer elements is used at thebias voltage. In transient analysis, nonlinear large signalcalculations can be performed. The scope of this paper islimited to the static and harmonic analyses.

In the model described above, we calculated collapsevoltage using a binary search algorithm. The bias voltagewas iterated until the maximum voltage that resulted ina convergent solution for the membrane displacement wasobtained. The collapse voltage of the silicon (Young’s mod-ulus, 150 GPa; Poisson’s ratio, 0.17; density, 2332 kg/m3)membrane (whose geometry is shown in Fig. 1) was foundto be 135.4 V for half metalization. The input impedanceof the device then was calculated at various bias voltagesusing harmonic analysis. A small amount of mechanicalloss was introduced in the membrane material to accountfor losses, such as coupling into the substrate and mate-rial damping in the silicon. The loss factor was adjusted sothat the mechanical Q of the membrane is on the order of100, which is close to the experimental measurements inthe frequency range of interest. Fig. 2 shows the electricalinput impedance for different bias voltages. The mechan-ical loss of the membrane material resulted in a nonzeroreal part, as depicted in Fig. 2(a). The impedance curves ofFig. 2(a) and (b) have two characteristic frequencies. Themagnitude of the impedance has a minimum (at fs) and amaximum (at fa). These two frequencies are the series res-onant and the parallel resonant frequencies, respectively.The maximum membrane displacement is achieved at theseries resonant frequency, as seen in Fig. 3(c). Due to the

yaralioglu et al.: fem analysis to investigate cmut 2187

Fig. 2. (a) Real part of the electrical input impedance. (b) Imaginarypart of the electrical input impedance. (c) Average displacement overthe membrane. Note that the frequency range of (a) is different fromthat used in (b) and (c).

Fig. 3. Equivalent circuit model. C is the clamped capacitance of theCMUT. C0 is the active capacitance. n is the turns ratio. mm andkm are the mass and the spring constant of the equivalent parallelplate capacitor to the CMUT membrane. R indicates the combinedlosses in the membrane.

spring softening, the series resonant frequency [Fig. 2(b),approximately where the imaginary part is zero] and thepeak displacement [Fig. 2(c)] shift down with increased bi-ased voltage [8], as expected. Note that the displacementof Fig. 3(c) is calculated with zero source resistance.

The native mechanical resonant frequency of the mem-brane determines the parallel resonant frequency of theelectrical input impedance. The resonant frequency of themembrane of Fig. 1 is 21.71 MHz. Another interesting fea-ture of the impedance curves in Fig. 2(a) and (b) is that, asthe bias voltage increases, parallel resonant frequency alsoshifts toward smaller frequencies, primarily due to the in-herent parasitic capacitance of the membrane. In the nextsubsection, we will investigate this observation further anddecompose the membrane capacitance into active and par-asitic capacitances, using the coupling coefficient, k2

t , cal-culation.

1. Equivalent Circuit Model and k2t Calculation: Fig. 3

shows a typical equivalent circuit, whose input impedanceis similar to the impedance results depicted in Fig. 2(a)and (b). Using simple equations describing the turns ra-tio and the mechanical impedance of a spring-mass model[9], this circuit accurately predicts the electrical inputimpedance for a parallel plate capacitor. However, becausethe electrostatic force in a CMUT membrane is appliedonly over the electrode area, and the membrane has a cer-tain deflection profile rather than a constant displacement,the elements of the equivalent circuit need correction [9].Recently, Lohfink et al. [10] proposed an alternative ap-proach to calculate the element values of the circuit: ratherthan directly using the mechanical impedance of the mem-brane; their method finds an equivalent spring-mass modelfor the membrane impedance around the first resonantfrequency of the membrane. The same approach will beused in this study, with slight variations. We will find an


Fig. 4. Average membrane displacement for half and full metalizationand equivalent parallel plate capacitor. Collapse voltages are 135.4 Vand 116.8 V for half and full metalization, respectively.

equivalent parallel plate capacitor whose displacement isequal to the average displacement of the membrane, andwe will assume the gap height of the equivalent parallelplate transducer is equal to three times the maximum av-erage displacement of the membrane just prior to collapse.The average membrane displacement is given by:

x(V ) =

R∫0

2πru(r)dr

πR2 , (2)

where R is the radius of the membrane, u(r) is the displace-ment profile under the applied bias voltage. However, thedisplacement of a parallel plate transducer depends onlyon the ratio of the applied voltage to the collapse voltage[12]. The real root, which is less than one third of the gapheight, of the following polynomial provides the displace-ment of a parallel plate capacitor:

V

Vcoll=

√27x

4d3eff

(deff − x) , (3)

where deff is the gap height. Collapse occurs when thedisplacement is equal to one-third of the gap. Fig. 4 showsthe displacement as a function of the bias voltage for themembrane depicted in Fig. 1 and for the equivalent par-allel plate capacitor. The parallel plate capacitor predictsthe average membrane displacement with a maximum er-ror of 3% (which occurs when the bias voltage is close tozero). Because the approximation is calculated for criti-cal displacement, the error decreases as the bias voltageincreases.

