Electromechanical coupling factor of capacitive micromachined ultrasonic transducers Alessandro Caronti, a) Riccardo Carotenuto, and Massimo Pappalardo Dipartimento di Ingegneria Elettronica, Università Roma Tre, Via della Vasca Navale 84, 00146 Roma, Italy Received 15 February 00; revised 15 May 00; accepted 14 October 00 Recently, a linear, analytical distributed model for capacitive micromachined ultrasonic transducers CMUTs was presented, and an electromechanical equivalent circuit based on the theory reported was used to describe the behavior of the transducer IEEE Trans. Ultrason. Ferroelectr. Freq. Control 49, 159 168 00. The distributed model is applied here to calculate the dynamic coupling factor k w of a lossless CMUT, based on a definition that involves the energies stored in a dynamic vibration cycle, and the results are compared with those obtained with a lumped model. A strong discrepancy is found between the two models as the bias voltage increases. The lumped model predicts an increasing dynamic k factor up to unity, whereas the distributed model predicts a more realistic saturation of this parameter to values substantially lower. It is demonstrated that the maximum value of k w, corresponding to an operating point close to the diaphragm collapse, is 0.4 for a CMUT single cell with a circular membrane diaphragm and no parasitic capacitance 0.36 for a cell with a circular plate diaphragm. This means that the dynamic coupling factor of a CMUT is comparable to that of a piezoceramic plate oscillating in the thickness mode. Parasitic capacitance decreases the value of k w, because it does not contribute to the energy conversion. The effective coupling factor k eff is also investigated, showing that this parameter coincides with k w within the lumped model approximation, but a quite different result is obtained if a computation is made with the more accurate distributed model. As a consequence, k eff, which can be measured from the transducer electrical impedance, does not give a reliable value of the actual dynamic coupling factor. 003 Acoustical Society of America. DOI: 10.111/1.157958 PACS numbers: 43.38.Bs, 43.38.Ar SLE I. INTRODUCTION a Telephone: 39 06 55177081; fax: 39 06 5579078; electronic mail: caronti@uniroma3.it Historically, the concept of the coupling factor of an electromechanical transducer, usually called a k factor, was introduced to characterize its ability to convert electrical energy into mechanical energy, and vice versa. Following the early work by Mason concerned with piezoelectric crystals,,3 the designers of transducers started to use the coupling factor both to characterize different piezomaterials and as an index of performance of transducers under practical configurations. Actually, coupling coefficients not only depend on the type of material, but also on the stress distribution, electric field, and geometry of the piezoelement. Indeed, the coefficients known as material coupling factors k mat e.g., k t, k p, k ij ), refer to a one-dimensional geometry and to a static or quasistatic energy transformation cycle, and can be easily computed as a combination of appropriate elastic, dielectric, and piezoelectric constants. In recent years, capacitive micromachined ultrasonic transducers have shown to be a promising alternative to piezoelectric transducers, especially in ultrasound imaging and nondestructive testing. 4,5 The ease of fabrication, integration with custom electronics, wide bandwidth in immersion operation, and large dynamic range of the CMUT have indicated that this technology is potentially better than piezoelectric technology for the realization of two-dimensional -D phased arrays for 3-D imaging. 6,7 For electrostatic transducers, including CMUTs, only a k factor has to be defined if the geometry of the membrane is assumed, as usual, to be circular; further, it is common use to refer to a dynamic energy transformation cycle rather than to a static one. Basic references for the dynamic coupling factor of electrostatic transducers are the book by Kinsler et al. 8 and the book by Hunt. 9 In Ref. 8, the coupling factor is defined in a general way as a ratio of energies, in Ref. 9 it is explicitly defined as a combination of parameters. The two approaches substantially give the same result, except for a small discrepancy that will be discussed in detail in the present work. Also relevant to the present work are recent papers reporting calculation of quasistatic coupling coefficients for electrostrictive ceramics. 