342 JOURNAL OF MICROELECTROMECHANICAL SYSTEMS, VOL. 13, NO. 2, APRIL 2004

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1 342 JOURNAL OF MICROELECTROMECHANICAL SYSTEMS, VOL. 13, NO. 2, APRIL 2004 Analysis and Analytical Modeling of Static Pull-In With Application to MEMS-Based Voltage Reference and Process Monitoring Luis Alexandre Rocha, Edmond Cretu, and Reinoud F. Wolffenbuttel Abstract The pull-in voltage of one- and two-degrees-of-freedom (DOF) structures has been symbolically and numerically analyzed with respect to drive mode dependence and hysteresis. Moreover, the time and temperature stability has been investigated and tested. Modeling results have been applied in the design of both folded-spring-suspended 1-DOF structures and single-side-clamped 2-DOF beams with a nominal pull-in voltage in the 5 10 V range and fabricated in an epi-poly process. Asymmetrically driven structures reveal pull-in close to the value predicted by the model (V pi 1-DOF is 4.65 V analytically simulated and 4.56 V measured; V pi 2-DOF is 9.24 V analytically simulated, 9.30 V in FEM and 9.34 V measured). Also the hysteresis is in close agreement (release voltage, V r, 1-DOF is 1.41 V analytically simulated and 1.45 V measured; V r 2-DOF is 9.17 V analytically simulated, 9.15 V in FEM and 9.27 V measured). In symmetrically operated devices the differences between the computed and measured V pi and V r are much larger and are due to process dependencies, which make these devices very suitable for process monitoring. The 2-DOF asymmetrically operated device is the most suitable for MEMS-based voltage reference. The stability in time is limited by charge build-up and calls for a 100-hour initial burn-in. Temperature dependence is 100 V K at V pi 5V, however, is calculable and thus can be corrected or compensated. [978] Index Terms Analytical modeling, dc voltage reference, MEMS stability, pull-in reproducibility, pull-in voltage. I. INTRODUCTION MICROELECTROMECHANICAL systems (MEMS) benefit from the relatively intensive coupling between different energy domains (electrical, mechanical, thermal) at the microscale level [1], [2]. One important phenomenon observed in the coupling between electric and mechanical domains is pull-in. This inherent instability in electromechanical devices has been subject of several studies. Since it depends mainly on dimensions, residual stress level and design, it has been used to characterize structural materials in surface micromachining processes [3], [4]. Unlike the case of the comb drive, which is based on area-varying capacitors, the design of vertical electrostatic actuators relies on gap-width varying capacitors and the Manuscript received December 17, 2002; revised December 15, This work is supported by the Netherlands Technology Foundation (STW) under Grant DEL Subject Editor K. D. Wise. L. A. Rocha and R. F. Wolffenbuttel are with the Delft University of Technology, Faculty EEMCS, Department for Microelectronics, Delft 2628 CD, The Netherlands ( l.rocha@ewi.tudelft.nl). E. Cretu is with the Delft University of Technology, Faculty EEMCS, Department for Microelectronics, Delft 2628 CD, The Netherlands and also with the Melexis, Tessenderloo, Belgium. Digital Object Identifier /JMEMS pull-in phenomenon has to be considered [5]. Pull-in causes the displacement range due to electrostatic force to be limited to one-third of the gap between the electrodes, in case of a motion perpendicular to the capacitor plate orientation. This effect also limits the dynamic range of capacitive accelerometers operating in the feedback mode. Charge drive (current drive with a series capacitance), rather than direct voltage drive can be used to circumvent pull-in, however, at the expense of attainable maximum force for given device dimensions [6]. Recently, pull-in has been proposed for use as voltage reference [7], [8]. Since silicon has good and stable mechanical properties [9], a voltage reference based on the pull-in voltage should be feasible. The simplestmicromechanicalsystem suitable for studying the stability of the pull-in voltage is composed of two electrodes, one movable and connected to a suspension beam with a certain spring constant (seefig. 