Electrode size also affects the maximum error. For fullmetalization in small biases, maximum error is almost10%. The parallel plate can better approximate the mo-tion of the membrane for small electrode coverage. How-ever, the bending effects of the membrane become more

Fig. 5. Ratio of parasitic capacitance to active capacitance (C =Cp(x) + C0(x)).

dominant as the electrode coverage increases. Because thetransducers usually are biased close to the collapse voltage,the error in the approximation is acceptable.

Previously, we calculated the coupling coefficient, k2T ,

of a transducer [12], and showed that, by adding an ap-propriate amount of parasitic capacitance to the parallelplate capacitance, a parallel plate transducer can model aCMUT membrane. The coupling coefficient of a parallelplate transducer is given by:

k2T =

2x

deff − x + (deff − 3x)Cp(x)C0(x)

, (4)

where C0(x) is the active (equivalent parallel plate) capac-itance, and Cp(x) is the parasitic capacitance [12]. For aparallel plate capacitor, (4) can be expressed in terms ofnormalized displacement, x/deff , which can be calculatedfrom (3) for a given normalized bias voltage. However, thetotal device capacitance of the CMUT membrane shouldbe equal to the sum of the active capacitance, C0(x), andthe parasitic capacitance, Cp(x). The total capacitanceand coupling coefficient of the membrane depicted in Fig. 1were calculated using ANSYS, as described in [12]. Sub-stituting these results in (4) revealed active and parasiticcapacitances at a given bias voltage. (It is important toknow the collapse voltage accurately for the above cal-culations.) The ratio of the parasitic capacitance to theactive capacitance is shown in Fig. 5 for various electrodecoverages. The ratio increases as the metalization on themembrane surface increases. This result matches very wellwith our previous finding that, at a given bias voltage, thecoupling coefficient of a CMUT membrane decreases as theradius of the electrode increases.

To this point, we have defined a method to break the de-vice capacitance into equivalent parallel plate, C0(x), andparasitic, Cp(x), capacitances. Other element values of the


Fig. 6. Imaginary part of input impedance calculated using ANSYSand the equivalent circuit approach. Bias voltage is 120 V.

equivalent circuit are simple to calculate. The spring con-stant of the equivalent parallel plate capacitor is given by:

km =27V 2

collC0(x) (deff + x)8d3

eff

. (5)

The equivalent mass can be calculated from the resonantfrequency using the relation:

mm =km

(2πf0)2 , (6)

where f0 is the resonant frequency of the membrane andthe turns ratio is:

n = C0(x)Vbias

deff − x. (7)

Note that element values need to be recalculated at eachbias voltage. The calculated input impedance is depictedin Fig. 6. We obtained a very good match between ANSYSand the equivalent circuit calculations.

As suggested by the above analysis, the parasitic capac-itance that is always present in the membrane transduceris responsible for the downshift of the parallel resonantfrequency. This relationship can be understood easily byusing the equivalent circuit shown in Fig. 3(b), where allthe acoustical elements were reflected to the electrical side.This circuit [Fig. 3(b)] has two parallel branches. At theresonant frequency, the sum of the mechanical membraneimpedance and the spring softening capacitance is zero, re-sulting in minimum electrical impedance. At the parallelresonant frequency, the mechanical membrane impedanceis zero, and the combination of two capacitances (the elec-trical clamped capacitance and the negative spring soft-ening capacitance) and with resistance creates maximumelectrical impedance. For the parallel plate capacitor, thedevice capacitance is equal to the negative spring softening

capacitance [Fig. 3(b)]. In this case, the parallel resonancein which the imaginary and the real parts are maximizedoccurs at the resonance of the membrane. However, be-cause of parasitics, if the total clamped electrical capaci-tance is greater than the magnitude of the negative capaci-tance, the parallel resonant frequency shifts down. Addingthe external parasitic capacitance would result in a furtherdownshift of the parallel resonant frequency. The inherentparasitics in the membrane capacitance occur because dis-placement is not constant over the membrane. Even with-out additional external parasitics, with the increased biasvoltage, the membrane’s inherent parasitics will result ina downshift of the parallel resonant frequency.

2. Optimization of Electrode Radius: Optimization ofthe metalization coverage when there is excessive exter-nal parasitic capacitance (which is the case for most ofthe CMUT devices), is another use of coupling coefficientcalculation. The excessive external parasitics are causedprimarily by the metal lines connecting the top electrodesof the membranes. There is usually a high dielectric mate-rial of either silicon oxide or silicon nitride between theselines and the bottom substrate. When there are no externalparasitics, half metalization seems to be the optimum forelectrode radius [11], [12], because it is a good compromisebetween bandwidth and collapse voltage [11]. For the elec-trode sizes larger than half coverage parasitics increase ata faster rate than active capacitance (or equivalent parallelplate capacitance) does. With 50% metalization (Fig. 5),the total capacitance is slightly larger than the active ca-pacitance, and the parasitics are very small. With 100%metalization, while the total capacitance increases fourtimes, the active capacitance increases only twice. There-fore, when there are no external parasitics, 50% coverage isoptimum in terms of coupling efficiency, as shown in Fig. 7.With external parasitics (Cp ext), increasing the electroderadius toward full metalization also increases the couplingcoefficient—a result of the increased ratio of active capaci-tance to total parasitics (Cp(x)+Cp ext). Therefore, whenthere is an excessive amount of external parasitics (as highas 300%), full metalization results in a better coupling co-efficient.

