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

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1 ieee transactions on ultrasonics, ferroelectrics, and frequency control, vol. 52, no. 12, december Finite-Element Analysis of Capacitive Micromachined Ultrasonic Transducers Goksen G. Yaralioglu, Member, IEEE, A.SanliErgun,Member, IEEE, and Butrus T. Khuri-Yakub, Fellow, IEEE Abstract In this paper, we present the results of finiteelement analysis performed to investigate capacitive micromachined ultrasonic transducers (CMUTs). Both threedimensional (3-D) and 2-D models were developed using a commercially available finite-element modeling (FEM) software. Depending on the dimensionality of the model, the membranes were constructed using plane or shell elements. The electrostatic gap was modeled using many parallel plate transducers. An axisymmetric model for a single membrane was built; the electrical input impedance of the device then was calculated in vacuum to investigate series and parallel resonant frequencies, where the input impedance has a minimum and a maximum, respectively. A method for decomposing the membrane capacitance into parasitic and active parts was demonstrated, and it was shown that the parallel resonant frequency shifted down with increased biased voltage. Calculations then were performed for immersion transducers. Acoustic wave propagation was simulated in the immersion medium, using appropriate elements in a 3-D model. Absorbing boundaries were implemented to avoid the reflections at the end of the medium mesh. One row of an array element, modeled with appropriate boundary conditions, was used to calculate the output pressure. The results were compared with a simpler model: a single membrane in immersion, with symmetry boundary conditions on the sidewalls that cause the calculations to reflect the properties of an infinitely large array. A 2-D model then was developed to demonstrate the effect of membrane dimensions on the output pressure and bandwidth. Our calculations revealed that the small signal transmit pressure was inversely proportional to the square root of gap height. We also compared FEM results with analytical and experimental results. I. Introduction Invented at Stanford University [1] [3], capacitive micromachined ultrasound transducers (CMUTs) have become very popular over the last decade for use in medical imaging [4], [5]. They easily can compete with their piezoelectric counterparts in terms of bandwidth, dynamic range, and sensitivity [6]. Moreover, ease of fabrication for complex device geometries (such as two-dimensional (2-D) arrays) has made CMUTs an even more attractive alternative to piezoelectric transducers. Recently, images obtained from a 2-D CMUT array have been demonstrated [7]. The increased interest in the CMUT technology and applications has accelerated modeling efforts for these devices. Initially, an equivalent circuit method was used to Manuscript received September 13, 2004; accepted May 12, The authors are with the Ginzton Laboratory, Stanford University, Stanford, CA ( goksenin@stanford.edu). predict output pressure and bandwidth [3], [8]. In its simplest form, the equivalent circuit has two ports. The electrical port contains the clamped capacitance of the transducer; the mechanical port is composed of the mechanical impedance of the membrane and the negative spring softening capacitance (caused by electromechanical interaction). In immersion, the mechanical side is terminated by the radiation impedance of the medium. However, this approach is not very accurate as it does not include the hydrodynamic mass loading of the immersion medium [9]. Lohfink et al. [10] recently demonstrated a method for accurate modeling of fluid loading. By defining an equivalent parallel plate capacitor for a CMUT membrane at each bias voltage, turns ratio was calculated precisely. However, the equivalent circuit approach still lacks the modeling of both cross talk between the membranes and the effect of higher order resonances. By using finite-element modeling (FEM) intensively, more accurate calculations can be made to evaluate the performance of the CMUT devices. Commercial and custom-developed FEM codes have been used for both static and dynamic analysis. Static FEM models have been used to accurately calculate collapse voltage and device capacitance [11] [13]. Bozkurt et al. [11] developed a FEM model in which the electrode coverage was optimized for optimum bandwidth. Recently, Bayram et al. [14] calculated average membrane displacement for different operation regimes. Dynamic FEM models also have been developed and used to estimate the output pressure and membrane displacement by performing time-domain analysis [15]. For the evaluation of cross talk, more intensive models were introduced to demonstrate propagating waves along the water-wafer interface [16]. Additionally, harmonic analysis was used to investigate the cross talk between the membranes in a 2- D model in which elements are assumed to be composed of infinitely long membranes [17]. In a similar geometry, a single membrane was used to model propagating modes in a periodic membrane structure [18] by assigning appropriate phase difference at the boundaries. Previously, we demonstrated output pressure calculation using 3-D FEM models [9]. In the simplest FEM construction, we modeled a single cell of a circular membrane array that was distributed uniformly in lateral directions over a 2-D rectangular grid. Appropriate boundary conditions were applied on the sidewalls of the model, so that the single membrane was replicated in the lateral directions. This model assumes that all the cells in the array operate with the same phase. More extensive models in /$20.00 c 2005 IEEE

2 2186 ieee transactions on ultrasonics, ferroelectrics, and frequency control, vol. 52, no. 12, december 2005 Fig. 1. Axisymmetric membrane and the electrostatic elements. I and J are the bottom and the top nodes of one of the electrostatic elements, respectively. The top electrode is assumed to be on the bottom surface of the membrane. The same membrane geometry has been used in all of the calculations throughout the paper unless otherwise noted. clude a 3-D model of an array element in which the element has a finite width in one dimension. In this paper, we will first model the membrane with an electrostatic gap, which will be included in the model through transducer elements. We will then show 3-D models for the CMUT transducer in immersion, and compare them with a 2-D axisymmetric model. Using the 2-D model, we will demonstrate the effect of membrane dimensions and present design steps for a given bandwidth and output pressure. The FEM results then will be validated by comparing them with analytical and