The Design, Fabrication, and Testing of

36 JOURNAL OF MICROELECTROMECHANICAL SYSTEMS, VOL. 3, NO. I, MARCH 1994 The Design, Fabrication, and Testing of Corrugated Silicon Nitride Diaphragms Patrick R. Scheeper, Wouter Olthuis, and Piet Bergveld Abstract-Silicon nitride corrugated diaphragms of 2 mm x 2 mm x lpm have been fabricated with having depths of 410, or pm. The diaphragms with 4-pm-deep 8 circular corrugations, 14 corrugations show a measured mechanical sensitivity (increase in the deflection over the increase in the applied pressure) which is 25 times larger than the mechanical sensitivity of flat diaphragms I of equal size and thickness. Since this gain in sensitivity is due to reduction of the initial stress, the sensitivity can only increase in Fig. I. Schematic cross-sectional view of a circular corrugated diaphragm the case of diaphragms with initial stress. and characteristic parameters. A simple analytical model has been proposed that takes the influence to-run reproducibility is determined by the reproducibility of that the presence of initial tensile stress into account. The model predicts of corrugations increases the sensitivity of the diaphragms, because the initial diaphragm stress is reduced. The the initial conditions of the reactor (affected by contamination, model also predicts that for corrugations with a larger depth adsorbed water, previously deposited layers) and the accuracy the sensitivity decreases, because the bending stiffness of the of the instruments which control the reactor temperature, the corrugations then becomes dominant. These predictions have pressure and the mass flow of the process gases. Variations of been confirmed by experiments. a factor 2 of the initial stress have been measured. The application of corrugated diaphragms offers the possibility to control the sensitivity of thin diaphragms by geometrical It is advantageous if the mechanical sensitivity of the mi- parameters, thus eliminating the effect of variations in the initial crophone diaphragms is not determined by deposition process stress, due to variations in the diaphragm deposition process parameters. A possible method to achieve this may be the andlor the influence of temperature changes and packaging stress. application of a corrugated diaphragm, as shown in Fig. 1. In this figure, the corrugated diaphragm is provided with a flat center zone. It has first been shown by Jerman [8] S I. INTRODUCTION ILICON (condenser) microphones are provided with a that corrugated diaphragms can be made accurately in silicon thin diaphragm. Several diaphragm materials have been using micromachining techniques. It has been calculated that applied, as for example Mylar [I], [2], silicon nitride [31, 141, a corrugated zone in a diaphragm can reduce stress with a and silicon [SI, [6]. The size of the diaphragms is typically factor 1000-10000 [9]. Therefore, one of the applications between 1 x 1 mm2 and 2 x 2 mm2, with a typical thickness of corrugated diaphragms is the decoupling of a mechnical between 150 nm and lpm. The sensitivity of these thin sensor from its encapsulation [9], [lo], in order to reduce diaphragms to a sound pressure (typically between 20 pPa and the influence of temperature changes and packaging stress. 20 Pa) is strocgly dependent on the stress in the diaphragm. Recently, Spiering er al. [ 111 have demonstrated a reduction The tensile stress in thin silicon nitride films is typically of of thermal stress with at least a factor 120 in a corrugated the order of lo8 N/m2 [7]. The stress in silicon diaphragms is diaphragm. determined by the diaphragm fabrication process. The tensile Flat diaphragms show a nonlinear relation between the stress in heavily boron-doped silicon diaphragms is equal to deflection and the applied pressure. For relatively small deflec- 7 x lo7 N/m [6]. Bergqvist tions, this relation is approximately linear. The nonlinearity for er al. [5] fabricated silicon diaphragms with a low doping level and a negligible stress. large deflections is caused by stress due to stretching of the In general, thin film microphone diaphragms show an initial diaphragm. It has been experimentally shown that corrugated stress which strongly determines the sensitivity of the mi- diaphragms have a larger linear range than flat diaphragms [8], crophone. The initial stress can be controlled within certain [12], because of the achieved reduction of the radial stress in limits by the parameters of the deposition process. The run- the diaphragm. The reduced influence of thermal stress and packaging stress, and the larger linear range compared with flat diaphragms, make the corrugated diaphragms attractive Manuscript received May 14. 