B. Membrane Radiating into Immersion Mediumin an Array

In a CMUT transducer, membranes that are similar tothe one analyzed in the previous section are grouped to-gether to form array elements. The geometry of the ele-ment determines the different imaging modalities. A typ-ical 1-D transducer array is composed of long rectangu-lar elements. The length of the array element usually ismuch longer than its width. The width of the element is setto half-wavelength at the maximum operation frequency.Therefore, a typical element is composed of hundreds ofCMUT cells in the elevation, but it is usually only four tosix cells in width. The high aspect ratio of the element al-lowed us to model only one row of the CMUT cells, greatly


Fig. 7. Coupling coefficient as a function of parasitic capacitance. Thecurves were calculated at 80% of the collapse voltages. The devicecapacitance for half metalization is 0.91×10−14 F. The device capac-itances of 75% and full metalization are 0.2×10−13 and 0.34×10−13,respectively. The collapse voltages are 135.4 V, 119.0 V, and 116.8 Vfor half to full metalization.

reducing the model size and computation time [9]. Symme-try boundary conditions were applied on the bottom andthe top edges of the CMUT row. This replicated the modelinfinitely many times in the elevation direction and gener-ated an infinitely long element at which all the rows weredriven by the same phase. Because the row was also sym-metric with respect to the centerline along the elevation,the model size was further reduced. Therefore, modeling ofone half of the row is sufficient. The mesh of the half row ofthe element is shown in Fig. 8. The pitch, w, between themembranes, was 32 µm. The element was six membranes inwidth and only half of them were modeled. The membraneswere meshed by shell elements (SHELL63). The nonmem-brane regions were clamped on the membrane plane. Afiner mesh size was used over the membrane region to accu-rately calculate the resonant frequencies and mode shapesof the membranes. The mesh size was set to 3 µm in thefluid region so that at the highest frequency (50 MHz)there were 10 elements per wavelength in water. The elec-trostatic elements were attached to the membrane nodeswithin the electrode region. The fluid region was composedof 3-D acoustic elements (FLUID30). The interaction be-tween the solid and fluid parts was activated by using theappropriate ANSYS flags. To eliminate the reflections fromthe end of the mesh, the envelope of the fluid region wascovered by absorbing elements (FLUID130). The first stepof the calculations was to bias the transducer elements tothe specified voltage. Next, prestressed harmonic analysis,which uses the small signal equivalent of the transducerelements at the bias point, was performed. During the fre-quency sweep, the distance between the membranes andthe absorbing elements were adjusted so that the radius ofthe model was equal to the wavelength. Therefore, at eachfrequency point the fluid region was remeshed. This was

Fig. 8. 3-D mesh of an array element. The image is obtained when thefrequency is 10 MHz. Therefore, the absorbing boundary on the topimage is 150 µm (wavelength) away from the origin. Bottom imageshows the zoomed area around the membranes. Clamped boundaryconditions were applied on the shell elements as shown. The elec-tromechanical elements are not shown for clarity.

required for the absorbing element to perform sufficientlywell to eliminate the reflecting waves from the boundary.However, for high frequencies in which the radius is lessthan the length of the row, the radius was simply set tothree times the pitch (3w).

We used the same membrane geometry discussed pre-viously. The membrane was biased at 80% of the collapsevoltage; then harmonic analysis was performed by apply-ing 1 V peak harmonic voltage at the top electrode. Thesource impedance was assumed to be zero. Fig. 9 depictsthe average displacement over the membranes. Because ofacoustic coupling through the immersion medium, the dis-placement of the each membrane is different. The coupling,which degrades the output pressure, is more pronounced,especially around the resonant frequency.

Electrical input impedance is another important param-eter that can be calculated using the model. The inputimpedance depicted in Fig. 10 was obtained by using thetotal current flowing through to the capacitors and was


Fig. 9. Average displacement over the membranes. Membrane 1 isthe one closest to the center and membrane 3 is the most outer one.

Fig. 10. Electrical input impedance of the three membranes operatingin an array element.

calculated only for half of the row. The input impedanceof the row is half of the impedance shown in Fig. 10. Thetotal impedance of the array element is the parallel com-bination of all the rows, and it can be found by dividingthe input impedance of the single row by the total numberof the rows.

C. Single Membrane with Symmetry Boundaries

Assuming symmetry boundaries in the width dimen-sion will further simplify the modeling of CMUT arrays.The corresponding transducer will be infinitely large inboth lateral dimensions. In this case, the model will becomposed of a membrane and a rectangular fluid columnabove it, as shown in Fig. 11. This model assumes all theneighboring cells are driven by the same phase, and have

Fig. 11. Single cell 3-D model. Electrostatic elements attached to theelectrode are not shown for clarity. The electrode region is at thecenter of the membrane as shown above.

Fig. 12. Average displacement over the membrane. The bias voltageis 80% of the collapse voltage (135.4 V).

the same displacement profile. The width (or the pitch),w, of the fluid column was 32 µm. As described in the pre-vious section, the membrane was first biased to a certainfraction of a collapse voltage; then, harmonic analysis wasperformed.