10 1 In Ref. 10 the method of computing the coupling coefficient for an electrostrictor is based on a generalization of the IEEE Standard on Piezoelectricity, 13 while the definition proposed in Ref. 1 generalizes that advanced by Berlincourt et al. 14 in a way that produces a zero value of the coupling coefficient for an unbiased electrostrictive material. Significantly, a universal equivalent circuit that is applicable to electrostrictive, piezoelectric and electrostatic transducers, based on the linearized 3-D theory of lead magnesium niobate PMN, is derived in Ref. 11, and the theoretical coupling coefficients for piezoelectric and electrostatic transducers are recovered J. Acoust. Soc. Am. 113 (1), January 003 0001-4966/003/113(1)/79/10/$19.00 003 Acoustical Society of America 79
as special cases. In particular, the expression of the coupling factor of a single-sided electrostatic transducer given by Hunt is obtained with an appropriate correspondence of parameters. Although the lumped equivalent circuit given in Ref. 11, as well as Hunt s theory, can describe the first-order behavior of the electrostatic transducer, the flexural bending of the diaphragm is not adequately taken into account for coupling factor evaluation. In this paper, starting from a definition based on converted and stored energies involved in a vibration cycle, the dynamic coupling factor k w of a lossless CMUT is calculated by using both lumped and distributed parameter models. The use of a distributed model is essential to account for the flexural vibration of the membrane in the electrostatic cell. Actually, it is found that k w approaches a peak value of about 0.4 as the bias voltage increases, unlike the lumped model prediction of an increasing k factor up to unity. The classical effective coupling factor k eff, which can be easily measured from the transducer electrical impedance, is also investigated. It is demonstrated that k eff computed with the lumped model coincides with k w, but this identity drops if a more realistic calculation is performed by using the distributed model. II. COUPLING FACTOR DEFINITIONS As mentioned in the Introduction, Kinsler et al. 8 define the k factor of an electrostatic transducer as the square root of the ratio of stored mechanical energy to total energy stored in a lossless vibration cycle, and derive a simple expression of k w using a lumped parameter model. Hunt 9 explicitly defines the coupling factor as a combination of parameters resulting in a slightly different expression, as will be discussed later. According to the latest IEEE Standard on Piezoelectricity, 13 the coupling factors are nondimensional coefficients that are useful to characterize a piezoelectric material under a particular stress and electric field configuration in the conversion of stored energy into mechanical or electric work. A graphic illustration of the electromechanical conversion is reported, showing that the static piezoelectric coupling factor, denoted here as k mat, can be defined, with reference to a static or quasistatic lossless transformation cycle of a piezoelectric specimen, as the square root of the ratio of the converted energy W c to the total stored energy per unit volume W, k mat W c W. In dynamic conditions, that is, when the element is in oscillation, it is still possible to define the k factor as a ratio of energies, 15 k w E c E tot, where E c is the converted energy and E tot is the total energy involved in a vibration cycle. A very common k factor, the only one that can be easily measured from the electrical impedance, is the effective coupling factor, 13 1 FIG. 1. Schematic cross section of a CMUT cell. k eff f p f s, 3 f p where f s and f p are the frequencies of maximum and minimum conductance, respectively. As demonstrated by Lamberti et al., 15 the piezoelectric coupling factor computed according to Eq. coincides, for a lossless specimen, with the effective coupling factor given by Eq. 3, so that the same physical meaning can be attributed to k eff. In addition, k w is proportional to k mat according to the following relation: k w 8 k mat. 4 These results are also important for CMUTs for two reasons: first, by means of Eq. the coupling factor k w can also be computed