1) andtheother onarigidsupportingsubstrate. Since the electrostatic force in a vertical field is inversely proportional to the square of the deflection and the restoring force of the beamis,toafirstapproximation,linearwithdeflection,anunstable system results in case of a deflection,, beyond a critical value,. The pull-in voltage,, is defined as the voltage that is requiredtoobtainthiscriticaldeflection[5]. Foraglobalstableequilibrium, the second derivative of the potential energyof the system with respect to deflection should be positive:, thus results from and is determined by the beam material, the beam dimensions, residual stress, and the electrodes dimensions (electrostatic energy). The approach is similar tothestructuralanalysisusedinthecaseofbucklingphenomenaof columns [10], extended to cope with stability analysis of electromechanical systems. The total potential energy has two terms, a mechanical and an electrical one. The mechanical potential energy,, is the elastic energy stored in the deformation of the mechanical structure. It has two components: the build-in strain energy component,, and the bending energy resulting from external applied forces,. The electric energy component,, is given by the energy stored into the equivalent capacitor plus the energy of the voltage supply [11]. Micromanufactured silicon beams usually exhibit residual stress (longitudinal and vertical), which is part of. The longitudinal stress may cause the buckling of the beam, and strongly influences the time and temperature dependence of the load-displacement curve. The effect is demonstrated in the load-deflection characteristic of a double-sided clamped structure, since this stress level cannot be released in an elongation. The pull-in voltage of a double-sided clamped structure /04$ IEEE

2 ROCHA et al.: ANALYTICAL MODELING OF STATIC PULL-IN 343 Fig. 1. Sketch of the basic eletromechanical device. reduces with compressive stress, which makes it unsuitable for use as a voltage reference. Moreover, the reproducibility would be limited by long-term drift due to stress relaxation. In the application presented here the pull-in voltage is exploited for the realization of a dc voltage reference. For long-term stability the longitudinal residual stress should not affect. Therefore, a single-side anchored beam with the other end free-standing should be used or the beam should be suspended using folded tethers at each end [12]. Both approaches ensure that the build-in strain energy component caused by longitudinal stress is zero. Even if these precautions have been implemented, a number of sources of uncertainty remain. These are a direct consequence of basic device operation. The pull-in voltage shows a temperature coefficient. Obviously, thermal expansion of the single-side clamped beam has to be considered, as the basic operation is the force balance between a surface effect (the electrostatic force) and a bulk effect (the compliance of the beam). This implies that the pull-in voltage is necessarily depending on the dimensions of the beam. The compliance of the beam varies inversely proportional with beam length, which increases with temperature. In addition there is the temperature dependence of the modulus of elasticity (Young s modulus, E) in silicon. The modulus of elasticity is included linearly in the expression for the compliance and thus in the pull-in voltage. As will be demonstrated here, these combined effects result in a negative temperature coefficient, TC, for. In addition to temperature effects, the basic device operation also gives rise to time dependencies. Since device operation depends on electrostatic actuation, it is prone to parasitic charge build-up. Surface charges play an important role on electrical stability behavior. These are primarily: 1) charge introduced during device manufacturing and 2) charges trapped in dielectric during device operation (in silicon these are often the native oxides layers on top of the electrodes). The surface charges yield a remanent electrostatic force [13], which might cause, especially in the case of trapped charges, a polarity dependent drift in long-term-operated micromechanical structures [11]. In this paper, a comprehensive study of the effect of all the sources of uncertainty on the pull-in voltage is performed. One and two-degrees-of-freedom (2-DOF) models are derived for different actuation modes, and compared with experimental data obtained on different test structures. The bistable regime of the Fig. 2. Fig. 3. Sketch of the 1-DOF structure. Single-side clamped structure. pull-in at different driving modes is also studied and the results of modeling are verified using pull-in measurements. Long-term measurements have been performed to monitor the drift mechanism due to oxide charging. Thermal cycling has been carried out to verify the predicted susceptibility to changes in ambient temperature. II. DEVICE OPERATION Two different structures have been used for studying pull-in. The first is a classic gap-varying, one-degree-of-freedom (1-DOF) structure, with folded springs (see Fig. 2). The second analyzed device is a single-sided clamped beam structure (see Fig. 3), with 2-DOF.

