Analytical calculation of collapse voltage of CMUT membrane ...

2004 IEEE Ultrasonics SymposiumAnalytical Calculation of Collapse Voltage of CMUTMembraneAmin Nikoozadeh, Baris Bayram, Goksen G. Yaralioglu, and Butrus T. Khuri-YakubE. L. Ginzton Laboratory, Stanford University, Stanford, CAAbstract— Because the collapse voltage determines the operatingpoint of the capacitive micromachined ultrasonic transducer(CMUT), it is crucial to calculate and control this parameter.One approach uses parallel plate approximation, where a parallelplate motion models the average membrane displacement. Thisusually yields calculated collapse voltage 25 percent higher thanthe actual collapse voltage. More accurate calculation involvesfinite element method (FEM) analysis. However, depending onthe required accuracy, the computation time may require manyhours.In this paper, we propose a fast numerical algorithm for thecalculation of collapse voltage. The algorithm uses the parallelplate method to approximate the force distribution over themembrane, and then applies an analytical solution for the plateequation, loaded by the approximated force distribution. Usingthis method, we are able to calculate the collapse voltage in acouple of seconds, within 0.1 percent accuracy. We report on thecollapse voltage calculation results using our method for fourdifferent design structures. While computation time of ourmethod is about three orders of magnitude less than the finiteelement method, the percentage error of collapse voltagecalculation is, nevertheless, less than four percent in all the designstructures. The proposed algorithm is also suitable for theinclusion of any external force distribution on the membrane,such as atmospheric pressure.Keywords- Capacitive micromachined ultrasonic transducer,CMUT, collapse voltage.I. INTRODUCTIONCollapse voltage of a capacitive micromachined ultrasonictransducer (CMUT) is a critical parameter for employing thedevice at the optimum operating point. The operating DC biasvoltage determines the performance of the transducer. It alsodetermines the operating regime at which the device isoperated, such as conventional and collapsed mode [1].Therefore, accurate knowledge of the collapse voltage isimperative.Initial attempts to calculate collapse voltage depended onparallel plate approximation [2]. In this approach, a pistontransducer models the membrane displacement, and thedisplacement profile of the membrane is neglected. The pistonis held over an electrostatic gap by a spring whose complianceis determined by the average spring constant of the membrane.The collapse occurs when the electrostatic force gradientovercomes the gradient of mechanical restoring force exertedby the spring. The collapse occurs as soon as the displacementreaches one third of the gap. This method usually predicts acollapse voltage 30 to 40 percent greater than the actual value.Finite Element Method (FEM) simulation is considered themost reliable and accurate method to compute the collapsevoltage. On the other hand, FEM analysis requires iterationbetween electrostatic and mechanical solutions. Depending onthe mesh size, FEM calculation may take several hours tocomplete.In this paper, we present a semi-analytical method for thecalculation of membrane displacement and collapse voltage.The method depends on the known solution of the equation ofmotion for the membrane. We assume that the membrane isclamped at the edges. The electrostatic gap is divided intomany parallel plate capacitors. The electrostatic pressure isassumed to be constant within the each segment. Our methodprovides reasonably accurate results in much less time thanFEM analysis.II. DESCRIPTION OF THE METHODThe proposed method, which uses vertical segmentation ofthe gap, an analytical solution of the motion for the membrane,and the method of superposition and iteration, is explained inthe following sections.A. SegmentationTo find the collapse voltage and displacement profile of themembrane, the electrostatic force between the two electrodes inthe CMUT structure must be known. To approximate theelectrostatic force acting on the membrane, we divide theelectrostatic gap into several vertical segments. Each segmentis modeled as a usual parallel plate capacitor. Provided that thenumber of segments is large enough, the electrostatic force isassumed to be constant over the each segment. Fig. 1 showsthe cross section of a circular membrane with partialmetallization. As shown in this figure, the region underneaththe top electrode is divided into N segments. Due to thesymmetry, only half of the membrane is shown. Note that eachsegment is a circular ring.abb i1 … i … NFigure 1. Method of segmentation applied on a circular membrane0-7803-8412-1/04/$20.00 (c)2004 IEEE.2562004 IEEE International Ultrasonics, Ferroelectrics,and Frequency Control Joint 50th Anniversary Conference