1993; revised November 4, 1993. Subject for specific applications, as for instance pressure sensors [8] Editor, N. de Rooij. This work was supported by the Dutch Foundation for Fundamental Research on Matter (FOM). and Fabry-Perot interferometers [ 131, [ 141. P. R. Scheeper was with the MESA Research Institute, University of The application of corrugated diaphragms in microphones %ente, 7500 AE Enschede, The Netherlands. He is now with Briiel and or capacitive pressure sensors may offer the possibility to Kjaer, 2850 Naerum, Denmark. W. Olthuis and P. Bergveld are with the MESA Research Institute, control the mechanical sensitivity of the diaphragm by means University of Twente, 7500 AE Enschede, The Netherlands. of the dimensions of the corrugations, which are often easier IEEE Log Number 92 15320. to control than the parameters of a deposition process. In 1057-7157/94$04.00 0 1994 IEEE SCHEEPER et al.: DESIGN OF CORRUGATED SILICON NITRIDE DIAPHRAGMS 37 this paper, the application of corrugated diaphragms for use in microphones is studied. Haringx [15] and Di Giovanni [16] have presented analytical models in which the effect of initial stress in the diaphragm is not included. The results of a finite element models describing the reduction of (thermal) stress by a corrugated zone were presented by Spiering et al. [9], by Ding [17], and by Zhang and Wise [SI. The deflection due to a homogeneous pressure was calculated by Ding [17] and by Zhang and Wise [18]. Unfortunately, the finite element calculations are only valid for a specific diaphragm geometry and size, and therefore the results cannot be applied to other diaphragms. In a recent paper, Spiering et al. [ 113 show that the flat center zone of a square diaphragm 150 r I 0 5 IO with a deep circular corrugation (deeper than 25pm) can be modeled as a stress-free circular membrane with a clamped edge and free radial movement. In this paper, it will be shown that for a diaphragm with shallow corrugations, the initial stress has a considerable effect on the mechanical behavior of the diaphragm. An analytical model will be presented in Section I1 for circular corrugated diaphragms with initial stress. In Section 111, the fabrication of corrugated diaphragms is discussed and in Section IV results of measurements are shown. 11. THE MECHANICAL SENSITIVITY OF CORRUGATED DIAPHRAGMS The center deflection, WO, of a flat, circular diaphragm with clamped edges and without initial stress, due to a homogeneous pressure, P, can be calculated from [8] P = 5.33--- Ed hi WO + 2.83--------- Ed hi WO” (1 - U’) R: hd (1 - u2) R; hi (1) where Ed, U, Rd, and hd are the Young’s modulus, the Pois- son’s ratio, the radius and the thickness of the diaphragm, respectively. It can be seen from (1) that if (wo/hd) << 1 the relation between the center deflection and the applied pressure is ipproximately linear. For larger values of (WO/&) the relation is nonlinear. In (1) it is assumed that the initial diaphragm stress can be neglected. Haringx [I51 presented a model to calculate the deflection of circular, stress-free corrugated diaphragms. Di Giovanni 1161 presented another set of equations, which are equal to the equations presented by Haringx for small diaphragm deflections, but provide a more accurate solution for large deflections (see the discussion on Fig. 11 in Section IV): where (3) -” ] (4) rig. 2. Example of a pressure-deflection curve of a stress-free circular flat and corrugated diaphragm without a flat zone. For the value of the parameters, see text. and for a sinusoidal corrugation profile where H is the depth of the corrugations, 1 is the corrugation spatial period, s is the corrugation arc length (see Fig. l), and q is the corrugation profile factor, which is larger than 1 for corrugated diaphragms. For flat diaphragms (H = O), q is equal to 1. In Section 111 it will be shown that the corrugated di- aphragms that have been fabricated show a rectangular cor- rugation profile. However, for shallow corrugations (H << l), the shape of the corrugations has only little influence on the profile factor q [15], 1161. In (2)-(5), it is