for this kind of transducer, and a comparison with the piezoelectric coupling factor based on the same definition is then possible; second, k w can be calculated by using both lumped and distributed models, and it will be shown that more realistic results are obtained with the latter. Calculations of electromechanical coupling coefficients for CMUTs according to definitions and 3 will be presented in the next section. III. COUPLING FACTOR OF CMUTS A schematic of a CMUT single cell, consisting of a metallized diaphragm top electrode stretched over a heavily doped silicon substrate bottom electrode, is shown in Fig. 1. Many such elements are electrically connected in parallel to make the transducer, as can be seen from the portion of a CMUT shown in Fig.. The details of the fabrication process from different research groups can be found in Refs. 16 18. The dynamic coupling factor can be computed by considering an idealized vibration cycle in which the energy is converted either to electric or mechanical work, the result being the same as demonstrated in Ref. 13 for a static transformation cycle of a piezoelectric specimen. In the following subsections, a lossless dynamic cycle in which the conversion of energy is from mechanical to electrical work is considered. With reference to the equivalent circuit of a CMUT driven in transmission shown in Fig. 3, 1 this type of conversion takes place when the transducer does not radiate energy, i.e., Z r 0, and it is disconnected from the electrical source e.g., Z s ). In these conditions, the 80 J. Acoust. Soc. Am., Vol. 113, No. 1, January 003 Caronti et al.: Capacitive micromachined ultrasonic transducers
FIG. 4. Lumped equivalent circuit of a single cell for the coupling factor calculation. FIG.. Top view of a CMUT, consisting of a -D array of circular membranes. The regions in light gray are membrane electrodes and upper interconnections, the holes around each membrane are used to form the cavity below by selective etching of a sacrificial material. CMUT is in free oscillation and part of the energy of vibration, in each lossless cycle, is converted into electrostatic energy and stored in the transducer capacitance. The dynamic coupling factor will be computed for a single electrostatic cell by using both lumped and distributed parameter models; the calculation for an ideal CMUT with all identical elements, including parasitic capacitance, is also reported. A. Lumped parameter model Consider a CMUT single cell with no internal losses, driven by a sinusoidal voltage V V ac e j t and working in vacuum, so that no acoustic energy is radiated. In order to calculate the electromechanical coupling factor we make use of the equivalent circuit of Fig. 4, which is obtained from the circuit of Fig. 3 by setting Z s and Z r 0, and using lumped parameters in place of Z Eb and Z m, whose real parts are vanishing in the absence of losses. In the circuit of Fig. 4, C 01 is the static capacitance of the cell, L e is the electrical inductance corresponding to the mass of the diaphragm, and C e is the electrical capacitance associated to the mechanical compliance. These parameters can be obtained by expanding the expression for the mechanical impedance around the fundamental resonance frequency of the diaphragm, 19 that is, around the point where the device is FIG. 3. Electromechanical equivalent circuit of a transmitting CMUT. Z s is the source impedance, Z Eb is the blocked electrical impedance, Z m is the mechanical impedance, and Z r is the radiation impedance. According to phasor notation, boldface is used to represent quantities in the frequency domain. intended to operate. Depending on the structural model used for the diaphragm, analytical expressions for the mechanical impedance are reported in Ref. 1. If the diaphragm is modeled as a membrane, the lumped inductance, and capacitance are found to be L e m m 1 1.446 ss 1 1, 5 C m e 1 c m 0.9568 1 8, 6 where m and c m are the effective mass and compliance of the membrane around the resonance frequency, 1 is the transformation factor of the cell, s is the surface mass density, is the membrane tension per unit length, and S 1 is the area of the diaphragm. If the diaphragm is modeled as a plate, the lumped parameters are L e p 1.883 ss 1 1 C p e 0.977S 1 1 19 D, 8 where D is the flexural rigidity of the plate. The transformation factor is