2004 IEEE Ultrasonics Symposiumthe thickness of the plate to the outer radius. The smaller thisratio, the more accurate is the solution.D. Superposition and IterationWith the analytical closed-form solution for the equation ofmotion of a concentrically loaded circular plate, and thelinearity of the equation, we can use the method ofsuperposition to find the total solution due to the total forcedistribution. We employ the analytical solution to find aclosed-form solution for each of the forces, and thensuperimpose all the solutions.Because the electrostatic force is dependent on the gapheight, we need to iterate the solution. In the first iteration, theelectrostatic force is determined over the non-deflectedmembrane. The atmospheric pressure is added to theelectrostatic force, and with this set of forces, we calculate thefirst estimate of the displacement profile. The next forcedistribution is found by using the new deflection profile. Thesesteps are repeated until the solution converges within anacceptable error tolerance. The displacement of the center ofthe membrane is used as a criterion to determine if the solutionconverges to a final result.We used the geometry depicted in Fig. 3 to compare FEMand our calculation method. The membrane is underatmospheric pressure. We assumed the bottom electrode coversthe surface of the substrate. In this case, the effect of parasiticcapacitance is enhanced.Deflection (µm)0-0.02-0.04-0.06-0.08-0.1ModelANSYS-0.120 5 10 15 20Radial distance (µm)Figure 4. Displacement profile of the membrane. The bias voltage is 150V,which is less than the collapse voltage.The collapse voltage is calculated using a binary searchalgorithm. As mentioned above, the displacement of the centerof the membrane is used as a criterion to determine whether thesolution converges or diverges. When the bias voltage ishigher than the collapse voltage, the displacement of the centerof the membrane diverges quickly. Fig. 5 compares thedisplacement of the center of the membrane versus number ofiterations for different DC bias voltages.10 µmSilicon Ox.( ε = 3.78)rVacuum gap20 µmSilicon( ε =11.7)rCenter displacement (µm)105V102V100VFigure 3. CMUT geometry. Top and bottom electrodes are shown by blacklines and assumed to be infinitesimally thin in ANSYS calculation.Fig. 4 compares the displacement profile of the middleplane of the membrane calculated using our method to theprofile calculated using the FEM analysis. Although thecomputation time is significantly smaller for our algorithm, ourresult is analogous to the FEM simulation, within an acceptableerror for practical purposes.E. Collapse voltageWhen the electrostatic force gradient overcomes therestoring mechanical force, the membrane will collapse ontothe substrate.Iteration numberFigure 5. Center displacement as function of iteration number for variousbias voltages. The collapse voltage is 104.5V.According to these plots, in all the cases where centerdisplacement converges to a finite value, the second derivativeof displacement (with respect to the number of iterations) isalways negative. For the divergence case, the second derivativeof the displacement (with respect to the number of iterations)becomes zero at a point, and then positive afterwards. Thisobservation can be used as a criterion to distinguish betweenconvergence and divergence, and to decide whether the appliedvoltage is above or below the collapse voltage. Therefore,using the binary search, we can find the collapse voltage withthe desired accuracy.0-7803-8412-1/04/$20.00 (c)2004 IEEE.2582004 IEEE International Ultrasonics, Ferroelectrics,and Frequency Control Joint 50th Anniversary Conference

2004 IEEE Ultrasonics SymposiumIII.RESULTSWe compared the results of our algorithm with FEMsimulation results, and found that 100 segments over the wholemembrane are adequate to calculate the collapse voltageaccurately. Our calculation took a few seconds; the samecalculation using FEM analysis would take several hours.Table 1 compares the result of our method to FEMsimulations for four different design configurations. All ofdevices are made from Silicon Nitride (Si 3 N 4 ), and have thesame membrane thickness of 1µm, insulator layer thickness of0.1µm, and gap distance of 1µm. In order to make a relativelybroad comparison, we changed the radius of membrane, radiusof top electrode, and location of top electrode. Location of topelectrode refers to the relative location of top electrode withrespect to the membrane: number “1” means that electrode iscompletely on top of the membrane; number “0” means thatelectrode has been placed underneath the membrane.Table 1 shows that our results match the FEM simulation.For all design cases, the results obtained using our method arewithin 5% of those obtained with FEM analysis. However, thecomputation time required for our method was approximatelythree orders of magnitude less than for FEM analysis.REFERENCES[1] Baris Bayram, Edward Hæggström, Goksen 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.[2] I. Ladabaum, Xuecheng 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. 678-690,May 1998.[3] S. Timoshenko and S. Woinowsky-Krieger, “Theory of Plates andShells”, McGraw-Hill book company, Inc., 2 nd ed. 1959.[4] W.P. Mason, “Electromechanical Transducers and Wave Filters”, D.Van Nostrand Company Inc., London, 1948.[5] F.V. Hunt, “Electroacoustics; the analysis of transduction, and itshistorical background”, Cambridge: Harvard University Press, 2 nd ed.1982.[6] Butrus T. Khuri-Yakub et al., “Silicon Micromachined UltrasonicTransducers,” Jpn. J. Appl. Phys. 39, 2883-2887 (May 2000).[7] Yaralioglu et al., “Calculation and measurement of electromechanicalcoupling coefficient of cMUT transducers,” IEEE Trans. Ultrason.,Ferroelect., Freq. Contr., vol. 50, pp. 449-456, April 2003.TABLE I.COMPARISON OF THE METHOD WITH ANSYSDesign parametersFour different design structures# 1 # 2 # 3 # 4Radius of membrane (µm) 50 50 50 25Radius of top electrode (µm) 50 10 50 25Top electrode position 1 1 0 1Our collapse voltage (volts) 166.7 352.5 131.8 637.7FEM collapse voltage (volts) 164.8 350.3 126 630.4Percentage error 1.15 0.64 4.6 1.16Our computation time (sec) ~ 2 ~ 2 ~ 2 ~ 4FEM computation time (hours) ~ 4 ~ 4 ~ 4 ~ 4IV. CONCLUSIONWe developed a semi-analytical method to calculate thedisplacement profile of a circular membrane. The same semianalyticalalgorithm was employed to compute the collapsevoltage of a circular CMUT membrane.Our tool is extremely useful for the design of cMUTdevices. By calculating the displacement profile, we can obtainother accurate and useful information, such as devicecapacitance and output pressure. . It is easy to include theatmospheric pressure or, in general, the pressure of themedium, as well as residual stresses. As shown in Table 1, theproposed algorithm is quickly calculated, and the accuracy isacceptable for most practical applications.In future work, we will include the fringing capacitance,and also improve the accuracy of the algorithm by consideringthe effect of shearing stresses and lateral pressures ondeflection of the membrane. We will also extend the samealgorithm to any membrane shape, such as square andhexagonal, where the analytical solution exists.0-7803-8412-1/04/$20.00 (c)2004 IEEE.2592004 IEEE International Ultrasonics, Ferroelectrics,and Frequency Control Joint 50th Anniversary Conference