OVER THE last several years, extensive efforts have

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1 192 JOURNAL OF MICROELECTROMECHANICAL SYSTEMS, VOL. 7, NO. 2, JUNE 1998 A High-Sensitivity -Axis Capacitive Silicon Microaccelerometer with a Torsional Suspension Arjun Selvakumar, Member, IEEE, and Khalil Najafi, Senior Member, IEEE Abstract This paper presents a new z-axis high-sensitivity silicon-micromachined capacitive accelerometer fabricated using a three-mask dissolved-wafer process (DWP). It employs capacitive sensing using overlap-area variations between comb electrodes and a torsional suspension system to provide high sensitivity without compromising bandwidth, full-scale range, or the pull-in voltage ceiling. Excellent electrical sensitivity is obtained by using high-aspect-ratio comb fingers with narrow air gaps of 2 m and a large overlap area of 12 m m. Torsional suspension beams 150 m long with a cross-sectional area of 12 m 2 3 m are used to improve the mechanical gain. Simulations of the capacitance between sense fingers show a highly linear region over a wide 14-m tip deflection range. Accelerometers were fabricated and yielded sensitivities of mv/g, a nonlinearity less than 0.2% over a range of 04 to+3 g, a full-scale range of 04 to+6 g, and pull-in voltages greater than 8 V. A 3-dB cutoff frequency of 35 Hz was measured in air. The calculated thermomechanical noise in the sensor is 0.28 mg over this bandwidth. [293] Index Terms Capacitive sensing, high-sensitivity accelerometer, microaccelerometer, silicon microelectromechanical sensor, torsional suspension. I. INTRODUCTION OVER THE last several years, extensive efforts have been devoted to the development of micromachined accelerometers for applications as diverse as automotive safety and ride control [1], inertial navigation and guidance [2], vibration sensing in machine health monitoring, motion detection in virtual reality headsets, etc. Many of these applications stand to benefit from micromachined sensors that have a high sensitivity and a wide dynamic range while still being low cost and mass produced. Predominantly, these micromachining efforts have concentrated on using bulk-silicon [2], [3] or surface-micromachining [4] techniques to fabricate the sensor. A majority of these sensors utilize a capacitivesense mechanism wherein the deflection of the inertial mass produces a change in the interelectrode air gap of the capacitor. Thus, the change in capacitance provides a measure of the acceleration in a direction perpendicular to the plane of the sensor capacitor. Sensing capacitance in this manner achieves relatively high sensitivity by merely reducing the air gap Manuscript received August 11, 1997; revised February 16, Subject Editor, J. Fluitman. This work was supported by ARPA under Contracts JFBI and DABT63-95-C-0111 and by NSF NYI under Grant ECS A. Selvakumar is with Input/Output, Inc., Stafford, TX USA ( Arjun_Selvakumar@i-o.com). K. Najafi is with the Center for Integrated Sensors and Circuits, University of Michigan, Ann Arbor, MI USA. Publisher Item Identifier S (98) and increasing the overlap area between the sense electrodes. However, reducing the air gap reduces the full-scale range of the sensor and increases the squeeze film damping, thus reducing the bandwidth of the device. In addition, since the readout circuitry applies a voltage across the sense plates of the sensor, narrow gaps, large areas, and low spring constants (required for high sensitivity) can cause the plates to pull in at low voltages [5] due to the electrostatic force of attraction. Other undesirable qualities of decreasing the gap spacing are degradation of the output linearity and promotion of stiction between large released surfaces. In this paper, we will discuss and demonstrate a -axis accelerometer employing a new capacitive-sense mechanism [6] that achieves high sensitivity without compromising fullscale range or bandwidth. The sensor operation is based on a coplanar sense electrode movement wherein the change in capacitance is caused by the variation in the interelectrode overlap area rather than the air gap. This coplanar motion enables the design of sensors with a wide full-scale range since the size of the air gap no longer limits the movement range of the electrodes. Another intrinsic property of this motion is that squeeze film damping in the interelectrode air gap is avoided, and, hence, the bandwidth of the sensor is improved dramatically. Finally, with this coplanar approach, the motion sensitive plane is perpendicular to the air gap, and, therefore, the pull-in instability is improved raising the pull-in voltage ceiling, and stiction