assumed that the corrugated diaphragm is not provided with a flat center zone, in contrast to the corrugated diaphragm which will be presented in Section I11 (see also Fig. 1). This can be compensated for in the factor s/l, representing the ratio of the real distance between the diaphragm center and the edge, measured along the corrugation profile, and the diaphragm radius: where N is the number of corrugations. Note that (6) is only valid for a rectangular corrugation profile. In Fig. 2, typical load-deflections curves are shown for a stress-free flat and a stress-free (shallow) corrugated di- aphragm without flat zone, according to (1) and (2). In this example, a Poisson’s ratio of 0.3, a Young’s modulus of 3 x lo1’ N/m2, a radius of 1 mm, a diaphragm thickness of 1 pm, and a corrugation depth of 5 pm have been assumed. The assumed Young’s modulus is based on the processes used in our labs, yielding values of 3-4 x 10l1 N/m2. If only small deflections are considered (smaller than 3.6 pm in Fig. 2), it can be seen that the corrugated diaphragm is stiffer than a flat diaphragm. This has been confirmed by experiments, as can, for instance, be found in [ 151. The larger stiffness of the corrugated diaphragm is caused by the larger jexural rigidity in the tangential direction. The flexural rigidities in the radial direction, which depend on the thickness of the diaphragm material, are equal for the flat and the corrugated diaphragm. Note that bending occurs in the radial as well as the tangential 38 JOURNAL OF MICROELECTROMECHANICAL SYSTEMS. VOL. 3, NO. 1, MARCH 1994 estimate can be made by considering the cubic terms in the pressure-deflection curves. The stress, due to stretching of the middle plane of a diaphragm, can be estimated by considering the pressure- deflection curve of a diaphragm with a large initial stress @I: (a) (b) Fig. 3. A top view (a) and a cross-sectional view (b) of a deflected circular diaphragm. where r~d is the initial stress in the diaphragm. This equation has been rewritten so that it can be seen that the second term between the brackets represents the stress that is caused by direction. This can be elucidated with Fig. 3(a). Fig. 3(a) shows stretching of the middle plane of the diaphragm. Equation a top view of a circular stress-free diaphragm. The points (l), the pressure-deflection curve for a flat diaphragm without A, B, and C are at equal distances from the diaphragm center initial stress, can be rewritten in a similar way as (7). The 0. Fig. 3(b) shows a diametrical cross-sectional view of the second term between the brackets is then exactly the same as diaphragm. The z axis is an axis of rotational symmetry. It can in (7) and also represents the stress that is caused by stretching be seen from Fig. 3(b) that the diaphragm shows a curvature of the middle plane of the diaphragm. in the radial direction. For small deflections of the stress-free Rewriting (2), the pressure-deflection curve for a corrugated diaphragm, the points A and C are displaced vertically and diaphragm without initial stress, gives indicated as A‘ and C’. In the undeflected state, the normals from the points A, B, and C are all parallel to each other and perpendicular to the plane of the diaphragm. In the deflected state, the normals from the points A’ and C’ intersect the z axis at point D. Because of the rotational symmetry, the normals Comparing the second term between the brackets of (8) and from all points at the same distance from the diaphragm center (7), it can be seen that the stress, due to stretching of the middle 0 form a conical surface with apex D. Therefore, considering plane, has been reduced with a factor bp/2.83 in the corrugated diaphragm. Thus, if the behavior of a corrugated diaphragm the normals of the points A’ and B’, it can be concluded that with initial stress is described using superposition of a stress- the diaphragm also bends in the tangential direction. free corrugated diaphragm and a flat diaphragm with initial In the case of large diaphragm deflections (larger than 3.6 stress, a flat diaphragm with an initial stress of adbp/2.83 pm in Fig. 2), the tensile stress that occurs due to stretching of a diaphragm can no longer be neglected. The resulting instead of ffd should be considered. Since superposition is only valid for linear systems, only the linear terms of (7) and (8) increase in the stiffness of a diaphragm causes