defined as 1 1 C 01V DC, 9 d 0 where V DC is the bias voltage and d 0 is the effective distance between the electrodes, including the static displacement of the diaphragm. Suppose now that the diaphragm is in free oscillation with a velocity v(t), the electrical source being disconnected from the transducer. The coupling factor can be easily computed by setting in the inductor an initial current I 0 at t 0, representing the amplitude of the transverse velocity, and calculating the transient of the circuit of Fig. 4. The electrical current in the loop results, i t I 0 cos 0 t, where 0 is the natural frequency, 0 1 L e C s, 7 10 11 J. Acoust. Soc. Am., Vol. 113, No. 1, January 003 Caronti et al.: Capacitive micromachined ultrasonic transducers 81
FIG. 5. Dynamic coupling factor of a CMUT single cell with membrane and plate diaphragm, computed in the lumped parameter approximation. and C s is the series capacitance of C e and C 01. The kinetic energy is given by KE t 1 L ei 0 cos 0 t, and the voltage across the capacitance C 01 is 1 v C0 t I 0 sin 0 C 0 t. 13 01 When the diaphragm passes through the position of static deflection, the strain energy variation with respect to the bias point is zero and all the energy is in kinetic form. In this situation, the total energy involved in a vibration cycle can be calculated from Eq. 1 as E tot KE max 1 L ei 0. 14 On the other hand, when the diaphragm has its maximum displacement, the kinetic energy is zero and the electrostatic energy stored in C 01, representing the energy converted from mechanical into electrical form, is given by see Eq. 13 EE 1 01 C I 0 0 C 01 C s C 01 1 L ei 0. According to Eq., the dynamic coupling factor is k w EE E tot C s C 01 C e C e C 01. 15 16 A plot of k w is shown in Fig. 5 as a function of the bias voltage V DC for a membrane and a plate diaphragm. The physical parameters of the CMUT element used for simulation are listed in Table I CMUT III. The spring-softening effect, 0 that is, the increasing compliance of the diaphragm with increasing bias voltage, has not been taken into account. As can be seen, the coupling factor of a membrane is higher than that of a plate, owing to the fact that a clamped membrane has a higher average displacement. The difference gets lower as the bias voltage, hence the displacement, decreases. The electrostatic spring-softening can be included in the lumped model by roughly decreasing the stiffness of the diaphragm by the quantity 1 /C 01, 8,9,0 so that the mechanical compliance changes from c m to TABLE I. Parameters of the CMUTs used for coupling factor calculations. Parameters CMUT I a CMUT II b CMUT III c CMUT IV d Membrane radius a ( m) 5.5 0.0 0.0 5.0 Membrane thickness d m ( m) 0.5 0.6 0.45 0.6 Membrane collapse voltage V cr V) 155 11 8 74 Gap height d g ( m) 1.0 0.35 0.40 0.35 Resonance frequency f R (MHz).3 6.9 5. 4.4 Number of elements 9510 1510 1510 930 Total surface area mm 100.0 3.4 3.4 3.4 a Data from Ref. 16. b Unpublished data. c Data from Ref. 17. d Unpublished data. 8 J. Acoust. Soc. Am., Vol. 113, No. 1, January 003 Caronti et al.: Capacitive micromachined ultrasonic transducers
FIG. 6. Dynamic coupling factor of a CMUT single cell including or not the electrostatic spring softening. A lumped parameter approximation is used for computation. c m c m 1 1 c m C 01, and the coupling factor given by Eq. 16 becomes k w 1 c m 1 c m C 01 1 c m C 01 C e C 01. 17 18 This relation, that was derived according to Eq., coincides with the electromechanical coupling coefficient of a single electrostatic unit as explicitly defined by Hunt Ref. 9, p. 181. An ambiguity exists in the literature between Refs. 9 and 8, because in Ref. 8, p. 351 the reported expression of the coupling factor is the same as Eq. 16, which does not include the spring-softening effect. A comparison between coupling factors including or not this effect is shown in Fig. 6. Note that the discrepancy becomes important as the bias voltage increases. As far as effective coupling factor is concerned, according to Eq. 3 the frequencies f s and f p, in the absence of losses, coincide with the frequencies f r and f a of infinite and zero admittance, respectively. The input electrical admittance of the circuit of Fig. 4 is 1 Y Ei j C 01. 19 L e 1/ C e Setting Y Ei and Y Ei 0, the resonance and antiresonance frequencies turn out to be f r 1/( L e C e ) and f a 1/( L e C s ), respectively. The substitution into Eq. 3 yields k eff f a f r f a C e C e C 01. 