The calculated average displacement (Fig. 12) is verysimilar to the displacement obtained using the model de-scribed in the previous section, especially for frequenciesabove 15 MHz. The periodicity of the cell reveals itself asa sudden change (observed at about 47.5 MHz) at slightlyabove the frequency in which the wavelength in water isequal to the pitch (f = 1500 m.sec/32µm = 46.87 MHz).Below this frequency, the acoustic wave propagation in thefluid column is perpendicular to the transducer plane; as itmoves away from the membrane, the field pattern quicklyconverges to that of a plane wave. Above this frequency,however, the wavelength in water is less than the period-


Fig. 13. Electrical input impedance of a single cell operating in aninfinitely large transducer. Bias voltage is 80% of the collapse voltage.

icity, which results in diffraction of waves in water. There-fore, where the wavefronts are parallel to the transducerplane, there are waves being sent to the higher orders aswell as to the 0th order. For example, the angle of the firstdiffracted order is given by [19]:

θ = arctan(

λ

w

), (8)

where w is the periodicity or the pitch of the array.We also have calculated the input impedance of the

transducer (Fig. 13). This calculation was performed fora single transducer; therefore, the impedance is expectedto be three times larger than the impedance shown inFig. 10. There is another important difference between thetwo impedances: for the membranes operating in an array,as the frequency decreases, the real part goes to zero; inthe case of a single cell, the real part converges to a certainvalue.

This discrepancy can be explained easily using theequivalent circuit. At DC, the real part of the electricalinput impedance of the circuit, shown in Fig. 3 is given by:

Re {Zin} =R/n2(

1 +Ckm

n2 − C

C0

) . (9)

In immersion, the radiation impedance terminates themechanical port; therefore R is equal to the real part of theradiation impedance. For the infinitely large plane trans-ducer, the real part of the radiation impedance is constant;for membranes operating in an array element, the real partof the radiation impedance decreases to zero as the fre-quency decreases. Therefore, the real part of the inputimpedance cannot be accurately predicted for low frequen-cies using the infinitely large transducer model. However,the model can be used for the output pressure calcula-tions. Fig. 14 compares the pressure over the membrane

Fig. 14. Average pressure over the membrane plane. For both models,the bias voltage is 80% of the collapse voltage.

Fig. 15. Average pressure over the membrane plane for models withvarious widths.

plane obtained using the two models. As shown above, be-cause of acoustic interaction between the membranes, thedisplacement of the membranes operating in an array witha finite width can be different, which results in strong dipsin the output pressure. Except for this discrepancy, thetwo calculations matched quite well.

Using the infinitely large transducer model, we investi-gated the effect of membrane separation by changing thewidth of the square region surrounding the membrane, asshown in Fig. 15. As the width increases, the bandwidthof the transducer decreases due to the change in the radia-tion impedance. The real part of the radiation impedancedecreases because the real part is scaled by the ratio ofactive area to total area. This increases the quality factorof the membranes. The imaginary part of the radiation


Fig. 16. Output pressure over the membrane plane. The ratio of theactive membrane area to the total area is set equal to each other.Bias voltages are the same and set to the 80% of the collapse voltage.

impedance increases, due to the increased fluid loading onthe membrane. With a larger, nonactive region around themembrane, the fluid also can be moved to the side with-out any radiation, which increases the hydrodynamic massof the fluid. Thus, for high bandwidth, it is important todensely pack the membranes, so as to eliminate dead re-gions between them. This becomes especially important forhigh-frequency devices in which the membranes are only afew microns in diameter. The decrease of the resonant fre-quency is another effect of the increased periodicity. Thisis again due to increased volume of the fluid interactingwith the membrane, as explained above.

D. Axisymmetric Model for Immersion

A further simplification of the infinitely large transduceris an axisymmetric model in which the membrane radiatesinto a cylindrical wave guide [10]. The complexity of theresulting 2-D model is very low, compared to the 3-D mod-els discussed earlier. The constructed FEM model is verysimilar to the model used for an infinitely large transducer,except that plane elements were used in the FEM mesh.The outer radius, rout, of the fluid column was chosen sothat both 3-D and 2-D models have the same amount ofnonactive region:

rout =w√π

, (10)

where w is the width of the total area or it also can beinterpreted as the pitch of the array. Fig. 16 compares theoutput pressures obtained using the 3-D and 2-D models.The 2-D model predicts slightly higher output pressureand more bandwidth. This is due to the fact that the 2-Dmodel assumes a better packing density. But, 2-D FEMis a simple and computationally very efficient model; it

Fig. 17. Pressure on the center line.

can be used for any CMUT transducers whose membranesare distributed arbitrarily, as long as the active and totalareas are defined appropriately. The appropriate selectionof active and total area was demonstrated in [10]. Thismodeling approach can provide important information onvarious features of a CMUT transducer such as outputpressure and bandwidth.

For example, Fig. 17 plots the pressure distributionalong a line passing through the center of the membranefor 3-D and 2-D models. The absorbing boundary, whichusually is positioned one wavelength away from the mem-brane, was placed further away for demonstration pur-poses. As shown in Fig. 17, the waves quickly convergeto plane waves. To demonstrate the propagated waves,the real part of the pressure distribution also is added toFig. 17. The imaginary part is simply 90 degrees out ofphase with the real part.