the nonlinear are used. Thus, the following equation is obtained: behavior. Since the corrugated diaphragm shows a smaller tensile rigidity in the radial direction than a flat diaphragm, the stress and the nonlinearity in a corrugated diaphragm are (9) smaller than in a flat diaphragm. The corrugated diaphragm has a smalleg stiffness and shows a larger deflection than a The mechanical sensitivity of a circular diaphragm is defined flat diaphragm, considering the high-pressure range (higher as than 50 Pa in Fig. 2). The large-deflection behavior has been demonstrated by Jerman for doped silicon corrugated diaphragms [8] and by Van Mullem for polyimide corrugated Therefore, the mechanical sensitivity of the corrugated di- diaphragms [ 121. aphragm with initial stress, for small deflections, is given The deflection of a corrugated diaphragm with initial stress by can be approximated by means of an analytical expression, which is a superposition of the linear model of a corrugated diaphragm without initial stress, and the linear (flat) mem- brane model which assumes a high initial tensile stress. The superposition is based on the assumption that the corrugated The behavior of a corrugated diaphragm is determined by diaphragm can be modelled as a fictitious flat diaphragm, the profile factor q. The number of corrugations, N, has only which locally has the same radial and tangential flexural little influence on the profile factor (see (5) and (6)), whereas rigidity as the corrugated diaphragm [15]. An initial stress q is nearly proportional with the corrugation depth, H (see is assumed to act on the fictitious flat diaphragm. The initial (5)). Therefore, the corrugation depth is the most effective stress is reduced by the presence of corrugations in the di- parameter to influence the behavior of corrugated diaphragms. aphragm. However, an estimate must be made of the reduction In Fig. 4 the mechanical sensitivity for small deflections of the initial stress caused by the corrugations. Since the of corrugated diaphragms without a flat zone is shown as a nonlinear behavior of diaphragm is caused by stress, and this function of the corrugation depth, H, for diaphragms with an nonlinearity is reduced by the presence of corrugations, the initial stress of lo7 N/m2(b) and lo8 N/m2(c> (equation (1 l)), SCHEEPER ef al.: DESIGN OF CORRUGATED SILICON NITRIDE DIAPHRAGMS 39 'j E ,j 0 5 10 15 Corrugation depth [urn] Fig. 4. The mechanical sensitivity of a corrugated diaphragm (without a flat (C) zone) without initial stress (a) and with an initial stress of lo7 N/m2 (b) and Fig. 5. Schematic representation of the fabrication process of cormgated lo8 N/mZ (c). iiaphragms with a flat zone. and for the stress-free case (a) (equation (2)). A Poisson's ratio of 0.3, a Young's modulus of 3 x 10I1N/m2, a radius of 1 mm, and a diaphragm thickness of 1 pm have been assumed. Note that for flat diaphragms (H = 0) the mechanical sensitivity reduces drastically if the initial stress increases. For stress-free corrugated diaphragms the mechanical sen- sitivity decreases if the corrugation depth increases. For the corrugated diaphragms with initial stress the mechanical sensi- tivity increases for small corrugation depths, due to reduction of the initial stress. For relatively large corrugation depths, the mechanical sensitivity decreases for increasing corrugation depths and approaches asymptotically the curve for stress- free corrugated diaphragms. The mechanical sensitivity of the diaphragm is then determined by geometrical parameters and Fig. 6. SEM photograph of a square silicon nitride diaphragm with a square not by the initial stress. It can be seen in the example from Fig. corrugation pattern. 4 that this is achieved for a corrugation depth larger than 10 pm. The ratio of the mechanical sensitivities lo7 of the dia hragms with an initial stress of N/m2 (b) and lo8 N/m B (c) has (c) Square diaphgms are made by anisotropic etching been reduced from a factor 10, for flat diaphragms, to a factor from the reverse side of the wafer in a 33 wt% KOH 1.10, for diaphragms with a corrugation depth of 10 pm. solution (73\"C), using the LPCVD silicon nitride on the Therefore, in this example it may be expected that variations reverse side of the wafer serves as an etch mask. in the initial stress in the diaphragm material have a negligible This process was developed by Spiering et al. [ 