0 Thus, as anticipated in the Introduction, the effective coupling factor k eff, within the validity of the lumped approximation, has the same value as the dynamic coupling factor k w without spring softening see Eq. 16. It is interesting to note that if k eff is computed as ( f a f r )/f r, the result coincides with k w as given by Eq. 18, including the springsoftening effect. A better evaluation of the variation of the coupling coefficients with the polarizing voltage is provided by the distributed model, as discussed in the next section. B. Distributed parameter model In the previous section we computed the dynamic coupling factor of a single electrostatic cell by using a lumped parameter circuit. In this section we calculate the coupling factor by using a more accurate distributed model of CMUTs, 1 including parasitic capacitance and the springsoftening effect. Consider a CMUT with no losses, consisting of n identical microelements electrically connected in parallel, and driven by a sinusoidal voltage V V ac e j t. With reference to the equivalent circuit of Fig. 3, the input electrical admittance can be expressed by the sum of two contributions: Y Ei I V Y Eb Y Em, 1 where Y Eb is the blocked admittance and Y Em is the motional admittance. 8 At each antiresonance frequency, Y Ei 0 and the charge on the electrodes remains unchanged while the diaphragm is oscillating, because the signal source is electrically insulated from the transducer. This condition permits us to evaluate the dynamic coupling factor, as done in the previous section. Again, the total energy involved in a vibration cycle is the kinetic energy at its maximum value that can be obtained by multiplying by n the kinetic energy of a single membrane, J. Acoust. Soc. Am., Vol. 113, No. 1, January 003 Caronti et al.: Capacitive micromachined ultrasonic transducers 83
FIG. 7. Dynamic coupling factor of a CMUT single cell with a membrane and plate diaphragm and no parasitic capacitance. Simulations based on lumped and distributed models are shown. KE max n 1 s S 1 t ds 1, max where a is the radius and k s membrane, is the wave number of the where (r,t) is the symmetric transverse displacement, whose analytical expression depends on the diaphragm model. For a clamped circular membrane without any interaction with the surroundings, the transverse displacement in frequency domain is 1 k s 1 0V DC / s s d 3 0. 4 r, 1V J 0 k s r J 0 k s a k s a, 3 J 0 k s a For a clamped circular plate, the transverse displacement has the expression 1 r, 1 V S 1 D J 0 K s r J 0 K s a I 1 K s a I 0 K s r I 0 K s a J 1 K s a K s a 4 J 0 K s a I 1 K s a I 0 K s a J 1 K s a, 5 where the parameter K s is given by K s 4 s D 0V DC 3. 6 Dd 0 The integral in has been computed numerically for both membrane and plate models. The electrical energy stored in a vibration cycle is EE 1 C 0a C p V ac, 7 where C 0a nc 01 is the active capacitance of the transducer and C p is the parasitic capacitance. The active capacitance of a single cell has been evaluated numerically, including the static deformation produced by the polarizing voltage and neglecting the fringing field. Thus, also the value of C 0a slightly depends on the bias voltage. Antiresonance frequencies can be calculated from Eq. 1 setting I dq/dt 0. If mechanical and electrical losses, as well as radiation in the surrounding medium, are neglected, the blocked admittance is given by 1 Y Eb j C 0a C p, and the motional admittance is 8 Y Em Im Z m, 9 where (C 0a V DC )/d 0 is the transformation factor of the CMUT, and Z m nz m1 is the mechanical impedance of the totality of membranes. Numerical solutions of Eq. 1 with I 0 have been obtained for several values of the bias voltage, and the corresponding antiresonance frequencies have been used to evaluate the total energy; see Eqs. 6. Finally, k w has been computed according to Eq. with E c given by Eq. 7. Figure 7 shows a comparison between the lumped and 84 J. Acoust. Soc. Am., Vol. 113, No. 1, January 003 Caronti et al.: Capacitive micromachined ultrasonic transducers
