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

    Finite-Element Analysis of Capacitive Micromachined 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 a commercially available finite-element modeling (FEM) soft- ware. 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 paral- lel 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 bi- ased voltage. Calculations then were performed for immer- sion 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 bound- ary conditions, was used to calculate the output pressure. The results were compared with a simpler model: a single membrane in immersion, with symmetry boundary condi- tions 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 di- mensions on the output pressure and bandwidth. Our cal- culations 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 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 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 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 and applications 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: goksenin@stanford.edu).

    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 spring softening capacitance (caused by electromechanical inter- action). 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 ac- curate 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 model- ing (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 cal- culate 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 mem- brane displacement for different operation regimes. Dy- namic FEM models also have been developed and used to estimate the output pressure and membrane displacement by performing time-domain analysis [15]. For the evalua- tion 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 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 membrane array that was distributed uniformly in lateral directions over a 2-D rectangular grid. Appropriate boundary condi- tions were applied on the sidewalls of the model, so that the single membrane was replicated in the lateral direc- tions. This model assumes that all the cells in the array operate 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. 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 ef- fect 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 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 electrostatic elements (TRANS126). The electrostatic membrane first was modeled; the model then was extended for immersion devices in which fluid-solid interaction was performed ac- curately by ANSYS. For immersion devices, the absorbing boundaries 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 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 ele- ments used in the model. The thickness and the radius of the circular membrane were 1.3 µm and 15 µm, respec- tively. The gap was 0.2 µm, and the top electrode was on the bottom surface of the membrane. The membrane was composed 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 electromechanical transducer elements (TRANS126), which are basically par- allel plate capacitors, the electrostatic forces applied be- tween the top and the bottom electrodes were included in the model. The elements apply opposite electrostatic at- traction 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 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, 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 num- ber 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 de- scribed in [12], [13] and the above approach, we compared the deflection profiles computed. When the top and bot- tom 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 sig- nal equivalent of the transducer

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