experimental results. II. FEM Calculations The 2-D and 3-D models considered in this study were constructed using a commercially available FEM package (ANSYS8.0, ANSYS Inc., Canonsburg, PA). In 2-D models, the membrane was built using axisymmetric plane elements; in 3-D models, shell elements were used. The electrical ports were included in the model using electrostatic elements (TRANS126). The electrostatic membrane first was modeled; the model then was extended for immersion devices in which fluid-solid interaction was performed accurately by ANSYS. For immersion devices, the absorbing boundaries at the end of the liquid mesh also were implemented using available elements (FLUID30, FLUID130). The scope of this paper is limited to the harmonic analysis in which the alternating current (AC) signals are assumed to be small relative to the direct current (DC) bias, and in 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 elements used in the model. The thickness and the radius of the circular membrane were 1.3 µm and 15 µm, respectively. The gap was 0.2 µm,andthetopelectrodewason the bottom surface of the membrane. The membrane was composed of plane elements (PLANE42) that are axisymmetric 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 electromechanical transducer elements (TRANS126), which are basically parallel plate capacitors, the electrostatic forces applied between the top and the bottom electrodes were included in the model. The elements apply opposite electrostatic attraction forces to the nodes to which they are attached. In the model, the top nodes of these elements were attached to the section of the membrane at which the electrode is located; the bottom nodes were simply clamped. The area of the i th component is given by: A i =2πr i r, (1) where r i is the radius of the center of the element, and r is the mesh size. This approach divides the electrostatic gap into many segments; within each segment, the electrostatic pressure is modeled by a small parallel plate capacitor. Therefore, the electrostatic field is assumed to be constant within each capacitor, and the fringing fields at the rim of the electrode were disregarded. If the number of segments is sufficiently high (approximately 25 over the electrode region), the membrane s static deflection can be calculated very accurately. Using both ANSYS as described in [12], [13] and the above approach, we compared the deflection profiles computed. When the top and bottom electrodes were equal in size, both calculations for the displacement profile were in agreement within 1%. The described model can be used for static, harmonic, and transient analysis. In harmonic analysis, the small signal equivalent of the transducer elements is used at the bias voltage. In transient analysis, nonlinear large signal calculations can be performed. The scope of this paper is limited to the static and harmonic analyses. In the model described above, we calculated collapse voltage using a binary search algorithm. The bias voltage was iterated until the maximum voltage that resulted in a convergent solution for the membrane displacement was obtained. The collapse voltage of the silicon (Young s modulus, 150 GPa; Poisson s ratio, 0.17; density, 2332 kg/m 3 ) membrane (whose geometry is shown in Fig. 1) was found to be V for half metalization. The input impedance of the device then was calculated at various bias voltages using harmonic analysis. A small amount of mechanical loss was introduced in the membrane material to account for losses, such as coupling into the substrate and material damping in the silicon. The loss factor was adjusted so that the mechanical Q of the membrane is on the order of 100, which is close to the experimental measurements in the frequency range of interest. Fig. 2 shows the electrical input impedance for different bias voltages. The mechanical loss of the membrane material resulted in a nonzero real part, as depicted in Fig. 2(a). The impedance curves of Fig. 2(a) and (b) have two characteristic frequencies. The magnitude of the impedance has a minimum (at f s )anda maximum (at f a ). These two frequencies are the series resonant and the parallel resonant frequencies, respectively. The maximum membrane displacement is achieved at the series resonant frequency, as seen in Fig. 3(c). Due to the

3 yaralioglu et al.: fem analysis to investigate cmut 2187 Fig. 3. Equivalent circuit model. C is the clamped capacitance of the CMUT. C 0 is the active capacitance. n is the turns ratio. m m and k m are the mass and the spring constant of the equivalent parallel plate capacitor to the CMUT membrane. R indicates the combined losses in the membrane. spring softening, the series resonant frequency [Fig. 2(b), approximately where the imaginary part is zero] and the peak displacement [Fig. 2(c)] shift down with increased biased voltage [8], as expected. Note that the displacement of Fig. 3(c) is calculated with zero source resistance. The native mechanical resonant frequency of the membrane determines the parallel resonant frequency of the electrical input impedance. The resonant frequency of the membrane of Fig. 1 is MHz. Another interesting feature of the impedance curves in Fig. 2(a) and (b) is that, as the bias voltage increases, parallel resonant frequency also shifts toward smaller frequencies, primarily due to the inherent parasitic capacitance of the membrane. In the next subsection, we will investigate this observation further and decompose the membrane capacitance into active and parasitic capacitances, using the coupling coefficient, k 2 t,calculation. Fig. 2. (a) Real part of the electrical input impedance. (b) Imaginary part of the electrical input impedance. (c) Average displacement over the membrane. Note that the frequency range of (a) is different from that used in (b) and (c). 1. Equivalent Circuit Model and kt 2 Calculation: Fig. 3 shows a typical equivalent circuit, whose input impedance is similar to the impedance results depicted in Fig. 2(a) and (b). Using simple equations describing the turns ratio and the mechanical impedance of a spring-mass model [9], this circuit accurately predicts the electrical input impedance for a parallel plate capacitor. However, because the electrostatic force in a CMUT membrane is applied only over the electrode area, and the membrane has a certain deflection profile rather than a constant displacement, the elements of the equivalent circuit need correction [9]. Recently, Lohfink et al. [10] proposed an alternative approach to calculate the element values of the circuit: rather than directly using the mechanical impedance of the membrane; their method finds an equivalent spring-mass model for the membrane impedance around the first resonant frequency of the membrane. The same approach will be used in this study, with slight variations. We will find an