191, who fab- effect on the resulting mechanical sensitivity of a corrugated ricated corrugated diaphragms with a square boddiaphragm diaphragm with a corrugation depth of at least 10 pm. Another structure in the center. advantage of the application of corrugated diaphragms is that Since the diaphragms are fabricated on < 100 > silicon, if the initial stress is equal to 108N/m2 (c), the mechanical only square diaphragms can be etched. However, the cormga- sensitivity is increased with a factor of about 10. tions can be etched in square or circular patterns. Fig. 6 shows an example of a corrugated diaphragm with corrugations in a 111. THE FABRICATION OF CORRUGATED DIAPHRAGMS square pattern. The corrugation depth is about 19 pm. It was found that this type of diaphragm was very vulnerable and the The fabrication process of LPCVD silicon nitride corrugated diaphragms were damaged very easily. This is probably caused diaphragms with a flat zone is schematically shown in Fig. by high local stresses at the sharp corners of the corrugations. S(aj(c): Fig. 7 shows a 2 x 2mm2 square diaphragm with corruga- (a) Corrugations with a depth of 4, < 10, 14, and 19 pm have tions in a circular pattern. A cross-sectional view of a detail of been etched in a 3-in p-type, 100 >-oriented silicon the diaphragm is shown in Fig. 8. It was found that this type wafer using a 0.5-pm-thick evaporated aluminum etch of diaphragm was robust and could withstand the handling mask and reactive ion etching (RIE) in an SFc-02 and dicing of the wafer. Therefore, this type of diaphragm plasma. was used for further testing. The diaphragms with different (b) After removal of the aluminum etch mask, 1 pm of low corrugation depths were fabricated on separate wafers. Each pressure chemical vapor deposited (LPCVD) Silicon wafer was provided with flat diaphragms, which were used as nitride is deposited on both sides of the wafer. Square a reference. windows are etched in the silicon nitride on the reverse Note that all Corrugated diaphragms are provided with 8 side of the wafer using RE in a CHF5/02 plasma. corrugations and a flat center zone. The flat zone is meant JOURNAL OF MICROELECTROMECHANICAL SYSTEMS, VOL. 3, NO. 1, MARCH 1994 ‘ 6t 0 A t .AA bb00 Fig. 7. Top view of a 2 mm x 2 mm x 1 pm silicon nitride diaphragm with a circular corrugation pattern. The diaphragm is backside illuminated. Fig. 8. SEM photograph of a detail as of the cross-sectional view of a corrugated diaphragm, shown in Fig. 7. The silicon substrate has not been etched for obvious reasons of mechanical integrity. for reliable deflection measurements, because the deflection is measured by means of a mechanical scan with a surface profiler. The width of the corrugations is 25 pm and the separation between two corrugations is also 25 pm, giving a corrupt:?-- period of 50 pm, as can be seen from Fig. 8. The excelA. step coverage of the LPCVD silicon nitride is also shown Fig. 8. w. MEASUREMENT RESULTS The mechanical sensitivity of all diaphragms was measured using the bulge test, as for instance described by Bromley et al. [20]. For this purpose, the samples with a diaphragm were mounted on special holders. A homogeneous pressure was applied to the samples using a Wallace & Tiernan Chlorator FA-235-G pressure controller. The diaphragm deflection was measured using a DEK-TAK 3030 surface profiler. A stylus force of 1 x lo-’ N was used for the flat diaphragms, whereas the stylus force was 5 x N for the corrugated diaphragms. In the latter case, a larger force was necessary to provide 0 5 10 15 corrugation depth [wm] Fig. 9. The measured static center deflection of corrugated diaphragms as a function of the corrugation depth. sufficient damping of the movement of the stylus when it scanned the corrugations. It was found that the corrugated diaphragms, which were not loaded by any external force, showed a spontaneous static deflection. The finite element calculations of Spiering et al. [9], Ding [17], and Zhang and Wise [18] have indicated that the presence of a tensile or compressive stress in the corrugated diaphragm causes this spontaneous deflection. Note that this spontaneous deflection is different from buckling, which is a spontaneous deformation due to a too high compressive stress. Buckling occurs only in case of compressive stress and only if the compressive stress has exceeded a critical value. The direction into which a