3 344 JOURNAL OF MICROELECTROMECHANICAL SYSTEMS, VOL. 13, NO. 2, APRIL 2004 Both structures allow two types of actuation: symmetric and asymmetric. In the case of asymmetric actuation, the voltage is applied between two sets of electrodes only; the upper set on the one side of the moving bar and the lower set on the other side (for the 2-DOF structure), yielding an unbalanced electrostatic force counteracted by the restoring elastic force. In case of a symmetric drive, the voltage is applied across all four sets of electrodes (see Fig. 2 and 3). An intuitive analysis suggests that the symmetric drive would yield a larger pull-in voltage as compared to driving the same device asymmetrically. In the asymmetrically operated device, the electrostatic energy is counteracted by the elastic beam energy until the beam collapses at the pull-in threshold. In the symmetric case, however, the electrostatic fields are in a first-order approximation balanced. Beyond a certain voltage level the beam nevertheless collapses. Inevitable asymmetries in the structure provide the onset for pull-in, which can occur only for voltage levels beyond which the electrostatic energy provided by one set of electrodes cannot be compensated by that of the other set of electrodes plus the elastic energy of the beam. Consequently, pull-in of the symmetrically driven beam is expected to be more abrupt and to take place at a larger value of the voltage applied as compared to asymmetric drive. joining the folded-beam segments are rigid, an analytical expression for can be found [16]: where: is the moment of inertia of the beams, is the Young s Modulus, and, and are the thickness, width, and length, respectively, of each beam. By solving the system an expression for the pull-in voltage is derived 2) Symmetric Drive: The total energy of the system when all four sets of fixed electrodes have the same electrical potential is: (2) (3) (4) III. PULL-IN MODEL Two methodologies are generally used for pull-in analysis. 1) The dynamic system approach the electromechanical system is described by a set of differential equations. The stability analysis is therefore an analysis of the stability of the equilibrium points of the equivalent dynamic system (indirect Lyapunov method) [14]. 2) The variational approach, in which the equilibrium points and their stability are determined by studying the variations of the total potential energy [15]. The equilibrium points are given by. These are stable if is a local minimum, which is determined by the eigenvalues of. The pull-in voltage is the value of the applied voltage for which the physical equilibrium point loses its stability. A quasistatic regime for the pull-in analysis is assumed, the forces acting on the system are derived from a potential and the points of interest are the equilibrium positions and not the motion describing the transition between equilibrium points. Therefore, the variational approach is used here. A. 1-DOF Pull-In Model 1) Asymmetric Drive: The total energy of the electromechanical system shown in Fig. 2 can be expressed as The stability at the boundary conditions given by (3) leads to a pull-in voltage value This confirms that pull-in of the symmetrically driven beam requires indeed a 30% larger value of the voltage applied, as compared to asymmetric drive. This is a fundamental issue that is often overlooked in literature. B. 2-DOF Pull-In Model Two parameters ( and ) fully determine the configuration of the electromechanical system under study (see Fig. 3 and Fig. 4). The total energy of the system can be written as where: (5) (6) (7) represents the mechanical energy and the electrical energy stored in the capacitor. The pull-in voltage can be found analytically by solving the equation (8) (1) where denotes the capacitance at the initial position and the spring constant of the structure. Assuming that the trusses in variable. The values of the state variables (, ) correspond to the equilibrium position determined from (9)