4 2188 ieee transactions on ultrasonics, ferroelectrics, and frequency control, vol. 52, no. 12, december 2005 Fig. 4. Average membrane displacement for half and full metalization and equivalent parallel plate capacitor. Collapse voltages are V and V for half and full metalization, respectively. equivalent parallel plate capacitor whose displacement is equal to the average displacement of the membrane, and we will assume the gap height of the equivalent parallel plate transducer is equal to three times the maximum average displacement of the membrane just prior to collapse. The average membrane displacement is given by: R 2πru(r)dr 0 x(v )= πr 2, (2) where R is the radius of the membrane, u(r) is the displacement profile under the applied bias voltage. However, the displacement of a parallel plate transducer depends only on the ratio of the applied voltage to the collapse voltage [12]. The real root, which is less than one third of the gap height, of the following polynomial provides the displacement of a parallel plate capacitor: V 27x = (d eff x), (3) V coll 4d 3 eff where d eff is the gap height. Collapse occurs when the displacement is equal to one-third of the gap. Fig. 4 shows the displacement as a function of the bias voltage for the membrane depicted in Fig. 1 and for the equivalent parallel plate capacitor. The parallel plate capacitor predicts the average membrane displacement with a maximum error of 3% (which occurs when the bias voltage is close to zero). Because the approximation is calculated for critical displacement, the error decreases as the bias voltage increases. Electrode size also affects the maximum error. For full metalization in small biases, maximum error is almost 10%. The parallel plate can better approximate the motion of the membrane for small electrode coverage. However, the bending effects of the membrane become more Fig. 5. Ratio of parasitic capacitance to active capacitance (C = C p(x)+c 0 (x)). dominant as the electrode coverage increases. Because the transducers usually are biased close to the collapse voltage, the error in the approximation is acceptable. Previously, we calculated the coupling coefficient, k 2 T, of a transducer [12], and showed that, by adding an appropriate amount of parasitic capacitance to the parallel plate capacitance, a parallel plate transducer can model a CMUT membrane. The coupling coefficient of a parallel plate transducer is given by: k 2 T = 2x d eff x +(d eff 3x) C p(x) C 0 (x), (4) where C 0 (x) is the active (equivalent parallel plate) capacitance, and C p (x) is the parasitic capacitance [12]. For a parallel plate capacitor, (4) can be expressed in terms of normalized displacement, x/d eff, which can be calculated from (3) for a given normalized bias voltage. However, the total device capacitance of the CMUT membrane should be equal to the sum of the active capacitance, C 0 (x), and the parasitic capacitance, C p (x). The total capacitance and coupling coefficient of the membrane depicted in Fig. 1 were calculated using ANSYS, as described in [12]. Substituting these results in (4) revealed active and parasitic capacitances at a given bias voltage. (It is important to know the collapse voltage accurately for the above calculations.) The ratio of the parasitic capacitance to the active capacitance is shown in Fig. 5 for various electrode coverages. The ratio increases as the metalization on the membrane surface increases. This result matches very well with our previous finding that, at a given bias voltage, the coupling coefficient of a CMUT membrane decreases as the radius of the electrode increases. To this point, we have defined a method to break the device capacitance into equivalent parallel plate, C 0 (x), and parasitic, C p (x), capacitances. Other element values of the

5 yaralioglu et al.: fem analysis to investigate cmut 2189 capacitance [Fig. 3(b)]. In this case, the parallel resonance in which the imaginary and the real parts are maximized occurs at the resonance of the membrane. However, because of parasitics, if the total clamped electrical capacitance is greater than the magnitude of the negative capacitance, the parallel resonant frequency shifts down. Adding the external parasitic capacitance would result in a further downshift of the parallel resonant frequency. The inherent parasitics in the membrane capacitance occur because displacement is not constant over the membrane. Even without additional external parasitics, with the increased bias voltage, the membrane s inherent parasitics will result in a downshift of the parallel resonant frequency. Fig. 6. Imaginary part of input impedance calculated using ANSYS and the equivalent circuit approach. Bias voltage is 120 V. equivalent circuit are simple to calculate. The spring constant of the equivalent parallel plate capacitor is given by: k m = 27V coll 2 C 0(x)(d eff + x). (5) 8d 3 eff The equivalent mass can be calculated from the resonant frequency using the relation: m m = k m (2πf 0 ) 2, (6) where f 0 is the resonant frequency of the membrane and the turns ratio is: V bias n = C 0 (x) d eff x. (7) Note that element values need to be recalculated at each bias voltage. The calculated input impedance is depicted in Fig. 6. We obtained a very good match between ANSYS and the equivalent circuit calculations. As suggested by the above analysis, the parasitic capacitance that is always present in the membrane transducer is responsible for the downshift of the parallel resonant frequency. This relationship can be understood easily by using the equivalent circuit shown in Fig. 3(b), where all the acoustical elements were reflected to the electrical side. This circuit [Fig. 3(b)] has two parallel branches. At the resonant frequency, the sum of the mechanical membrane impedance and the spring softening capacitance is zero, resulting in minimum electrical impedance. At the parallel resonant frequency, the mechanical membrane impedance is zero, and the combination of two capacitances (the electrical clamped capacitance and the negative spring softening capacitance) and with resistance creates maximum electrical impedance. For the parallel plate capacitor, the device capacitance is equal to the negative spring softening 2. Optimization of Electrode Radius: Optimization of the metalization coverage when there is excessive external parasitic capacitance (which is the case for most of the CMUT devices), is another use of coupling coefficient calculation. The excessive external parasitics are caused primarily by the metal lines connecting the top electrodes of the membranes. There is usually a high dielectric material of either silicon oxide or silicon nitride between these lines and the bottom substrate. When