buckled diaphragm deforms, and the shape of a buckled diaphragm cannot be exactly predicted. Spiering et al. [9], Ding [17], and Zhang and Wise [18] have shown that the direction and magnitude of the spontaneous deflection of a corrugated diaphragm can, in principle, be calculated if the dimensions and the stress are known. If Fig. 9 the measured values of the spontaneous center deflection are given as a function of the corrugation depth. These values have been obtained by measuring the centre deflection of the diaphragm with stylus forces between 1 x low5 N and 5 x lo-’ N, and subsequently extrapolating the deflections to the value which is expected at a stylus force of 0 N, using a linear least squares fit. All diaphragms showed an upward static deflection. The spreading of the measured deflections is mainly due to the vibration of the diaphragm, which occurs if relatively low stylus forces are used for scanning the corrugated diaphragm. Larger stylus forces have not been used to avoid possible damage to the diaphragms. Since all types of diaphragms show a similar pressure- deflection curve, as can be seen from the (l), (2), and (7), the measured values of the center deflection for different applied pressures were fitted to the curve p = A(w0,m - W.d3 + B(wo,m - 20s) (12) where A and B are curvefit coefficients and WO,and ~ w, are the measured diaphragm center deflection and the static center deflection, respectively. Note that w, is not the extrapolated spontaneous center deflection, as shown in Fig. 9, but is the result of the spontaneous center deflection, as well as the deflection due to the stylus force of 5 x N. It can SCHEEPER et al.: DESIGN OF CORRUGATED SILICON NITRIDE DIAPHRAGMS 41 U E\" 1''\"\"\"\"\"''\" *\" \" 0 5 10 15 20 corrugation depth [uml Fig. 10. The measured mechanical sensitivity of corrugated diaphragms as a function of the corrugation depth (markers) and theoretically expected values for an initial stress of 3.1 x 10\" N/m' (- - -) and for the stress-free case (-). be concluded from the (10) and (12) that the mechanical sensitivity for small diaphragm center deflections, where the relation between P and (~0- ,w,) ~i s approximately linear, is given by Equations (12) and (13) are used to calculate the mechanical sensitivity for small deflections from the measured pressure- deflection curves. In Fig. 10 the measured mechanical sensitivities are shown as a function of the corrugation depth. An initial stress of 3.1 x lo8 N/m2 and a Young's modulus of 4.0 x 10l1 N/m2 have been obtained from the measurements on flat diaphragms. The theoretically expected sensitivities are shown for corrugated diaphragms with and without initial stress, based on the measured stress and Young's modulus. A Poisson's ratio of 0.3 is assumed. For the calculations an effective diaphragm radius R = U/,/. has been used. For a diaphragm with a side length a of 2 mm the effective diaphragm radius becomes 1.13 mm. It can be concluded from Fig. 10 that the measured mechan- ical sensitivities are in a good agreement with the theoretically predicted values, although in the theory it was assumed that the diaphragms are circular and the stress reduction was only a rough estimate. A corrugation depth of 4 pm increases the mechanical sensitivity from 1.0 nm/Pa, for flat diaphragms, to about 25 nm/Pa. The mechanical sensitivities of the diaphragms with a corrugation depth of about 14 pm (15 nm/Pa) agree reasonably with the mechanical sensitivity predicted by the stress-free model. In the case that the initial stress has been reduced consider- ably (nearly stress-free diaphragm), the mechanical sensitivity is inversely proportional to the value of the Young's modulus (see (1 1)). However, the Young's modulus is less dependent on process parameters than the initial stress [21]. Fig. 11 shows the measured pressure-deflection curve of a diaphragm with 14 pm-deep corrugations (markers) and the theoretically predicted values according to the model of Di Giovanni [16] ((2)-(5)), which is valid for stress-free corrugated diaphragms. Although in the low-pressure region the model agrees reasonably with the measured values, the 12 10 -6 26 m n4 2 0 Fig. 11. The measured pressure-deflection curve for a diaphragm with 14-pm-deep corrugations (markers) and the theoretically expected values according to the model of Di Giovanni (161 (line). pressures corresponding to large deflections