between narrowly spaced electrodes is reduced. High sensitivity is obtained by using: 1) the mechanical advantage offered by a torsional beam suspension with a low spring constant in the axis along with 2) multiple interdigitated comb sense fingers with narrow gaps and large overlap areas between electrode fingers. It should be mentioned that unlike the devices presented by Rudolf [7] and by Cole [8], the capacitive sensing in this case is performed using interdigitated fingers rather than a parallelplate capacitor. As mentioned before, this helps increase the pull-in voltage, reduce stiction, increase the full-scale range, and simplify the fabrication process. Moreover, this design fabricated with the help of the dissolved-wafer process (DWP) technology [9] aids in independently choosing the width of the beam, which is a crucial and powerful design parameter in setting the sensitivity, full-scale range, and bandwidth of the device. The salient features mentioned above will be described in detail in the following sections. Starting with a description of the sensor structure and design, the discussion will then lead to the relevant analytical expressions for the mechanical /98$ IEEE

2 SELVAKUMAR AND NAJAFI: SILICON MICROACCELEROMETER WITH TORSIONAL SUSPENSION 193 (a) (b) Fig. 1. (a) A z-axis capacitive torsional accelerometer fabricated using a three-mask DWP. The support beams are twisted by a torque produced by an acceleration acting on the offset center of mass. This twisting is manifested as a change in the overlap area of the interdigitated sense fingers attached to the inertial mass. (b) Cross-sectional view along the mass and the sense fingers. and bending deformations are significantly small compared to the dimensions of the suspension beams that they effectively operate in the linear deformation range. The first assumption permits one to use the parallel-plate approximation to express the capacitance variation between the two sets of electrodes as a function of the sense electrode rotation. The verification of this assumption is carried out in the following section on modeling using FastCap [10], a capacitance extraction FEA program. The capacitance data was calculated for varying overlap positions of the electrodes that would be caused by the rotation of the inertial element about the suspension beam axis. Calculations were made from the complete overlap position to the overlap resulting from the leading edge of the sense fingers deflecting by 20 m. The capacitance change for sense fingers of length, with an air gap and relative permittivity, is derived for a trapezoidal overlap area using the parallelplate approximation expression. For small angles of rotation, the change in capacitance can be expressed as (1) and electrical behavior of the sensor. This design analysis is substantiated by both electrostatic and mechanical simulation of the sensor structure obtained through finite-element analysis (FEA) and reported in the subsequent modeling section. This paper will then briefly cover the fabrication technology, followed by the fabrication and test results in the experimental results section. Finally, the relevant conclusions will be drawn from this effort. II. SENSOR STRUCTURE AND DESIGN A capacitive -axis torsional accelerometer featuring the ideas mentioned above is illustrated in Fig. 1(a). The device is fabricated using a three-mask DWP technology. It consists of a 12- m-thick boron-doped silicon inertial mass suspended 7.5 m above the glass substrate by two narrow high-aspectratio 12 m 3- m-wide torsion beams anchored to the glass substrate. A large number of 300- m-long capacitivesense fingers extend from the end of the proof mass, opposite the support beams. These moving fingers along with fingers anchored to the substrate form an interdigitated pair of sense electrodes separated by a 2- m air gap. An acceleration acting on the offset center of mass of the structure produces a torque on the suspension beams, thereby twisting them by an angle [Fig. 1(b)]. The rotation of the mass and the attached sense fingers away from the rest position results in a reduction of the overlap area leading to a reduction in the capacitance between the sense fingers while maintaining a constant air gap. Due to the high-aspect-ratio geometry of the interdigitated electrodes and their rotational motion, the derivation of the capacitance versus displacement relationship can be quite involved. For the ease of evaluating the sensor with the help of closed-form analytical expressions, the following assumptions were made: 1) the sense fingers operate in a linear overlap area capacitance variation regime with a constant