there are no external parasitics, half metalization seems to be the optimum for electrode radius [11], [12], because it is a good compromise between bandwidth and collapse voltage [11]. For the electrode sizes larger than half coverage parasitics increase at a faster rate than active capacitance (or equivalent parallel plate capacitance) does. With 50% metalization (Fig. 5), the total capacitance is slightly larger than the active capacitance, and the parasitics are very small. With 100% metalization, while the total capacitance increases four times, the active capacitance increases only twice. Therefore, when there are no external parasitics, 50% coverage is optimum in terms of coupling efficiency, as shown in Fig. 7. With external parasitics (C p ext ), increasing the electrode radius toward full metalization also increases the coupling coefficient a result of the increased ratio of active capacitance to total parasitics (C p (x)+c p ext ). Therefore, when there is an excessive amount of external parasitics (as high as 300%), full metalization results in a better coupling coefficient. B. Membrane Radiating into Immersion Medium in an Array In a CMUT transducer, membranes that are similar to the one analyzed in the previous section are grouped together to form array elements. The geometry of the element determines the different imaging modalities. A typical 1-D transducer array is composed of long rectangular elements. The length of the array element usually is much longer than its width. The width of the element is set to half-wavelength at the maximum operation frequency. Therefore, a typical element is composed of hundreds of CMUT cells in the elevation, but it is usually only four to six cells in width. The high aspect ratio of the element allowed us to model only one row of the CMUT cells, greatly

6 2190 ieee transactions on ultrasonics, ferroelectrics, and frequency control, vol. 52, no. 12, december 2005 Fig. 7. Coupling coefficient as a function of parasitic capacitance. The curves were calculated at 80% of the collapse voltages. The device capacitance for half metalization is F.Thedevicecapacitances of 75% and full metalization are and , respectively. The collapse voltages are V, V, and V for half to full metalization. reducing the model size and computation time [9]. Symmetry boundary conditions were applied on the bottom and the top edges of the CMUT row. This replicated the model infinitely many times in the elevation direction and generated an infinitely long element at which all the rows were driven by the same phase. Because the row was also symmetric with respect to the centerline along the elevation, the model size was further reduced. Therefore, modeling of one half of the row is sufficient. The mesh of the half row of the element is shown in Fig. 8. The pitch, w, between the membranes, was 32 µm. The element was six membranes in width and only half of them were modeled. The membranes were meshed by shell elements (SHELL63). The nonmembrane regions were clamped on the membrane plane. A finer mesh size was used over the membrane region to accurately calculate the resonant frequencies and mode shapes of the membranes. The mesh size was set to 3 µm inthe fluid region so that at the highest frequency (50 MHz) there were 10 elements per wavelength in water. The electrostatic elements were attached to the membrane nodes within the electrode region. The fluid region was composed of 3-D acoustic elements (FLUID30). The interaction between the solid and fluid parts was activated by using the appropriate ANSYS flags. To eliminate the reflections from the end of the mesh, the envelope of the fluid region was covered by absorbing elements (FLUID130). The first step of the calculations was to bias the transducer elements to the specified voltage. Next, prestressed harmonic analysis, which uses the small signal equivalent of the transducer elements at the bias point, was performed. During the frequency sweep, the distance between the membranes and the absorbing elements were adjusted so that the radius of the model was equal to the wavelength. Therefore, at each frequency point the fluid region was remeshed. This was Fig D mesh of an array element. The image is obtained when the frequency is 10 MHz. Therefore, the absorbing boundary on the top image is 150 µm (wavelength) away from the origin. Bottom image shows the zoomed area around the membranes. Clamped boundary conditions were applied on the shell elements as shown. The electromechanical elements are not shown for clarity. required for the absorbing element to perform sufficiently well to eliminate the reflecting waves from the boundary. However, for high frequencies in which the radius is less than the length of the row, the radius was simply set to three times the pitch (3w). We used the same membrane geometry discussed previously. The membrane was biased at 80% of the collapse voltage; then harmonic analysis was performed by applying 1 V peak harmonic voltage at the top electrode. The source impedance was assumed to be zero. Fig. 9 depicts the average displacement over the membranes. Because of acoustic coupling through the immersion medium, the displacement 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 parameter that can be calculated using the model. The input impedance depicted in Fig. 10 was obtained by using the total current flowing through to the capacitors and was

7 yaralioglu et al.: fem analysis to investigate cmut 2191 Fig. 9. Average displacement over the membranes. Membrane 1 is the one closest to the center and membrane 3 is the most outer one. Fig. 11. Single cell 3-D model. Electrostatic elements attached to the electrode are not shown for clarity. The electrode region is at the center of the membrane as shown above. Fig. 10. Electrical input impedance of the three membranes operating in an array element. calculated only for half of the row. The input impedance of the row is half of the impedance shown in Fig. 10. The total impedance of the array element is the parallel combination of all the rows, and it can be found by dividing the input impedance of the single row by the total number of the rows. C. Single Membrane with Symmetry Boundaries Assuming symmetry boundaries in the width dimension will further simplify the modeling of CMUT arrays. The corresponding transducer will be infinitely large in both lateral dimensions. In this case, the model will be composed of a membrane and a rectangular fluid column above it, as shown in Fig. 11. This model assumes all the neighboring cells are driven by the same phase, and have Fig. 12. Average displacement over the membrane. The bias voltage is 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 previous section, the membrane was first biased to a certain fraction of a collapse voltage; then, harmonic analysis was performed. The calculated average displacement (Fig. 12) is very similar to the displacement obtained using the model described in the previous section, especially for frequencies above 15 MHz. The periodicity of the cell reveals itself as a sudden change (observed at about 47.5 MHz) at slightly above the frequency in which the wavelength in water is equal to the pitch (f = 1500 m.sec/32µm =46.87 MHz). Below this frequency, the acoustic wave propagation in the fluid column is perpendicular to the transducer plane; as it moves away from the membrane, the field pattern quickly converges to that of a plane wave. Above this frequency, however, the wavelength in water is less than the period-