distributed model calculations of the dynamic coupling factor of a single cell (n 1) with no parasitic capacitance (C p 0), both including the spring-softening effect. For low values of the bias voltage, the lumped model prediction approximates to that of the distributed model. For higher values, the coupling factor yielded by the distributed model is lower, because the pistoń-like motion of the diaphragm in the lumped model is more effective in the energy conversion than the actual flexural motion. In other words, because of the actual nonuniform energy distribution over the membrane area, a greater kinetic energy must be supplied in a vibration cycle to convert the same electrical energy as the piston motion does in the lumped model. As can be seen, a quite different prediction results from distributed model as the bias voltage reaches its critical value; indeed, the coupling factor has a saturation-like behavior with a peak value in proximity of the collapse voltage, in contrast with the increase nearly up to unity predicted by the lumped model. Since the CMUT performance improves with increasing bias voltage, the transducer is frequently biased near the collapse, and an investigation of this operating condition is then required. This will be done in the next section. IV. ANALYSIS OF THE COLLAPSE POINT A. Lumped parameter model When a polarizing voltage is applied to the diaphragm of an electrostatic cell, the displacement can be found by solving the static equilibrium equation, which is obtained by equating the restoring force to the electrostatic force. If the cell is represented as a parallel plate capacitor and the restoring force is assumed to be linear, the equilibrium equation can be written as 1 x c DC 0S 1 V DC m0 d 0 0 S 1 V DC d g x r DC, 30 d m where d m is the thickness, r is the relative permittivity and c m0 is the low-frequency mechanical compliance of the diaphragm, d g is the gap height and x DC is the static displacement. When the bias voltage reaches a critical value, the parabolic curve of the electrostatic force, given by the right-hand side of Eq. 30, becomes tangent to the straight line representing the elastic restoring force, given by the left-hand member. In this situation, the slope of the two curves must be the same at the point of tangency, and derivative of Eq. 30 with respect to x DC yields 1 0S 1 V DC 3. 31 c m0 d 0 Equations 30 and 31 define the collapse point of the diaphragm; 9,1 beyond the critical voltage, the electrostatic force overwhelms the restoring force and the diaphragm collapses over the fixed electrode. Solving Eqs. 30, 31 for x DC and V DC, the maximum equilibrium displacement is found to be x cr d e 3, 3 and the corresponding collapse voltage is V cr 8 7 d e 3 c m0 0 S 1, 33 FIG. 8. Dynamic coupling factor of a CMUT, computed with the distributed model, as a function of the resonance frequency shift normalized to the fundamental frequency R0 without bias effects. J. Acoust. Soc. Am., Vol. 113, No. 1, January 003 Caronti et al.: Capacitive micromachined ultrasonic transducers 85
FIG. 9. Coupling coefficients of a CMUT single cell with a plate diaphragm. where d e d m / r d g is the effective distance between the electrodes without polarization. Using these relations in Eq. 9 and substituting in Eqs. 6 and 8 with c m 0.9568 c m0 for a membrane (c m 0.977 c m0 for a plate, the parameters C (m) e, C (p) e turn out to be approximately equal to the static capacitance of the cell. For a membrane we have C m e C cr 01 V d e x cr 0.9568c m0 0.9568C cr 01, 34 while for a plate C (p) e 0.977 C (cr) 01, where C (cr) 01 0 S 1 /(d e x cr ) is the critical value of C 01. Thus, the coupling factor given by Eq. 18 approaches unity. Actually, as reported in Ref. 9, pp. 184 185, membrane collapse usually occurs at values of the normalized displacement x cr /d e substantially less than the theoretical limit given by Eq. 3. For example, in Ref. 9 was found, for an electrostatic device experimentally tested, that fall-in occurred at a value of x cr /d e around 0.. As noted, this premature collapse was likely to be caused by the fact that the central portion of the diaphragm reaches the critical spacing while the average static displacement x DC is still lower than its critical value. With this value of x cr, the critical parameter C e is found to be about 0.55 both for membrane and plate diaphragms, and the resulting maximum value of the coupling factor is 0.74. The validity of these results is, however, limited to the lumped parameter approximation. In the next paragraph, the coupling factor will be investigated near the critical point by using the distributed parameter model. FIG. 10. Coupling factor of several CMUTs with membrane diaphragm cells. Parasitic capacitance has not been considered in the simulations. 86 J. Acoust. Soc. Am., Vol. 113, No. 1, January 003 Caronti et al.: Capacitive micromachined ultrasonic transducers