fringe field contribution and 2) the torsional The twisting angle [11], assuming a rigid mass plate, is given by (2), where and are lengths and widths of the mass and beam, respectively, is the structure thickness, is the modulus of rigidity, is the mass, and is the -axis acceleration. The correction factor is a function of the cross-sectional aspect ratio of the beam and is given in Table I, and for this design, wherein the aspect ratio is Since the change in capacitance is not a very strong function of any one physical parameter, high sensitivity is achieved collectively by a number of factors: 1) a large mass and an increased sense area due to thick deep-boron-diffused fingers offered by the DWP; 2) narrow compliant suspension beams with low torsional stiffness dry etched with an aspect ratio greater than 8 : 1; and 3) small 2- m gaps between the sense fingers. Note that for a given mass size, the torsional design allows increasing the torque and by increasing. Further increases in can be derived from longer sense fingers as well as additional sets of fingers along the periphery of the inertial mass [not shown in Fig. 1(a)]. Although it would seem to be advantageous to use a crosssectional aspect ratio of 1.0 in order to obtain a small and hence a high sensitivity ( ), the effect of cross-axis sensitivity will be greatly increased. Acceleration along the and axes will produce rotational and translational movement of the mass, respectively, due to the flexural bending of the suspension beams around the axis and can be obtained from [12]. The cross-axis sensitivity in the direction is very low largely due to the geometry of the narrow sense fingers and also to the high spring constant given by (3) with Young s modulus of silicon. The resulting deflection in the axis gives rise to an insignificant change in (2)

3 194 JOURNAL OF MICROELECTROMECHANICAL SYSTEMS, VOL. 7, NO. 2, JUNE 1998 TABLE I TABULATED VALUES OF CORRECTION FACTOR FOR RECTANGULAR CROSS SECTIONS OF BEAMS UNDER TORSION [12] overlap area [the term in the bracket in (4)] Sensitivity in the direction arises from the moment produced by the acceleration acting on the suspended structure, which has an offset center of mass with respect to the suspension beams. The resulting lateral rotation of the mass produces a change in capacitance between the sense fingers through the angular deflection of the moving sense finger toward the fixed fingers through changes in the air gap between them. It is assumed that each beam undergoes bending in opposite directions rather than stretching to accommodate the moment applied to them. Again, using beam-bending theory, the moment developed by the acceleration acting on the offset center of mass produces a rotation of the mass given below by (5). The change in the capacitance can be calculated assuming that the change in the fringe field contribution is negligible and is given by (6), where and (3) (4) (5) For example, a conventional parallel-plate device with a rest capacitance pf, an air gap of 2 m, and a spring constant N/m (which is required to produce a sensitivity equivalent to the device presented in this paper) yields a pull-in voltage ceiling as low as 1.5 V. On the other hand, by using the design described in this paper, the sensing motion maintains a constant air gap and electrostatic forces generated on either side of the moving electrode oppose and hence balance each other. Hence, pull-in instability is avoided. Of course, processing nonidealities can produce minor variations in the air gap or the overlap area, but even in this scenario, the overall net force is suppressed by the action of the opposing electrostatic forces. Thus, for any given offset, this design will inherently have a larger pull-in voltage ceiling than one without the opposing forces. Additionally, the sensitivity need not be compromised and the design parameters can be arbitrarily varied without affecting the pull-in voltage. The rotational motion is limited only when the sense fingers traverse the recess depth and make contact with the underlying glass substrate. The recess etch depth can be made large (5 15 m) to provide a wide full-scale range. As mentioned before, there is an additional set of sense fingers on the far side of the inertial mass adjacent to the torsional beam such that when this set travels downwards the other set travels upwards and vice versa. This teeter totter design enables the substrate to serve as a shock stop in both and directions. In conventional silicon-glass parallel-plate capacitive accelerometers, the squeeze-film effect between the narrow gap (1 2 m) of the electrode plates is the dominant mechanism in