8 2192 ieee transactions on ultrasonics, ferroelectrics, and frequency control, vol. 52, no. 12, december 2005 Fig. 13. Electrical input impedance of a single cell operating in an infinitely large transducer. Bias voltage is 80% of the collapse voltage. Fig. 14. Average pressure over the membrane plane. For both models, the bias voltage is 80% of the collapse voltage. icity, which results in diffraction of waves in water. Therefore, where the wavefronts are parallel to the transducer plane, there are waves being sent to the higher orders as well as to the 0 th order. For example, the angle of the first diffracted order is given by [19]: ( ) λ θ =arctan, (8) w 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 for a single transducer; therefore, the impedance is expected to be three times larger than the impedance shown in Fig. 10. There is another important difference between the two impedances: for the membranes operating in an array, as the frequency decreases, the real part goes to zero; in the case of a single cell, the real part converges to a certain value. This discrepancy can be explained easily using the equivalent circuit. At DC, the real part of the electrical input impedance of the circuit, shown in Fig. 3 is given by: R/n 2 Re {Z in } = ( 1+ Ck m n 2 C ). (9) C 0 In immersion, the radiation impedance terminates the mechanical port; therefore R is equal to the real part of the radiation impedance. For the infinitely large plane transducer, the real part of the radiation impedance is constant; for membranes operating in an array element, the real part of the radiation impedance decreases to zero as the frequency decreases. Therefore, the real part of the input impedance cannot be accurately predicted for low frequencies using the infinitely large transducer model. However, the model can be used for the output pressure calculations. Fig. 14 compares the pressure over the membrane Fig. 15. Average pressure over the membrane plane for models with various widths. plane obtained using the two models. As shown above, because of acoustic interaction between the membranes, the displacement of the membranes operating in an array with a finite width can be different, which results in strong dips in the output pressure. Except for this discrepancy, the two calculations matched quite well. Using the infinitely large transducer model, we investigated the effect of membrane separation by changing the width of the square region surrounding the membrane, as shown in Fig. 15. As the width increases, the bandwidth of the transducer decreases due to the change in the radiation impedance. The real part of the radiation impedance decreases because the real part is scaled by the ratio of active area to total area. This increases the quality factor of the membranes. The imaginary part of the radiation

9 yaralioglu et al.: fem analysis to investigate cmut 2193 Fig. 16. Output pressure over the membrane plane. The ratio of the active 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 on the membrane. With a larger, nonactive region around the membrane, the fluid also can be moved to the side without any radiation, which increases the hydrodynamic mass of the fluid. Thus, for high bandwidth, it is important to densely pack the membranes, so as to eliminate dead regions between them. This becomes especially important for high-frequency devices in which the membranes are only a few microns in diameter. The decrease of the resonant frequency is another effect of the increased periodicity. This is again due to increased volume of the fluid interacting with the membrane, as explained above. D. Axisymmetric Model for Immersion A further simplification of the infinitely large transducer is an axisymmetric model in which the membrane radiates into a cylindrical wave guide [10]. The complexity of the resulting 2-D model is very low, compared to the 3-D models discussed earlier. The constructed FEM model is very similar to the model used for an infinitely large transducer, except that plane elements were used in the FEM mesh. The outer radius, r out, of the fluid column was chosen so that both 3-D and 2-D models have the same amount of nonactive region: r out = w π, (10) where w is the width of the total area or it also can be interpreted as the pitch of the array. Fig. 16 compares the output pressures obtained using the 3-D and 2-D models. The 2-D model predicts slightly higher output pressure and more bandwidth. This is due to the fact that the 2-D model assumes a better packing density. But, 2-D FEM is a simple and computationally very efficient model; it Fig. 17. Pressure on the center line. can be used for any CMUT transducers whose membranes are distributed arbitrarily, as long as the active and total areas are defined appropriately. The appropriate selection of active and total area was demonstrated in [10]. This modeling approach can provide important information on various features of a CMUT transducer such as output pressure and bandwidth. For example, Fig. 17 plots the pressure distribution along a line passing through the center of the membrane for 3-D and 2-D models. The absorbing boundary, which usually is positioned one wavelength away from the membrane, was placed further away for demonstration purposes. As shown in Fig. 17, the waves quickly converge to plane waves. To demonstrate the propagated waves, the real part of the pressure distribution also is added to Fig. 17. The imaginary part is simply 90 degrees out of phase with the real part. An important design criterion for the CMUT transducer is the dimensions of the membrane that populates the elements of the transducer. Membrane thickness and radius determine output voltage, bandwidth, and collapse voltage.fig.18 shows the mechanical membrane impedance for membranes with various radii (10, 15, and 20 µm). To determine the mechanical impedance, we apply a constant pressure over the membrane area, and calculate the average velocity using ANSYS. The mechanical membrane impedance is defined as the ratio of the integral pressure over the area (total force) to the average velocity. There is an analytical solution for the circular membrane impedance [20]. The thickness of the membranes was chosen so that they all have the same resonant frequency. The collapse voltages were 91.5 V and V for 10 µm and 20 µm membranes, respectively. The gap was 0.2 µm; the electrode radius was set to half of the membrane radius. The output pressure obtained using these membranes is plotted in Fig. 19 when the bias voltage was 80% of the collapse voltage. The ratio of the membrane area to the total