An important design criterion for the CMUT transduceris the dimensions of the membrane that populates the el-ements of the transducer. Membrane thickness and radiusdetermine output voltage, bandwidth, and collapse volt-age. Fig. 18 shows the mechanical membrane impedancefor membranes with various radii (10, 15, and 20 µm). Todetermine the mechanical impedance, we apply a constantpressure over the membrane area, and calculate the av-erage velocity using ANSYS. The mechanical membraneimpedance is defined as the ratio of the integral pres-sure over the area (total force) to the average velocity.There is an analytical solution for the circular membraneimpedance [20]. The thickness of the membranes was cho-sen so that they all have the same resonant frequency. Thecollapse voltages were 91.5 V and 177.6 V for 10 µm and20 µm membranes, respectively. The gap was 0.2 µm; theelectrode radius was set to half of the membrane radius.

The output pressure obtained using these membranes isplotted in Fig. 19 when the bias voltage was 80% of the col-lapse voltage. The ratio of the membrane area to the total


Fig. 18. Mechanical membrane impedance in vacuum.

Fig. 19. Average pressure over the membrane plane (total area). Thecollapse voltages of the three membranes are 91.5, 135.4, and 177.6 V.

area was set to the same value (0.7) for each case. Becausethe mechanical membrane impedance is small compared tothe radiation impedance over a broader frequency band,the softer membrane generated the maximum bandwidth.But, the output pressure is low compared to the othermembrane sizes because of the low bias voltage that isdetermined by the collapse voltage. The stiffer membranegenerated higher output pressure with a narrower band-width. Another important observation was that the fluidloading reduced the resonant frequency at the same ratefor all of the membrane sizes.

The output pressure also is determined by the gap. Inthe simplest form, neglecting nonactive regions, the out-put pressure of a CMUT can be calculated directly fromthe equivalent circuit model at the resonant frequency atwhich the impedances of the reactive elements are zero. At

Fig. 20. Output pressure averaged over the membrane plane for vari-ous gap heights. The membrane radius and thickness are 15 µm and1.4 µm, respectively. For each gap height the bias voltage is set to80% of the collapse voltage. The collapse voltages are 17, 47.9, 88.0,135.4, and 248.8 V for the gap height, increasing from 0.05 µm to0.3 µm.

that frequency, the output pressure per volt of a CMUTtransducer is given by:

P0 =n

A, (11)

where A is the membrane area. This calculation is a directresult of the equivalent circuit model shown in Fig. 3. Fora parallel plate capacitor, (11) reduces to:

P0 =ε0Vbias

(deff − x)2. (12)

The maximum bias that can be applied to the parallelplate capacitor is the collapse voltage, and it is given by:

Vcoll =

√8kmd3

eff

27Aε0. (13)

If (13) is substituted in (12), one gets the pressure pervolt:

P0 =

√3kmε0

2Adeff. (14)

Note that the above calculations are valid only for smallsignals. But same dependence of the output pressure to thegap is obtained if the CMUT is assumed to be biased tothe certain fraction of the collapse voltage. For the 20 µmmembrane, we calculated the output pressure as a func-tion of gap height. The results are depicted in Fig. 20. Assuggested by (14), the output pressure per volt is inverselyproportional with the square root of gap height. Choosinga small gap height maximizes the output pressure per volt.However, the total pressure that can be obtained from a


Fig. 21. Effect of substrate.

CMUT is limited by the gap height, assuming there is nolimitation on the amplitude of the excitation voltage.

The effect of coupling into the substrate also can bedemonstrated using the 2-D model. Instead of clampingthe membrane edge, we added a 500-µm thick silicon waferinto the model, and connected the membranes with thesubstrate through a silicon post. Due to the post, the col-lapse voltage slightly decreased to 126 V. The longitudinalwave velocity in silicon is 8315 m/s. The reflections in thewafer were revealed as peaks and dips at every 8.3 MHzas shown in Fig. 21. Note that the configuration usedenhances the effect of substrate resonances because themodel assumes the source is composed of infinitely manyCMUT membranes covering the whole plane. Therefore,the radiation impedance into the silicon is equal to theplane wave impedance. However, for 1-D arrays, in widthdirection, the source is usually smaller than the wavelengthin silicon. Therefore, the effect of the substrate resonancesis not as strong as shown in the above model.

III. Validation of the FEM Models

Accurate FEM results require careful selection of meshdensity. The models developed in this paper were calcu-lated using various mesh densities. The mesh sizes were it-erated until the calculated results converged. In addition,the FEM results were compared with problems in whichanalytical solutions exist. We also confirmed our resultswith experiments.

A. Radiation Impedance Calculations

The best way to validate the FEM code is to run acase in which an analytical solution exists. For validationpurposes, we used a radiation impedance calculation as atest vehicle for our FEM codes.

Fig. 22. Radiation impedance of a rectangular slit. The width of theslit is 170.4 µm.

To test the FEM model depicted in Fig. 8, radiationimpedance of a rectangular slit was calculated using themodel, and the results were compared with theoretical cal-culations. Fig. 22 shows the real and imaginary parts of theradiation impedance, calculated both analytically and us-ing the FEM. The width of the rectangular slit is 170.4 µm,and the length is assumed to be infinitely long. The immer-sion medium is water (v = 1500 m/s, ρ = 1000 kg/m3).To find the radiation impedance, a uniform velocity fieldwas applied over the slit, and the resulting average pres-sure on the radiating surface was calculated. As expected,the real part of the radiation impedance approaches theacoustic impedance of water, and the imaginary part di-minishes as the frequency increases. There is a good matchbetween theoretical and FEM curves. For small frequen-cies (<10 MHz), the source is smaller than the wave-length, which generates wave fronts parallel to the absorb-ing boundary; the absorbing boundary works very well toabsorb the incoming waves. However, above 10 MHz, theradiating beam is more directive, and the waves impingeon the absorbing boundary with a certain angle. ANSYSuses a second-order absorbing boundary; because the re-flection from the boundary increases as the incidence angleincreases, the absorbing boundary cannot absorb all the in-coming waves, and ripples are created on the FEM curves(Fig. 22).