increasing damping and reducing bandwidth. The glass frame in this device does not serve as the sense electrode, and thus the large recess gap employed reduces the damping between the mass plate and the substrate, increasing the bandwidth of the structure. Calculation of the squeeze-film damping was carried out assuming a simple parallel-plate translational motion of the sensor structure toward the glass substrate. Although in reality the proof mass rotates, the result based on assuming translational motion [13] provides an upper limit for the damping coefficient [given by (8)] and hence a conservative estimate for the bandwidth The cross-axis sensitivity can be substantial depending on the dimensions of the device. From (5), it can be observed that is a strong function of and. As the torsional sensitivity is not as strong a function of (1), can be reduced by increasing and decreasing without drastically affecting,. The switched capacitor readout electronics applies a voltage to the sense electrodes of the accelerometer during each clock cycle. In conventional parallel-plate accelerometers, the instability caused by operating at the pull-in voltage given by (7) necessitates reducing the sense capacitance and increasing the air gap and spring constant (6) (7) where is a form factor that takes into account the geometry, is the viscosity of air at room temperature and pressure, and are the width and length of the mass, respectively, and is the air gap separating the sensor structure and the glass substrate. For this design geometry (including the damping area contributed by the sense electrodes), is equal to Ns/m. This value is approximately two orders of magnitude smaller than that present in a conventional parallelplate capacitive design of an equivalent area and with a 2- m air gap. Apart from reducing damping, the large gap also helps to reduce the stiction associated with rinsing and drying of these structures compared to a sensor employing a single set of narrowly spaced parallel-plate electrodes with an equivalent capacitive-sense area. (8)

4 SELVAKUMAR AND NAJAFI: SILICON MICROACCELEROMETER WITH TORSIONAL SUSPENSION 195 (a) (b) Fig. 2. (a) The FastCap model used for extracting the capacitance between the moving sense fingers and the fixed electrodes. The modeled beam was 300 m long, 12 m high, and 6 m wide with a 2-m gap to the fixed outer electrode of the same dimensions except with a width of 3 m. (b) A coarsely meshed model is shown for clarity. III. MODELING The goal of this modeling section is to investigate the validity of the assumptions made in the previous section by carrying out both electrostatic simulation using FastCap and structural and modal analysis using ANSYS. A comparative study of the analytical and simulation results is made at the end of the section. A. Electrostatic Modeling Simulation using FastCap was carried out by reducing the interdigitated sense fingers to a single moving sense finger and its surrounding fixed electrode pair. The electrodes were modeled as hollow three-dimensional (3-D) surfaces separated by an air dielectric as shown in Fig. 2(a). Fig. 2(b) shows a similar model with dimensions and mesh sizes that are altered for ease of viewing the center electrode, which is rotated with respect to the outer electrodes. The total capacitance between the two electrodes was obtained for each incremental angular displacement of the moving electrode from complete overlap Fig. 3. Plot of the capacitance versus position data obtained from FastCap as well as by parallel-plate approximation expression (2). to a tip displacement of 20 m below the fixed electrode surface.

5 196 JOURNAL OF MICROELECTROMECHANICAL SYSTEMS, VOL. 7, NO. 2, JUNE 1998 (a) (b) (c) Fig. 4. The simulated response of the structure (a) (c) due to a 1-g acceleration along the axial directions z; x; and y, respectively. The mode shapes of the first three resonant frequencies match the response shown above in the same order (as expected). Fig. 3 is a plot of the capacitance between the fingers obtained from FastCap as well as the parallel-plate approximation method. The offset between the two curves should be examined keeping in mind that the data obtained through the analytical method (parallel-plate approximation) did not take into consideration the fringe field contribution along the length of the fingers. The capacitance data extracted by FastCap indicates two nonlinear regions: a brief range near the complete overlap position and that beyond the 16- m tip displacement. These regions sandwich a wide linear region (from 2to 16 m) wherein the fringe field contribution is largest, but remaining linear with respect to