10 2194 ieee transactions on ultrasonics, ferroelectrics, and frequency control, vol. 52, no. 12, december 2005 Fig. 18. Mechanical membrane impedance in vacuum. Fig. 20. Output pressure averaged over the membrane plane for various gap heights. The membrane radius and thickness are 15 µm and 1.4 µm, respectively. For each gap height the bias voltage is set to 80% of the collapse voltage. The collapse voltages are 17, 47.9, 88.0, 135.4, and V for the gap height, increasing from 0.05 µm to 0.3 µm. that frequency, the output pressure per volt of a CMUT transducer is given by: P 0 = n A, (11) where A is the membrane area. This calculation is a direct result of the equivalent circuit model shown in Fig. 3. For a parallel plate capacitor, (11) reduces to: P 0 = ε 0V bias (d eff x) 2. (12) Fig. 19. Average pressure over the membrane plane (total area). The collapse voltages of the three membranes are 91.5, 135.4, and V. area was set to the same value (0.7) for each case. Because the mechanical membrane impedance is small compared to the radiation impedance over a broader frequency band, the softer membrane generated the maximum bandwidth. But, the output pressure is low compared to the other membrane sizes because of the low bias voltage that is determined by the collapse voltage. The stiffer membrane generated higher output pressure with a narrower bandwidth. Another important observation was that the fluid loading reduced the resonant frequency at the same rate for all of the membrane sizes. The output pressure also is determined by the gap. In the simplest form, neglecting nonactive regions, the output pressure of a CMUT can be calculated directly from the equivalent circuit model at the resonant frequency at which the impedances of the reactive elements are zero. At The maximum bias that can be applied to the parallel plate capacitor is the collapse voltage, and it is given by: 8k m d 3 eff V coll =. (13) 27Aε 0 If (13) is substituted in (12), one gets the pressure per volt: 3k m ε 0 P 0 =. (14) 2Ad eff Note that the above calculations are valid only for small signals. But same dependence of the output pressure to the gap is obtained if the CMUT is assumed to be biased to the certain fraction of the collapse voltage. For the 20 µm membrane, we calculated the output pressure as a function of gap height. The results are depicted in Fig. 20. As suggested by (14), the output pressure per volt is inversely proportional with the square root of gap height. Choosing a small gap height maximizes the output pressure per volt. However, the total pressure that can be obtained from a

11 yaralioglu et al.: fem analysis to investigate cmut 2195 Fig. 21. Effect of substrate. Fig. 22. Radiation impedance of a rectangular slit. The width of the slit is µm. CMUT is limited by the gap height, assuming there is no limitation on the amplitude of the excitation voltage. The effect of coupling into the substrate also can be demonstrated using the 2-D model. Instead of clamping the membrane edge, we added a 500-µm thick silicon wafer into the model, and connected the membranes with the substrate through a silicon post. Due to the post, the collapse voltage slightly decreased to 126 V. The longitudinal wave velocity in silicon is 8315 m/s. The reflections in the wafer were revealed as peaks and dips at every 8.3 MHz as shown in Fig. 21. Note that the configuration used enhances the effect of substrate resonances because the model assumes the source is composed of infinitely many CMUT membranes covering the whole plane. Therefore, the radiation impedance into the silicon is equal to the plane wave impedance. However, for 1-D arrays, in width direction, the source is usually smaller than the wavelength in silicon. Therefore, the effect of the substrate resonances is not as strong as shown in the above model. III. Validation of the FEM Models Accurate FEM results require careful selection of mesh density. The models developed in this paper were calculated using various mesh densities. The mesh sizes were iterated until the calculated results converged. In addition, the FEM results were compared with problems in which analytical solutions exist. We also confirmed our results with experiments. A. Radiation Impedance Calculations ThebestwaytovalidatetheFEMcodeistoruna case in which an analytical solution exists. For validation purposes, we used a radiation impedance calculation as a test vehicle for our FEM codes. To test the FEM model depicted in Fig. 8, radiation impedance of a rectangular slit was calculated using the model, and the results were compared with theoretical calculations. Fig. 22 shows the real and imaginary parts of the radiation impedance, calculated both analytically and using the FEM. The width of the rectangular slit is µm, and the length is assumed to be infinitely long. The immersion medium is water (v = 1500 m/s, ρ = 1000 kg/m 3 ). To find the radiation impedance, a uniform velocity field was applied over the slit, and the resulting average pressure on the radiating surface was calculated. As expected, the real part of the radiation impedance approaches the acoustic impedance of water, and the imaginary part diminishes as the frequency increases. There is a good match between theoretical and FEM curves. For small frequencies (<10 MHz), the source is smaller than the wavelength, which generates wave fronts parallel to the absorbing boundary; the absorbing boundary works very well to absorb the incoming waves. However, above 10 MHz, the radiating beam is more directive, and the waves impinge on the absorbing boundary with a certain angle. ANSYS uses a second-order absorbing boundary; because the reflection from the boundary increases as the incidence angle increases, the absorbing boundary cannot absorb all the incoming waves, and ripples are created on the FEM curves (Fig. 22). Radiation impedance also was calculated for the geometry presented in Section II-C, as shown in Fig. 23. First, a uniform velocity distribution was applied over the square region that forms the bottom part of the fluid column (Fig. 11). Because this design represents an infinitely large source radiating into a fluid half space, the radiation impedance should equal the acoustic impedance of the immersion fluid (Z c =1.5 MRayl). Our calculation matched very well with this theoretical value. (A negligibly small imaginary part is attributable to the numerical errors of the FEM calculation.) In the case in which the velocity