Radiation impedance also was calculated for the ge-ometry presented in Section II-C, as shown in Fig. 23.First, a uniform velocity distribution was applied over thesquare region that forms the bottom part of the fluid col-umn (Fig. 11). Because this design represents an infinitelylarge source radiating into a fluid half space, the radiationimpedance should equal the acoustic impedance of the im-mersion fluid (Zc = 1.5 MRayl). Our calculation matchedvery well with this theoretical value. (A negligibly smallimaginary part is attributable to the numerical errors ofthe FEM calculation.) In the case in which the velocity


Fig. 23. Radiation impedance of a circular piston in square regionradiating into a rectangular fluid column. When the square surfaceis assumed to be moving with a constant velocity (full excitation),the radiation impedance is equal to the acoustic impedance of thefluid.

field is applied over a circular area within the square re-gion (Fig. 22), the real part of the impedance should bescaled by the ratio of the radiating region to the total re-gion. In our calculations, the active region was a circle in30-µm diameter placed in a 32 µm by 32 µm square area.The real part (as calculated in this configuration) matchesthe expected value of 1.04 MRayl. The imaginary part ofthe radiation impedance, however, is caused by fluid mov-ing back and forth, between the interior and exterior ofthe circular region. The linear slope of the imaginary partindicates a constant fluid loading on the circular region forlow frequencies. The sudden jump just above 46.87 MHzis caused by periodicity, as explained before.

B. Experiment

To test the FEM results, we also measured the outputpressure of a CMUT immersed in oil, using an optical in-terferometer (Polytec, OFV511/OFV2700, Physik Instru-mente, Auburn, MA). The interferometer was coupled to amicroscope, and the laser beam was focused on the air-oilinterface, using a 20X objective. The CMUT was immersed2.3 mm below the interface. The transducer was composedof hexagonal membranes. The diameter of a circle that canbe inscribed in the membrane was 28.8 µm. This particu-lar device was fabricated using silicon-on-insulator (SOI)bonding technique, where 1 µm thick SOI layer was bondedover cavities defined by silicon oxide, creating a 1-µm thicksilicon membrane. The post width was 5 µm. The cavityheight was 0.25 µm over a 0.1 µm insulation layer. Thecollapse voltage of the transducer was 145 V; it was biasedat 100 V. A negative 30 ns square pulse was applied to thetransducer via a bias-T.

The interferometer output was measured using an os-cilloscope. Using the sensitivity of the interferometer, the

Fig. 24. Half of the displacement at the air-oil interface.

Fig. 25. Calculated and experimental frequency response of a CMUT.

voltage waveform was converted into displacement. Be-cause the actual displacement without the air-oil inter-face should be half of the displacement measured at theinterface, the measurement result was divided by two.Fig. 24 shows half of the measured displacement. Thepressure spectrum was determined by calculating the fastFourier transform (FFT) of the displacement waveform,and multiplying it by the radian frequency and the acous-tic impedance of oil (v = 1480 m/s, ρ = 920 kg/m3). Thedisplacement waveform was properly windowed to reducethe artifacts in the FFT. To find the actual pressure onthe transducer, the pressure spectrum 2.3 mm away fromthe transducer surface was corrected for attenuation anddiffraction in oil. The resulting measured pressure spec-trum is shown in Fig. 25. The peak value of the pressure(determined by applying a tone burst at 10 MHz to thetransducer) was 13.2 kPa/V at 10 MHz.


A similar model to the one described in Section II-B wasdeveloped to calculate the pressure output of the trans-ducer. The circular membranes were replaced by hexago-nal cells. Fig. 25 also shows the calculated spectrum us-ing ANSYS. The peak calculated pressure is 15.0 kPa/V.There is a good match between the two spectrums. OurFEM model estimated the center frequency and the lowercut-off frequency accurately, but it slightly overestimatedthe higher cut-off frequency.

IV. Conclusions

In this paper, we have demonstrated comprehensiveFEM models for CMUTs. The electrostatic ports were in-corporated into the models using electromechanical trans-ducer elements. The scope of the paper was limited to har-monic analysis and assumed a linearized model for the elec-trostatic transducers. The calculations in vacuum showedthat the CMUT transducer has inherited parasitics be-cause the capacitance of a CMUT transducer cannot becompletely modeled by a parallel plate transducer. Theseparasitics result in a decrease in parallel resonant fre-quency as bias voltage increases. We also have proposeda method for the decomposition of membrane capacitanceinto active and parasitic parts by calculating the couplingcoefficient. Previously, it had been shown that half metal-ization improved bandwidth by reducing the inherent par-asitic capacitance of the membrane. However, our calcula-tions showed that, if there were external parasitic capaci-tance in addition to the inherent parasitics, full metaliza-tion would result in a better coupling coefficient.