are notably larger than predicted. For example, at a deflection of 90 pm the required pressure according to the model of Di Giovanni is 9.0 kPa, whereas 10.5 kPa was measured. The model of Haringx [15], which has been used by Jerman [8], predicts a pressure of 17 kPa. V. CONCLUSIONS Silicon nitride corrugated diaphragms of 2mm x 2mm x 1 pm have been fabricated. The diaphragms were provided with 8 circular corrugations with depths of 4, 10, or 14pm. It was found that the behavior of the corrugated diaphragms was strongly influenced by the initial stress. The measured mechanical sensitivity was considerably lower than the values that were predicted by the model of Di Giovanni [16], which assumed a stress-free diaphragm. A model has been proposed that takes the influence of tensile stress into account. This model is a super-position of the model of a stress-free corru- gated diaphragm and a diaphragm with a high initial tensile stress. The reduction of the initial stress was estimated from the change in the cubic terms in the load-deflection equations from the stress-free flat and corrugated diaphragm. The model predicts that the mechanical sensitivity increases for small corrugation depths, due to the reduction of the initial diaphragm stress. For large corrugation depths, the mechanical sensitivity decreases, due to the stiffening effect of the corrugations. Both effects have been confirmed by experiments. For large corrugation depths, the mechanical sensitivity is only weakly depending on the initial stress and becomes equal to the value predicted by the stress-free model. It can be concluded that analytical models have been pre- sented for corrugated diaphragms for three different cases. First, the models of Haringx [I51 and Di Giovanni [16] for stress-free (shallow) corrugated diaphragms. Next, the model of Spiering et al. [ 11 J for a diaphragm with a (single) deep corrugation with initial tensile stress, which can be considered as if it is stress-free. Finally, our model describes the intermediate case of diaphragms with shallow corrugations and with initial stress. All tested corrugated diaphragms showed a static center deflection without applying any load. This static center de- 42 JOURNAL OF MICROELECTROMECHANICAL SYSTEMS, VOL. 3, NO. 1, MARCH 1994 flection of 2 to 7pm should be taken into account when 1131 J. H. Jerman, D. J. Clift, and designing microphones with a corrugated diaphragm. A so- interferometer with a corrugated silicon diaphragm support.” in S. R. Malinson, “A miniature Fabry-Perot Dig. lution is to eliminate the static center deflection by fabricating Tech. Papers, IEEE Solid-state Sensor and Actuator Workshop, Hilton corrugations that are symmetrical about the central plane of Head Island, SC, June 1990, pp. J. 140-144. 141 N. F. Raley, D. R. Ciarlo, C. Koo, B. Beiriger, J. Trujillo, C. Yu, the diaphragm [ 181. G. Loomis, and C. R. Chow, “A Fabry-Perot microinterferometer for Summarizing, the mechanical sensitivity of diaphragms with visible wavelengths,” in Dig. Tech. Papers, IEEE Solid-State Sensor and initial tensile stress can be increased considerably by providing 151 J. A. Haringx, “Design of corrugated diaphragms,” Actuator Workshop, Hilton Head Island, SC, June 1992, pp. 17G173. ASME Trans., vol. the diaphragm with corrugations. Furthermore, corrugated 79, pp. 55-. 1957. diaphragms offer the possibility to make the mechanical sen- 161 M. Di Giovanni, Flat and Corrugated Diaphragm Design Handbook, 1st ed. New York: Marcel Dekker, 1982, pp. 255-262. sitivity of diaphragms independent of variations in the initial [17] X. Ding, “Behavior and application of silicon diaphragms with a boss stress, caused by variations of deposition process parameters. and corrugations,” in Dig. Tech. Papers, IEEE Solid-state Sensor and Actuator Workshop, Hilton Head Island, SC, June 1992, pp. 166-169. [ 1 SI Y. Zhang and K. D. Wise, “Performance of nonplanar silicon diaphragms AC~NOWLEDGMENT under large deflections,” in Dig. Tech. Papers, IEEE Micro Electro- mechanical Systems Workshop, Fort Lauderdale, The authors would like to thank P. Bakker for producing and 284-288. FL, Feb. 1993, pp. testing the corrugated diaphragms with a square corrugation [I91 V. L. Spiering, S. Bouwstra and J. H. J. Fluitman, “Realization of pattem, B. Otter for producing the SEM photographs, H. mechanical decoupling zones for package-stress reduction,” Sensors and Acruaiors A, vol. 37-38, pp. 8OW304, 1993. van Vossen for growing the LPCVD silicon nitride layers [20] E. L. Biomley, J. N. Randall, D. C. Flanders, and R. W. Mountain, and H. Voorthuyzen, V. Spiering, and R. Spiering for useful “A technique for the determination of stress in thin films,” J. Vac. Sci. suggestions and discussions. Technol. E, vol. 1, pp. 13-1366. 