12 2196 ieee transactions on ultrasonics, ferroelectrics, and frequency control, vol. 52, no. 12, december 2005 Fig. 23. Radiation impedance of a circular piston in square region radiating into a rectangular fluid column. When the square surface is assumed to be moving with a constant velocity (full excitation), the radiation impedance is equal to the acoustic impedance of the fluid. Fig. 24. Half of the displacement at the air-oil interface. field is applied over a circular area within the square region (Fig. 22), the real part of the impedance should be scaled by the ratio of the radiating region to the total region. In our calculations, the active region was a circle in 30-µm diameter placed in a 32 µm by32µm square area. The real part (as calculated in this configuration) matches the expected value of 1.04 MRayl. The imaginary part of the radiation impedance, however, is caused by fluid moving back and forth, between the interior and exterior of the circular region. The linear slope of the imaginary part indicates a constant fluid loading on the circular region for low frequencies. The sudden jump just above MHz is caused by periodicity, as explained before. B. Experiment To test the FEM results, we also measured the output pressure of a CMUT immersed in oil, using an optical interferometer (Polytec, OFV511/OFV2700, Physik Instrumente, Auburn, MA). The interferometer was coupled to a microscope, and the laser beam was focused on the air-oil interface, using a 20X objective. The CMUT was immersed 2.3 mm below the interface. The transducer was composed of hexagonal membranes. The diameter of a circle that can be inscribed in the membrane was 28.8 µm. This particular device was fabricated using silicon-on-insulator (SOI) bonding technique, where 1 µm thick SOI layer was bonded over cavities defined by silicon oxide, creating a 1-µmthick silicon membrane. The post width was 5 µm. The cavity height was 0.25 µm over a 0.1 µm insulation layer. The collapse voltage of the transducer was 145 V; it was biased at 100 V. A negative 30 ns square pulse was applied to the transducer via a bias-t. The interferometer output was measured using an oscilloscope. Using the sensitivity of the interferometer, the Fig. 25. Calculated and experimental frequency response of a CMUT. voltage waveform was converted into displacement. Because the actual displacement without the air-oil interface should be half of the displacement measured at the interface, the measurement result was divided by two. Fig. 24 shows half of the measured displacement. The pressure spectrum was determined by calculating the fast Fourier transform (FFT) of the displacement waveform, and multiplying it by the radian frequency and the acoustic impedance of oil (v = 1480 m/s, ρ = 920 kg/m 3 ). The displacement waveform was properly windowed to reduce the artifacts in the FFT. To find the actual pressure on the transducer, the pressure spectrum 2.3 mm away from the transducer surface was corrected for attenuation and diffraction in oil. The resulting measured pressure spectrum is shown in Fig. 25. The peak value of the pressure (determined by applying a tone burst at 10 MHz to the transducer) was 13.2 kpa/v at 10 MHz.

13 yaralioglu et al.: fem analysis to investigate cmut 2197 A similar model to the one described in Section II-B was developed to calculate the pressure output of the transducer. The circular membranes were replaced by hexagonal cells. Fig. 25 also shows the calculated spectrum using ANSYS. The peak calculated pressure is 15.0 kpa/v. There is a good match between the two spectrums. Our FEM model estimated the center frequency and the lower cut-off frequency accurately, but it slightly overestimated the higher cut-off frequency. IV. Conclusions In this paper, we have demonstrated comprehensive FEM models for CMUTs. The electrostatic ports were incorporated into the models using electromechanical transducer elements. The scope of the paper was limited to harmonic analysis and assumed a linearized model for the electrostatic transducers. The calculations in vacuum showed that the CMUT transducer has inherited parasitics because the capacitance of a CMUT transducer cannot be completely modeled by a parallel plate transducer. These parasitics result in a decrease in parallel resonant frequency as bias voltage increases. We also have proposed a method for the decomposition of membrane capacitance into active and parasitic parts by calculating the coupling coefficient. Previously, it had been shown that half metalization improved bandwidth by reducing the inherent parasitic capacitance of the membrane. However, our calculations showed that, if there were external parasitic capacitance in addition to the inherent parasitics, full metalization would result in a better coupling coefficient. The 3-D modeling of an array element showed that acoustic coupling between membranes creates dips in the output pressure spectrum. Although a simple 3-D model does not reveal these dips, the model still can be used for optimization of the output pressure spectrum. In this study, for example, we investigated the effect of fill factor on the bandwidth, and we found that the bandwidth depends on the fill factor. Thus, higher ratios of active area to total area will generate broader bandwidth. We also have shown that 2-D axisymmetric models can be used to predict device performance. The 2-D model, which is least expensive computationally, was used to determine the optimum gap height; as the gap height shrinks, the output pressure per volt increases. The collapse voltage also decreases with decreased gap height. Therefore, by fabricating devices with very small gaps, it is possible to obtain higher pressure per volt at lower bias voltages. Note that this is only valid for small signals. However, the total output pressure that can be obtained is, perforce, limited by the gap height assuming there is no limitation on the applied AC voltage. However, this is not generally the case, especially for CMUTs with integrated electronics. Standard integrated circuit techniques impose limitations on the amplitude of the pulses that can be used to excite the membranes. Therefore, it is important to maximize the output pressure per volt to get the maximum pressure for a given pulse amplitude. Increasing the collapse voltage results in higher output pressures per volt. One way to increase the collapse voltage is to fabricate devices with larger gaps. Although this design increases the collapse voltage (and