The 3-D modeling of an array element showed thatacoustic coupling between membranes creates dips in theoutput pressure spectrum. Although a simple 3-D modeldoes not reveal these dips, the model still can be usedfor optimization of the output pressure spectrum. In thisstudy, for example, we investigated the effect of fill factoron the bandwidth, and we found that the bandwidth de-pends on the fill factor. Thus, higher ratios of active areato total area will generate broader bandwidth.

We also have shown that 2-D axisymmetric models canbe used to predict device performance. The 2-D model,which is least expensive computationally, was used to de-termine the optimum gap height; as the gap height shrinks,the output pressure per volt increases. The collapse volt-age also decreases with decreased gap height. Therefore,by fabricating devices with very small gaps, it is possibleto obtain higher pressure per volt at lower bias voltages.Note that this is only valid for small signals. However, thetotal output pressure that can be obtained is, perforce,limited by the gap height assuming there is no limitationon the applied AC voltage. However, this is not generallythe case, especially for CMUTs with integrated electronics.Standard integrated circuit techniques impose limitationson the amplitude of the pulses that can be used to excitethe membranes. Therefore, it is important to maximize theoutput pressure per volt to get the maximum pressure fora given pulse amplitude.

Increasing the collapse voltage results in higher outputpressures per volt. One way to increase the collapse voltageis to fabricate devices with larger gaps. Although this de-sign increases the collapse voltage (and therefore the max-imum bias that can be applied), it does not increase theoutput pressure per volt. The collapse voltage should beincreased by using stiffer membranes. But this may reducethe bandwidth. Because there is a tradeoff between out-put pressure and bandwidth, the CMUT membrane designmust be considered both of these parameters.

In addition, when using the models simulating infinitelylarge transducers (both 3-D and 2-D) with symmetryboundaries around a single cell, care must be taken in cal-culating the electrical input impedance. In these models,the real part of the impedance converges to a specific valuethat is determined by the plane impedance of the medium,rather than vanishing toward low frequencies.

The models developed in this study also can be usedfor large signal simulations in which nonlinearity of theelectrostatic forces is included in the calculations. Smallsignal receive voltage calculations also can be performed.These are the topics of future research.

Acknowledgment

This work is supported by National Institutes of Health(NIH) and Office of Naval Research (ONR).

References

[1] M. I. Haller and B. T. Khuri-Yakub, “A surface micromachinedelectrostatic ultrasonic air transducer,” IEEE Trans. Ultrason.,Ferroelect., Freq. Contr., vol. 43, pp. 1–6, Jan. 1996.

[2] H. T. Soh, I. Ladabaum, A. Atalar, C. F. Quate, andB. T. Khuri-Yakub, “Silicon micromachined ultrasonic immer-sion transducers,” Appl. Phys. Lett., vol. 69, pp. 3674–3676,Dec. 1996.

[3] I. Ladabaum, X. Jin, H. T. Soh, A. Atalar, and B. T. Khuri-Yakub, “Surface micromachined capacitive ultrasonic transduc-ers,” IEEE Trans. Ultrason., Ferroelect., Freq. Contr., vol. 45,pp. 678–690, May 1998.

[4] D. M. Mills and L. S. Smith, “Real time in-vivo imaging withcapacitive micromachined ultrasonic transducer (cMUT) lineararrays,” in Proc. IEEE Ultrason. Symp., 2003, pp. 568–571.

[5] O. Oralkan, A. S. Ergun, J. A. Johnson, M. Karaman, U.Demirci, K. Kaviani, T. H. Lee, and B. T. Khuri-Yakub, “Ca-pacitive micromachined ultrasonic transducers: Next-generationarrays for acoustic imaging?,” IEEE Trans. Ultrason., Ferro-elect., Freq. Contr., vol. 49, pp. 1596–1610, Nov. 2002.

[6] D. M. Mills, “Medical imaging with capacitive micromachinedultrasonic transducer (cMUT) arrays,” in Proc. IEEE Ultrason.Symp., 2004, pp. 384–390.

[7] O. Oralkan, A. S. Ergun, C. Ching-Hsiang, J. A. Johnson, M.Karaman, T. H. Lee, and B. T. Khuri-Yakub, “Volumetric ul-trasound imaging using 2-D CMUT arrays,” IEEE Trans. Ultra-son., Ferroelect., Freq. Contr., vol. 50, pp. 1581–1594, Nov. 2003.

[8] F. V. Hunt, Electroacoustics: The analysis of transduction, andits historical background. Cambridge: Harvard Univ. Press, 1954.

[9] G. G. Yaralioglu, M. H. Badi, A. S. Ergun, and B. T.Khuri-Yakub, “Improved equivalent circuit and finite elementmethod modeling of capacitive micromachined ultrasonic trans-ducers,” in Proc. IEEE Ultrason. Symp., 2003, pp. 469–472.

[10] A. Lohfink, P.-C. Eccardt, W. Benecke, and H. Meixner,“Derivation of a 1D CMUT model from FEM results for lin-


ear and nonlinear equivalent circuit simulation,” in Proc. IEEEUltrason. Symp., 2003, pp. 465–468.

[11] A. Bozkurt, I. Ladabaum, A. Atalar, and B. T. Khuri-Yakub,“Theory and analysis of electrode size optimization for ca-pacitive microfabricated ultrasonic transducers,” IEEE Trans.Ultrason., Ferroelect., Freq. Contr., vol. 46, pp. 1364–1374,Nov. 1999.