1983. [21] S. Bouwstra, “A resonating microbridge mass flow sensor,” Ph.D. dissertation, Univ. of Twente, 1990, pp. 52-55. REFERENCES [I] D. Hohm and ducer,” J. R. Gerhard-Multhaupt, “Silicon-dioxide electret trans- Acousr. Soc. Amer., vol. 75, pp. 1297-1298, 1984. [2] A. J. Sprenkels, R. A. Groothengel, A. J. Verloop. and P. Bergveld, velopment of an electret microphone in silicon,” Sensors and Actuators, “De- Patrick R. Scheeper was bom in Nieuw Vennep, The Netherlands, on vol. 17. pp. 509-512, 19. October 25, 1965. He received the B.S. degree in applied physics from [3] D. Holm and G. Hess, “A subminiature condenser microphone with the Rooms Katholieke Hogere Technische School Rijswijk, Rijswijk, The silicon nitride membrane and silicon backplate,” J. Acousr. Soc. Amer., Netherlands, in 1988 and the Ph.D. degree from the Bio-Information Group, vol. 85, pp. 476-480, 19. Department of Electrical Engineering, University of Twente, The Netherlands, [4] W. Kuhnel and G. Hess. “Micromachined subminiature condenser in 1993. His current research is focused on the development of a microphone microphones in silicon,” Sensors andActuarors A. vol. 32, pp. 560-5, based on silicon technology for use in hearing aids. 1992. [SI J. Bergqvist, F. Rudolf, J. Maisano, F. Parodi. and M. Rossi, “A silicon condenser microphone with a highly perforated backplate,” in Dig. Tech. Papers, Transducers ’91, San Francisco, CA, June !991, pp. 266-269. 161 T. Bourouina, S. Spirkovitch, F. Baillieu, and C. Vauge, “A new microphone with a p+ silicon membrance,” Sensors and Actuutors A. vol. 31, pp. 149-152, 1992. Wouter Olthuis was born in Apeldoom, The Netherlands, on October [7] P. 23, 1960. He received the M.S. degree in electrical engineering from the nitride diaphragms for condenser microphones,” R. Scheeper, J. A. Voorthuyzen, and P. Bergveld, “PECVD silicon Sensors and Actuators University of Twente, Enschede, The Netherlands, in 1986, and the Ph.D. E, vol. 4, pp. 79-84, 1991. degree from the Biomedical Engineering Division of the Faculty of Electrical [SI J. H. Jerman, “The fabrication and use of micromachined corrugated sil- Engineering, University of Twente, Enschede, The Netherlands, in 1990. icon diaphragms,” Sensors and Actuators, vol. A21-A23, pp. 988-992, Currently, he is working as an Assistant Professor in the Biosensor Technology Group of the University of Twente. [9] 1990. V. L. Spiering, S. Bouwstra, R. M. E. J. Spiering, and M. Elwenspoek, “On-chip decoupling zone for package-stress reduction,” in Dig. Tech. Papers, Transducers ‘9I, San Francisco, CA, June 1991, pp. 982-985. [IO] H. L. Offereins, H. Sandmaier, B. Folkmer, U. Steger, and W. Lang. “Stress free assembly technique for a silicon based pressure sensor,” in Dig. Tech. Papers, Transducers ’91, San Francisco, CA, June 1991, pp. Het Bergveld was bom in Oosterwolde, The Netherlands, on January 26, 9869. 1940. He received the M.S. degree in electrical engineering from the Univer- [ 111 V. L. Spiering, S. Bouwstra, J. Burger, and M. Elwenspoek, “Membranes sity of Eindhoven, The Netherlands, in 1965 and the Ph.D. degree from the fabricated with a deep single corrugation for package stress reduction University of Twente, The Netherlands, in 1973. and residual stress relief,” in Dig. Tech. Papers, MicroMechanics Eu- Since 1965 he has been a member of the Biomedical Engineering Division rope Workshop (MME’93). Neuch\\^atel, Switzerland, Sept. 1993. pp. of the faculty of Electrical Engineering, University of Twente, and was in 1984 223-227. appointed as Professor in Biosensor Technology. He is one of the project [I21 C. J. van Mullem, K. J. Gabriel, and H. Fujita, “Large deflection leaders in the MESA Research Institute. His research subjects concem the performance of surface micromachined corrugated diaphragms,” in Dig. further development of ISFET’s and biosensors based on ISFET technology Tech. Papers, Transducers ‘91, San Francisco, CA, June 1991, pp. as well as silicon microphones. He has written more than 150 papers on these 1014-1017. topics.