displacement. Movement of the electrode from this intermediate region toward complete overlap results in the tapering off of the capacitance. This is because fringe fields shift from an electrode edge-surface coupling to a reduced electrode edge edge coupling. In the other extreme, the region beyond the 16- m displacement the overlap area between the electrodes becomes triangular in shape and increasingly nonlinear in nature. Using the parallelplate approximation, one can extract the capacitance versus overlap-area variation as a nonlinear relationship. In the intermediate range (2 16 m below the fixed electrode top surface), the parallel-plate approximation estimates the sensitivity with an error of about 9.6% from the FastCap simulation. This error can be acceptable for initial design iterations wherein feasibility calculations are needed. It is important to note that whether is positive or negative, the capacitance always decreases away from the maximum value which is at the complete overlap or null position (0 g). B. Mechanical Modeling The sensitivity including cross axis as well as natural frequencies were also obtained by finite-element method (FEM) simulations using ANSYS. Structural as well as modal analyses were performed to obtain this data, and Fig. 4 shows the models of an accelerometer under the effect of a 1-g acceleration in each of the three axes and with the undeformed shape shown as a dotted line. The model that is shown here had 150 m 3 m 12- m beams suspending an 800 m 400 m 12- m proof mass. The most sensitive as well as the lowest frequency was in the axis, followed by the rotation of the mass due to cross axis in the direction and lastly that due to an acceleration in the axis. The results of the analysis are shown in Table II and compare quite closely to analytical values with the exception of the cross-axis sensitivity in the -axis direction. The simulated values for the axis indicate a stiffer spring constant and a higher resonant frequency indicating that this mode has a cross-axis sensitivity less than that predicted analytically. The discrepancy between the simulated and analytical values of the cross-axis sensitivity is perhaps due to the stretching rather than bending of the beam producing the rotation of the proof mass from transverse -axis acceleration. IV. EXPERIMENTAL RESULTS The accelerometer was fabricated using the DWP [9] using three masks two on glass to create recess depths and to form metal interconnects and one on silicon to define the sensor structure. The process flow is illustrated in Fig. 5(a) (e). An unmasked deep boron diffusion is carried out to create a m boron etch stop [Fig. 5(a)]. This determines the sensor thickness. Electroplated nickel is then patterned on the wafer to act as the mask for the subsequent 15- m dry trench etch. The etch is carried out in an electron cyclotron reactor (ECR) [14], which can provide gaps as narrow as

6 SELVAKUMAR AND NAJAFI: SILICON MICROACCELEROMETER WITH TORSIONAL SUSPENSION 197 TABLE II COMPARISON OF THE ANALYTICAL AND SIMULATED VALUES OF THE DEFLECTION SENSITIVITY PER 1-g ACCELERATION AS WELL AS THE FIRST THREE RESONANT FREQUENCIES TABLE III THE CALCULATED PERFORMANCE OF THE TORSIONAL ACCELEROMETER Fig. 5. (a) (b) (c) (d) (e) The DWP flow using three masks one on silicon and two on glass. 2 m with aspect ratios as high as 20 using a chlorine etch recipe [Fig. 5(b)]. The glass wafer which will serve as the sensor substrate is etched in 7 : 3 hydrofluoric acid : nitric acid with a chromium/gold (300/4000 Å) masking layer to create a recess as deep as 7.5 m. The wafer is then patterned for a titanium/platinum/gold (300/300/700-Å) evaporation and lift off, which will yield the feedthrough from the silicon as well as the interconnects and the bond pads [Fig. 5(c)]. The glass and the silicon wafers are now electrostatically bonded together at 390 C and 800 V [Fig. 5(d)] and finally etched in ethylenediamine-pyrocatechol at 110 C to release the micromechanical structure [Fig. 5(e)]. Torsional accelerometers were designed with performance specifications given in Table III. They were designed for a sensitivity of 28 ff/g, with a bandwidth of 43 Hz, a full-scale range of 33 g, and cross-axis sensitivities of 0.7% and 0.043% in the - and -axis directions, respectively. It should be mentioned that this performance was based on a conservative design for proof of concept reasons. Based on this design and the DWP technology, it is straightforward to design sensors with a wide performance range. The accelerometers were fabricated, and Fig. 6 shows a scanning electron microscope (SEM) micrograph of a typical device with the silicon structure