therefore the maximum bias that can be applied), it does not increase the output pressure per volt. The collapse voltage should be increased by using stiffer membranes. But this may reduce the bandwidth. Because there is a tradeoff between output pressure and bandwidth, the CMUT membrane design must be considered both of these parameters. In addition, when using the models simulating infinitely large transducers (both 3-D and 2-D) with symmetry boundaries around a single cell, care must be taken in calculating the electrical input impedance. In these models, the real part of the impedance converges to a specific value that is determined by the plane impedance of the medium, rather than vanishing toward low frequencies. The models developed in this study also can be used for large signal simulations in which nonlinearity of the electrostatic forces is included in the calculations. Small signal 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 micromachined electrostatic ultrasonic air transducer, IEEE Trans. Ultrason., Ferroelect., Freq. Contr., vol. 43, pp. 1 6, Jan [2] H. T. Soh, I. Ladabaum, A. Atalar, C. F. Quate, and B. T. Khuri-Yakub, Silicon micromachined ultrasonic immersion transducers, Appl. Phys. Lett., vol. 69, pp , Dec [3] I. Ladabaum, X. Jin, H. T. Soh, A. Atalar, and B. T. Khuri- Yakub, Surface micromachined capacitive ultrasonic transducers, IEEE Trans. Ultrason., Ferroelect., Freq. Contr., vol. 45, pp , May [4] D. M. Mills and L. S. Smith, Real time in-vivo imaging with capacitive micromachined ultrasonic transducer (cmut) linear arrays, in Proc. IEEE Ultrason. Symp., 2003, pp [5] O. Oralkan, A. S. Ergun, J. A. Johnson, M. Karaman, U. Demirci, K. Kaviani, T. H. Lee, and B. T. Khuri-Yakub, Capacitive micromachined ultrasonic transducers: Next-generation arrays for acoustic imaging?, IEEE Trans. Ultrason., Ferroelect., Freq. Contr., vol. 49, pp , Nov [6] D. M. 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14 2198 ieee transactions on ultrasonics, ferroelectrics, and frequency control, vol. 52, no. 12, december 2005 ear and nonlinear equivalent circuit simulation, in Proc. IEEE Ultrason. Symp., 2003, pp [11] A. Bozkurt, I. Ladabaum, A. Atalar, and B. T. Khuri-Yakub, Theory and analysis of electrode size optimization for capacitive microfabricated ultrasonic transducers, IEEE Trans. Ultrason., Ferroelect., Freq. Contr., vol. 46, pp , Nov [12] G. G. Yaralioglu, A. S. Ergun, B. Bayram, E. Haeggstrom, and B. T. Khuri-Yakub, Calculation and measurement of electromechanical coupling coefficient of capacitive micromachined ultrasonic transducers, IEEE Trans. Ultrason., Ferroelect., Freq. Contr., vol. 50, pp , Apr [13] B. Bayram, G. G. Yaralioglu, A. S. Ergun, and B. T. Khuri- Yakub, Influence of the electrode size and location on the performance of a CMUT, in Proc. IEEE Ultrason. Symp., 2001, pp [14] B. Bayram, E. Haeggstrom, G. G. Yaralioglu, and B. T. Khuri- Yakub, A new regime for operating capacitive micromachined ultrasonic transducers, IEEE Trans. Ultrason., Ferroelect., Freq. Contr., vol. 50, pp , Sep [15] M. Kaltenbacher, H. Landes, K. Niederer, and R. Lerch, 3D simulation of controlled micromachined capacitive ultrasound transducers, in Proc. IEEE Ultrason. Symp., 1999, pp [16] G. Wojcik, J. Mould, P. Reynolds, A. Fitzgerald, P. Wagner, and I. Ladabaum, Time-domain models of MUT array cross-talk in silicon substrates, in Proc. IEEE Ultrason. Symp., 2000, pp [17] Y. Roh and B. T. Khuri-Yakub, Finite element analysis of underwater capacitor micromachined ultrasonic transducers, IEEE Trans. Ultrason., Ferroelect., Freq. Contr., vol. 49, pp , Mar [18] S. Ballandras, A. Caronti, W. Steichen, M. Wilm, V. Laude, T. Pastureaud, R. Lardat, and W. Daniau, Simulation of cmut radiating in water using a mixed finite element/boundary element approach, in Proc. IEEE Ultrason. Symp., 2002, pp [19] J. W. Goodman, Introduction to Fourier Optics. New York: McGraw-Hill, [20] W. P. Mason, Electromechanical Transducers and Wave Filters. London: D. Van Nostrand, Goksen Goksenin Yaralioglu (S 92 M 99) was born in Akhisar, Turkey, on May 13, He received his B.S., M.S., and Ph.D. degrees from Bilkent University, Turkey, in 1992, 1994, and 1999, respectively, all in electrical engineering. He is now working as an engineering research associate in E. L. Ginzton Laboratory, Stanford University. His current research interests include design, modeling and applications of micromachined ultrasonic transducers, and atomic force microscopy at ultrasonic frequencies. Arif Sanli Ergun (S 91 M 99) was born in Ankara, Turkey, in He received his B.Sc., M.Sc., and Ph.D. degrees in 1991, 1994, and 1999, respectively, all in electrical and electronics engineering from Bilkent University, Turkey. He is now at the E. L. Ginzton Laboratory, Stanford University, as an engineering research associate. His main research interests are acoustics, ultrasound, microelectromechanical systems (MEMS), and microwave 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 the American University of Beirut, the M.S. degree in 1972 from Dartmouth College, and the Ph.D. degree in 1975 from Stanford University, all in electrical engineering. He joined the research staff at the E. L. Ginzton Laboratory of Stanford University in 1976 as a research associate. He was promoted to senior research associate in 1978, and to a Professor of Electrical Engineering (Research) in He has served on many university committees in the School of Engineering and the Department of Electrical Engineering. Presently, he is the Deputy Director of the E. L. Ginzton Laboratory, and the associate chairman for graduate admissions in the electrical engineering department at Stanford. Professor Khuri-Yakub has been teaching both at the graduate and undergraduate levels for over 20 years, and his current research interests include in situ acoustic sensors (temperature, film thickness, resist cure, etc.) for monitoring and control of integrated circuits manufacturing processes, micromachining silicon to make acoustic materials and devices such as airborne and water immersion ultrasonic transducers and arrays, and fluid ejectors, and in the field of ultrasonic nondestructive evaluation and acoustic imaging and microscopy. Professor Khuri-Yakub is a fellow of the IEEE, a senior member of the Acoustical Society of America, and a member of Tau Beta Pi. He is associate editor of Research in Nondestructive Evaluation, a Journal of the American Society for Nondestructive Testing. Professor Khuri-Yakub has authored over 400 publications and has been principal inventor or co-inventor of 60 issued patents. He received the Stanford University School of Engineering Distinguished Advisor Award, June 1987, and the Medal of the City of Bordeaux for contributions to NDE, 1983.