[12] G. G. Yaralioglu, A. S. Ergun, B. Bayram, E. Haeggstrom, andB. T. Khuri-Yakub, “Calculation and measurement of electrome-chanical coupling coefficient of capacitive micromachined ultra-sonic transducers,” IEEE Trans. Ultrason., Ferroelect., Freq.Contr., vol. 50, pp. 449–456, Apr. 2003.

[13] B. Bayram, G. G. Yaralioglu, A. S. Ergun, and B. T. Khuri-Yakub, “Influence of the electrode size and location on the per-formance of a CMUT,” in Proc. IEEE Ultrason. Symp., 2001,pp. 949–952.

[14] B. Bayram, E. Haeggstrom, G. G. Yaralioglu, and B. T. Khuri-Yakub, “A new regime for operating capacitive micromachinedultrasonic transducers,” IEEE Trans. Ultrason., Ferroelect.,Freq. Contr., vol. 50, pp. 1184–1190, Sep. 2003.

[15] M. Kaltenbacher, H. Landes, K. Niederer, and R. Lerch, “3Dsimulation of controlled micromachined capacitive ultrasoundtransducers,” in Proc. IEEE Ultrason. Symp., 1999, pp. 1155–1158.

[16] G. Wojcik, J. Mould, P. Reynolds, A. Fitzgerald, P. Wagner, andI. Ladabaum, “Time-domain models of MUT array cross-talk insilicon substrates,” in Proc. IEEE Ultrason. Symp., 2000, pp.909–914.

[17] Y. Roh and B. T. Khuri-Yakub, “Finite element analysisof underwater capacitor micromachined ultrasonic transduc-ers,” IEEE Trans. Ultrason., Ferroelect., Freq. Contr., vol. 49,pp. 293–298, Mar. 2002.

[18] S. Ballandras, A. Caronti, W. Steichen, M. Wilm, V. Laude, T.Pastureaud, R. Lardat, and W. Daniau, “Simulation of cMUTradiating in water using a mixed finite element/boundary el-ement approach,” in Proc. IEEE Ultrason. Symp., 2002, pp.1048–1051.

[19] J. W. Goodman, Introduction to Fourier Optics. New York:McGraw-Hill, 1968.

[20] W. P. Mason, Electromechanical Transducers and Wave Filters.London: D. Van Nostrand, 1948.

Goksen Goksenin Yaralioglu (S’92–M’99)was born in Akhisar, Turkey, on May 13,1970. He received his B.S., M.S., and Ph.D.degrees from Bilkent University, Turkey, in1992, 1994, and 1999, respectively, all in elec-trical engineering. He is now working as anengineering research associate in E. L. Ginz-ton Laboratory, Stanford University. His cur-rent research interests include design, mod-eling and applications of micromachined ul-trasonic transducers, and atomic force mi-croscopy at ultrasonic frequencies.

Arif Sanli Ergun (S’91–M’99) was bornin Ankara, Turkey, in 1969. He received hisB.Sc., M.Sc., and Ph.D. degrees in 1991, 1994,and 1999, respectively, all in electrical andelectronics engineering from Bilkent Univer-sity, Turkey. He is now at the E. L. Ginz-ton Laboratory, Stanford University, as anengineering research associate. His main re-search interests are acoustics, ultrasound, mi-croelectromechanical systems (MEMS), andmicrowave electronics.

Butrus T. Khuri-Yakub (S’70–S’73–M’76–SM’87–F’95) was born in Beirut, Lebanon.He received the B.S. degree in 1970 from theAmerican University of Beirut, the M.S. de-gree in 1972 from Dartmouth College, and thePh.D. degree in 1975 from Stanford Univer-sity, all in electrical engineering. He joined theresearch staff at the E. L. Ginzton Laboratoryof Stanford University in 1976 as a researchassociate. He was promoted to senior researchassociate in 1978, and to a Professor of Elec-trical Engineering (Research) in 1982. He has

served on many university committees in the School of Engineeringand the Department of Electrical Engineering.

Presently, he is the Deputy Director of the E. L. Ginzton Labora-tory, and the associate chairman for graduate admissions in the elec-trical engineering department at Stanford. Professor Khuri-Yakubhas been teaching both at the graduate and undergraduate levels forover 20 years, and his current research interests include in situ acous-tic sensors (temperature, film thickness, resist cure, etc.) for moni-toring and control of integrated circuits manufacturing processes,micromachining silicon to make acoustic materials and devices suchas airborne and water immersion ultrasonic transducers and arrays,and fluid ejectors, and in the field of ultrasonic nondestructive eval-uation and acoustic imaging and microscopy.

Professor Khuri-Yakub is a fellow of the IEEE, a senior memberof the Acoustical Society of America, and a member of Tau Beta Pi.He is associate editor of Research in Nondestructive Evaluation, aJournal of the American Society for Nondestructive Testing. Profes-sor Khuri-Yakub has authored over 400 publications and has beenprincipal inventor or co-inventor of 60 issued patents. He receivedthe Stanford University School of Engineering Distinguished Advi-sor Award, June 1987, and the Medal of the City of Bordeaux forcontributions to NDE, 1983.