suspended on top of the glass substrate. The suspension beams, sense electrodes, and suspended proof mass can be seen in the center of the micrograph. Stress gradients due to nonsymmetric boron diffusion produced a curvature at either ends of the teeter totter structure curving downward to the glass substrate. This curvature manifested itself in such a way as to produce a null position offset in the sense capacitance-deflection curve of the sensor. This effect resulted in removing the nonlinearity that would have been present in the proximity of the ideal 0-g capacitancedeflection curve. Additionally, the capacitive sensitivity is also reduced slightly due to the decreased overlap sense capacitance area and a decrease in the effective full-scale range of the sensor, which is determined by the limited rotational motion of the teeter totter structure. Fig. 7 is a close-up view near the supports of the inertial mass plate. A torsional suspension beam can be seen along with fixed and moving sense fingers. Testing of the accelerometers was carried out using a switched capacitor integrated circuit [15] offering a gain of 15 mv/ff. The test results are summarized and tabulated in Table IV. The sensitivity was measured with a 2-g flip test conducted on two accelerometers as shown in Fig. 8. A total 40 ff of capacitance change over 2 g was measured on these devices with linear output characteristics. This is 29% less than the design value for this sensor and is attributed to the stress induced curvature of the sense fingers effectively decreasing the capacitive sensitivity of the sensor, the fabricated interelectrode air gap being 15% larger than the designed value and tensile stress in the deep-boron-diffused suspension beams increasing the torsional stiffness of the suspension. The frequency response was measured using an Unholtz Dickie shaker table operating at a fixed acceleration of 1 g. The

7 198 JOURNAL OF MICROELECTROMECHANICAL SYSTEMS, VOL. 7, NO. 2, JUNE 1998 Fig g flip test of the accelerometer exhibiting a sensitivity of 300 mv/g. Fig. 6. A SEM micrograph of a typical torsional accelerometer with a 600 m m proof mass suspended by 300-m-long suspension beams over a 7.5-m air gap over the glass substrate. Fig. 9. Frequency response using a 61-g acceleration signal. The 3-dB cutoff frequency can be observed at 35 Hz. Fig. 7. Close-up of a torsional beam 150 m long, 3 m wide, and 12 m high and sense fingers 300 m long and 12 m high separated from each other by 2-m air gaps. frequency response is plotted in Fig. 9, and the 3-dB cutoff frequency was observed at 35 Hz. This compares within 18.4% of the design calculation of 43 Hz obtained for the secondorder system with a damping factor and a natural frequency Hz. The expression for the 3-dB cutoff frequency was derived from equating the transfer function of the accelerometer (second order over damped system) to 3 db and is given below in (9). The 1-Hz data points are the dc values obtained from the flip test for two identical devices The unavailability of a turntable necessitated the use of a shaker table to test full-scale range. The sensor was subjected to a sinusoidal input acceleration of fixed frequency (15 Hz) and an increasing amplitude ranging from 0.25 to 6 g. The (9) positive and negative peak values of the sensor output were tracked for each input excitation value. This was carried out for both the right-side-up and upside-down orientations of the sensor. Fig. 10 is a plot of the output voltage as a function of the absolute acceleration, which is the shaker table signal corrected for the earth s gravitational field. For example, an input acceleration of 4-g peak peak applied to the sensororiented right side up results in a swing from 1 g in the upward direction to 3 g in the downward direction. The sign convention used in the plot is positive and negative when the input acceleration is aligned along and against the earth s gravitational field, respectively. Hence, for each excitation signal, two data points are extracted the voltage values at the peak positive and negative absolute acceleration swings. Note that during the negative acceleration swing, the sense fingers deflect toward an increasing electrode overlap area resulting in a larger output voltage. The full-scale range was detected as a clipping of the output waveform peaks at absolute acceleration values of 4 and 6 g. This is due to the extremities of the sense fingers coming into contact with the glass substrate and the sensor structure being inhibited from further rotation. The