FIG. 11. Coupling factor of a CMUT with plate diaphragm cells for several values of the parasitic capacitance in percent of the active capacitance. B. Distributed parameter model As it is known, the condition for the fundamental resonance frequency R of a clamped circular membrane is ka.405, 8 which substituted into Eq. 4 for an electrostatically excited membrane yields Ra / s.405 0V DC a 3. 35 d 0 Note that, according to the spring-softening behavior, the resonance frequency decreases as the polarizing voltage increases. The absolute maximum value of V DC is the one that makes the right-hand side of Eq. 35 equal to zero and the collapse voltage must be proportional to this value, i.e., V cr d e 3 0 d 3 e a 8c m0 0 S 1, 36 where 0 1 is a numerical coefficient. As can be noted, Eq. 36 exhibits the same functional dependence as Eq. 33, which is derived according to the lumped model. If the coupling factor k w is calculated, in the way illustrated in the previous section, as a function of the relative amount R / R0 by which the fundamental frequency drops because of the bias voltage, as given by Eq. 35, the resulting plot is shown in Fig. 8. The effective coupling factor, computed according to Eq. 0 with f a and f r numerically evaluated, is also shown. As it is possible to see, unlike the lumped model predictions, for the distributed model the effective coupling factor k eff does not coincide with k w, so that a measure of k eff from the CMUT electrical impedance does not represent a reliable value of k w. Also note that, as soon as R changes of 15% of its nominal value, the coupling factor k w is at 85% of its maximum value. Figure 9 shows the behavior of k w and k eff of a single cell with a plate diaphragm as a function of V DC, both computed with lumped and distributed models. It is notable that all the k factors draw together for low values of the bias voltage. A result worthy of note is the one shown in Fig. 10, where k w is plotted for several CMUTs with circular membranes having different dimensions, stress, fundamental frequency, and collapse voltage, as reported in Table I. As one can see, the maximum value of the coupling factor yielded by the distributed model is about 0.4, regardless of the transducer design. This value seems to be reasonably a bounding value of the coupling factor k w of an electrostatic cell with a circular membrane. For a piezoceramic plate oscillating in thickness mode, the dynamic k factor is a reduced value of the thickness material coupling factor k t, which usually takes values in the range 0.45 0.50. If this range is used into Eq. 4, the corresponding range of k w turns out to be 0.40 0.45. Thus, the coupling factor of a CMUT with circular membranes is very close to that of a piezoceramic oscillating in the thickness direction. Finally, the coupling coefficient of a CMUT is worsened by the parasitic capacitance, for it does not contribute to the energy conversion. A plot of k w for several values of C p in percent of the active capacitance C 0a is shown in Fig. 11. V. CONCLUSIONS In this paper, a definition of the electromechanical coupling coefficient, involving the energies stored in a dynamic vibration cycle, has been used to calculate the dynamic coupling factor k w for lossless CMUTs by using both lumped and distributed parameter models. A strong discrepancy exists between the lumped and distributed model predictions of k w as the bias voltage reaches its critical value. The maximum value of k w, corresponding to membrane collapse, is close to unity for the lumped model, whereas a quite lower value of about 0.40 for a cell with membrane diaphragm J. Acoust. Soc. Am., Vol. 113, No. 1, January 003 Caronti et al.: Capacitive micromachined ultrasonic transducers 87
0.36 for plate diaphragm, which is approached with a saturation-like behavior, is predicted by the distributed model. Further, calculations of k w for several CMUTs with circular membranes, having different geometry and stress, have shown that the peak value, which is reached at different critical voltages, is always 0.40. This result permits us to state that the coupling factor of a CMUT is very close to that of a piezoceramic plate oscillating in thickness direction, for which the dynamic k factor has typically values between 0.40 and 0.45. The effective coupling coefficient k eff has also been investigated, showing that this parameter is a good estimation of the device coupling factor only for low values of the bias voltage, far from the critical point. Actually, k eff coincides with k w within the lumped model approximation, but when it is computed using a more accurate distributed model this coincidence drops and, for practical higher voltages, k eff must be abandoned as a reliable measure of the CMUT coupling factor. 1 A. Caronti, G. Caliano, A. Iula, and M. Pappalardo, An accurate model for capacitive micromachined ultrasonic transducers, IEEE Trans. Ultrason. Ferroelectr. Freq. Control 49, 159 168 00. W. P. Mason, An electromechanical representation of a piezoelectric crystal used as a transducer, Proc. IRE 3, 15 163 1935. 3 W. P. Mason, Piezoelectric Crystals and Their Applications to Ultrasonics Van Nostrand, New York, 1950. 4 M. I. Haller and B. T. Khuri-Yakub, A surface micromachined ultrasonic air transducer, IEEE Trans. Ultrason. Ferroelectr. Freq. Control 43, 1 6 1996. 5 X. C. Jin, I. Ladabaum, F. L. Degertekin, S. Calmes, and B. T. Khuri- Yakub, Fabrication and characterization of surface micromachined capacitive ultrasonic immersion transducers, IEEE/ASME J. Microelectromech. Syst. 8, 100 114 1999. 6 O. Oralkan, X. C. Jin, K. Kaviani, A. S. Ergun, F. L. Degertekin, M. Karaman, and B. T. Khuri-Yakub, Initial pulse-echo imaging results with one-dimensional capacitive micromachined ultrasonic transducer arrays, IEEE Ultrasonics Symposium IEEE, New York, 000, pp. 959 96. 7 U. Demirci, O. Oralkan, J. A. Johnson, A. S. Ergun, M. Karaman, and B. T. Khuri-Yakub, Capacitive micromachined ultrasonic transducer arrays for medical imaging: experimental results, IEEE Ultrasonics Symposium IEEE, New York, 001, pp. 957 960. 8 L. E. Kinsler, A. R. Frey, A. B. Coppens, and J. V. Sanders, Fundamentals of Acoustics Wiley, New York, 198. 9 F. V. Hunt, Electroacoustics. The Analysis of Transduction, and Its Historical Background American Institute of Physics, Woodbury, NY, 198. 10 C. L. Hom, S. M. Pilgrim, N. Shankar, K. Bridger, M. Massuda, and S. R. Winzer, Calculation of quasi-static electromechanical coupling coefficients for electrostrictive ceramic materials, IEEE Trans. Ultrason. Ferroelectr. Freq. Control 41, 54 551 1994. 11 J. C. Piquette and S. E. Forsythe, Generalized material model for lead magnesium niobate PMN and an associated electromechanical equivalent circuit, J. Acoust. Soc. Am. 104, 763 77 1998. 1 J. C. Piquette, Quasistatic coupling coefficients for electrostrictive ceramics, J. Acoust. Soc. Am. 110, 197 07 001. 13 ANSI/IEEE Std. 176/1987, IEEE Standard on Piezoelectricity IEEE, New York, 1987. 14 D. A. Berlincourt, D. R. Curran, and H. Jaffe, Piezoelectric and piezomagnetic materials and their function in transducers, in Physical Acoustics, edited by W. P. Mason Academic, New York, 1964, Vol. IA, pp. 169 70. 15 N. Lamberti, A. Iula, and M. Pappalardo, The electromechanical coupling factor in static and dynamic conditions, Acustica 85, 39 46 1999. 16 X. C. Jin, I. Ladabaum, and B. T. Khuri-Yakub, The microfabrication of capacitive ultrasonic transducers, IEEE/ASME J. Microelectromech. Syst. 7, 95 30 1998. 17 A. Caronti, H. Majjad, S. Ballandras, G. Caliano, R. Carotenuto, A. Iula, V. Foglietti, and M. Pappalardo, Vibration maps of capacitive micromachined ultrasonic transducers by laser interferometry, IEEE Trans. Ultrason. Ferroelectr. Freq. Control 49, 89 9 00. 18 O. Ahrens, D. Hohlfeld, A. Buhrdorf, O. Glitza, and J. Binder, A new class of capacitive micromachined ultrasonic transducers, in Ref. 6, pp. 939 94. 19 W. P. Mason, Electromechanical Transducers and Wave Filters Van Nostrand, New York, 1943. 0 I. Ladabaum, X. C. Jin, H. T. Soh, A. Atalar, and B. T. Khuri-Yakub, Surface micromachined capacitive ultrasonic transducers, IEEE Trans. Ultrason. Ferroelectr. Freq. Control 45, 678 690 1998. 1 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. Ferroelectr. Freq. Control 46, 1364 1374 1999. 88 J. Acoust. Soc. Am., Vol. 113, No. 1, January 003 Caronti et al.: Capacitive micromachined ultrasonic transducers