8 SELVAKUMAR AND NAJAFI: SILICON MICROACCELEROMETER WITH TORSIONAL SUSPENSION 199 Fig. 10. The measured full-scale range of the torsional accelerometer. The nonlinearity is within 0.2% over 04 to 3 g, and the full-scale range is 04 to 6 g. TABLE IV CALCULATED VERSUS MEASURED RESULTS OF THE TORSIONAL ACCELEROMETER boron-stress curvature of the sense fingers limited the range to 10 g (net) as opposed to the calculated value of 32.7 g. The offset center of mass device design combined with the stress curvature in the sense fingers produces an asymmetric output about the 0-g value. Although the damping present in the device reduces the response of the accelerometer slightly, the test still gave valuable information on the linear behavior as plotted in Fig. 10. Over a 7-g span between 4 and 3g, the nonlinearity is less than 0.2%. V. CONCLUSION A new torsional -axis accelerometer design using varying overlap-area interdigitated sense fingers is presented. By using this design, high sensitivity can be obtained without compromising features such as a wide bandwidth, sufficient pull-in voltage ceiling, and a large full-scale range. Analytical expressions to predict the sensor behavior have been derived, and simulations of the mechanical and electrical performance were carried out. Electrostatic simulations indicate that the effect of the fringe-field capacitance contribution along the long narrow sense fingers, in general, does not prevent obtaining a wide linear range of operation. The accelerometer was designed for and fabricated with a DWP using a glass and silicon wafer. Using only three masks, all the essential components of the structure were provided a suspended structure, opposing capacitive-sense electrodes, support frame, and shock stops in both directions of motion. Two identical devices were tested and had sensitivities of mv/g using a 2-g flip test and had pull-in voltages greater than 8 V while maintaining a sense gap of 2 m between the sense fingers. The measured 3-dB cutoff frequency was 35 Hz with the sensor operated in atmospheric pressure. The calculated thermomechanical noise in the sensor is 47 g Hz at atmospheric pressure for the current design. Over the 35-Hz measured sensor bandwidth, the total noise equates to 0.28 mg. A deep-boron-diffusionrelated stress gradient caused a curvature of the structure toward the substrate and, hence, reduced the rotational range of motion to 10 g (spanning from 4to 6 g). Nonetheless, the output nonlinearity was merely 0.2% over a 7-g range within this full-scale range. The temperature sensitivity of the sensor has not been measured due to the required complexity of the test setup, but it is expected to be relatively small because: 1) glass and silicon have close thermal expansion coefficients; 2) changes in the intrinsic stress of the suspension support due to temperature variation will not affect device sensitivity or offset significantly when operated in the torsional mode; and 3) the capacitive gap will not change with temperature. Shock testing has not been performed either, but, as was discussed before, the sensor has a built-in shock stop when the extremities of the sense fingers touch the glass substrate. The curvature in the proof mass due to the nonuniform boron concentration profile can be reduced or eliminated by more careful diffusion and annealing steps, thus producing a wider range of operation. It should also be mentioned that the design presented in this paper can be readily fabricated using surface micromachining employing polysilicon or metallic thin films. Furthermore, varying the design parameters of the device geometry highlighted in the analysis, a wide performance range can be achieved, thus being suitable for a broad range of applications. ACKNOWLEDGMENT The authors would like to thank the staff of the Solid State Electronics Laboratory for providing fabrication support, F. Ayazi for much needed help in testing the accelerometers, and N. Yazdi for help with the interface circuit. REFERENCES [1] G. A. MacDonald, A review of low cost accelerometers for vehicle dynamics, Sens. Actuators, vol. A21 A23, pp , [2] Y. de Coulon, T. Smith, J. Hermann, M. Chevroulet, and F. Rudolf, Design and test of precision servoaccelerometer with digital output, in Proc. 7th Int. Conf. Solid-State Sensors and Actuators (Transducers 93), June 1993, pp [3] W. Henrion, L. DiSanza, M. Ip, S. Terry, and H. Jerman, Wide dynamic range direct digital accelerometer, in IEEE Proc. Solid-State Sensor and Actuator Workshop, June 1990, pp [4] W. Kuehnel and S. Sherman, A surface micromachined silicon accelerometer with on-chip detection circuitry, Sens. Actuators, vol. A45, pp. 7 16, [5] W. C. Tang, Digital capacitive accelerometer, U.S. patent , Oct. 11, [6] A. Selvakumar, F. Ayazi, and K. Najafi, A high sensitivity z-axis torsional silicon accelerometer, in Tech. Dig. Int. Electron Devices Meet., 1996, pp [7] F. Rudolf, A micromechanical capacitive accelerometer with a two point inertial suspension, Sens. Actuators, vol. 4, pp , [8] J. C. Cole, A new sense element technology for accelerometer subsystems, in Proc. Int. Conf. Solid-State Sensors and Actuators (Transducers 91), June 1991, pp

9 200 JOURNAL OF MICROELECTROMECHANICAL SYSTEMS, VOL. 7, NO. 2, JUNE 1998 [9] Y. Gianchandani and K. Najafi, A bulk silicon dissolved wafer process for microelectromechanical devices, IEEE J. Microelectromech. Syst., vol. 1, no. 2, pp , [10] K. Nabors, S. Kim, and J. White, Fast capacitance extraction of general three-dimensional structures, IEEE Trans. Microwave Theory Tech., vol. 40, no. 7, pp , [11] S. Timoshenko, Elements of Strength of Materials, 5th ed. New York: Van Nostrand, 1968, pp [12] R. J. Roark, Formulas for Stress and Strain, 4th ed. New York: McGraw-Hill, 1965, p [13] J. B. Starr, Squeeze film damping in solid-state accelerometers, in Dig. IEEE Solid State Sensor and Actuator Workshop (Hilton Head 90), June 1990, pp [14] W. H. Juan and S. W. Pang, Control of etch profile for fabrication of Si microsensors, J. Vac. Sci. Technol., vol. A 14, no. 3, pp , May/June [15] N. Yazdi, A. Mason, K. Najafi, and K. D. Wise, A low power generic interface circuit for capacitive sensors, Dig. Solid State Sensor and Actuator Workshop (Hilton Head 96), June 1996, pp Arjun Selvakumar (S 96 M 97) was born in Madras, India, in He received the B.E. degree in electrical engineering in 1989 from the Regional Engineering College, Tiruchi, India, and the M.S. degree in 1991 and the Ph.D. degree in 1997, both in electrical engineering, from the University of Michigan, Ann Arbor. His doctoral research was on the design and development of highperformance silicon microelectromechanical sensors and actuators and their fabrication technology for automotive, environmental monitoring, and biomedical applications at the Center for Integrated Sensors and Circuits, University of Michigan. Since April 1997, he has been working for the Advanced Systems Group (ASG) of Input/Output, Inc., Stafford, TX, as a Product Development Engineer for the design and development of micromachined sensors and components. His responsibilities include leading the research and development of the micromachined seismic sensor product lines, directing the micromachining process design and development in the manufacturing division, and developing the testing and quality-control metrics for sensor packaging and readout electronics. Khalil Najafi (S 84 M 86 SM 97) was born in He received the B.S. degree in 1980, the M.S. degree in 1981, and the Ph.D. degree in 1986, all in electrical engineering from the University of Michigan, Ann Arbor. From 1986 to 1988, he was a Research Fellow. From 1988 to 1990, he was an Assistant Research Scientist. From 1990 to 1993, he was an Assistant Professor. Since September 1993, he has been an Associate Professor at the Center for Integrated Sensors and Circuits, Department of Electrical Engineering and Computer Science, University of Michigan. His research interests include development of microfabrication and micromachining technologies for solid-state integrated sensors and microactuators, analog and digital integrated circuits, implantable microtelemetry systems and transducers for biomedical applications and wireless communication, technologies and structures for MEMS and microstructures, hermetic packaging techniques for microtransducers, and low-power wireless sensing/actuating systems. He has been active in the field of solid-state sensors and actuators for more than ten years. He is an Associate Editor for the Journal of Micromechanics and Microengineering. Dr. Najafi was awarded a National Science Foundation Young Investigator Award from 1992 to He was the recipient of the Beatrice Winner Award for Editorial Excellence at the 1986 International Solid-State Circuits Conference and the Paul Rappaport Award for coauthoring the Best Paper published in the IEEE TRANSACTIONS ON ELECTRON DEVICES. In 1994, he received the University of Michigan s Henry Russel Award for outstanding achievement and scholarship and was selected by students as Professor of the Year in He has been involved in the program committees of several conferences and workshops dealing with solid-state sensors and actuators, including the International Electron Devices Meeting, the Hilton- Head Solid-State Sensors and Actuators Workshop, and the IEEE/ASME MEMS Workshop. He is the Editor for Solid-State Sensors for the IEEE TRANSACTIONS ON ELECTRON DEVICES.