4 ROCHA et al.: ANALYTICAL MODELING OF STATIC PULL-IN 345 For symmetric drive, the structure retains its zero-displacement equilibrium position until the pull-in is reached. Approximation of the potential by the first two terms of the Taylor series around the equilibrium position, and solving (8) using, yields the following analytical expression for the pull-in voltage of the symmetric drive (13) Fig. 4. Identification of the state variables used in the 2-DOF model. 1) Asymmetric Drive: In case of the asymmetric drive only two of the four electrodes are actuated. The mechanical energy is described by (10) where: is the Young s Modulus, I is the moment of inertia and is the length of the beam. The electrical energy (considering as zero-level energy) is described by (11) A local continuation method was implemented in Mathematica for tracking the coordinates (, ) of the equilibrium point with increasing voltage. The approach used to solve the problem is based on sweeping of the voltage from the initial value toward increasing positive values. For each voltage value the stability points are computed, by approximating the general potential in Taylor series around the previously computed equilibrium point,. denotes the normalized tip deflection. This makes it possible to track the evolution of the equilibrium point as a function of. For the computed values of,, the eigenvalues of the associated Hessian at that point are computed. If any of these has a negative value, then the equilibrium point is unstable, and the pull-in voltage is found. 2) Symmetric Drive: Unlike the case of the asymmetric drive, where an iteration procedure is required to solve (8), an approximate analytical expression for the pull-in can be derived for the symmetric mode of operation. The mechanical energy for the symmetric drive is the same as the one for the asymmetric drive and described by (10). The Electric energy is described by (12) where denotes the normalized inter-plate gap distance and the normalized zero-voltage overlapping length between the movable and fixed plates. Some remarks should be made with respect to this expression (13). The most significant is the resemblance with the pull-in formulas given by (4) and (6). However, in the present case the dominant cause of pull-in is not a resultant force, but rather a resultant moment,, with. The analytical expression (13) is very useful to a designer in a systematic sensitivity analysis with respect to different geometry parameters. C. Hysteresis Model One important aspect of the pull-in phenomenon is its bi-stable regime, also called hysteresis (see Fig. 8) [17]. When pull-in occurs, there is an unbalance between electrostatic and elastic forces. The resulting net force/moment drives the movable part toward the mechanical stopper. When it hits the fixed structure, a reaction force develops, and the static equilibrium is reestablished. For reliability reasons and for avoiding electrical short-circuits, MEMS structures are usually designed with stoppers to limit the displacement of the movable part. It should be emphasized that the hysteresis in such a MEMS device is not due to a parasitic or practical device limitation, such as sticking. Rather, it is a fundamental feature. The stopper position is not the cause of hysteresis, but it determines the hysteresis magnitude. The bi-stable regime associated with the pull-in phenomenon can be explained using the parallel-plate structure of Fig. 1. The stopper is positioned somewhere between the values of the normalized displacement at pull-in and full gap distance,, to prevent the beam from hitting the counter electrode and thus compromising reliability and short-circuiting the capacitor. Fig. 5 illustrates the variation of the electrostatic and elastic forces with the displacement. Pull-in occurs when the mechanical force can no longer balance the electrostatic one. After pull-in the structure will stop at the designed stopper position (in this example in the middle of the initial gap), where the electrostatic force equals the sum of the mechanical force with the reaction force of the stopper. After pull-in has occurred, in the new equilibrium position the unbalance between electrostatic force ( in Fig. 5) and the elastic one is higher than at the pull-in onset. A lower voltage is required to reach a balance between the electrostatic and mechanical force ( in Fig. 5). There are two distinct displacement solutions for this

5 346 JOURNAL OF MICROELECTROMECHANICAL SYSTEMS, VOL. 13, NO. 2, APRIL 2004 Fig. 6. Photograph of the folded spring structure. Fig. 5. Explaining hysteresis. equilibrium of forces (the reaction force of the stopper becomes zero ). The larger displacement solution is unstable, so the movable part, after the voltage decreases below a critical release voltage, will jump back to a lower equilibrium position. The hysteresis amplitude is reduced when moving the displacement constraint, as imposed by the mechanical stopper position, closer to the critical displacement at pull-in. The goal of the hysteresis model is to find the release voltage,, at which the mechanical force equals the electrostatic one at the stop position. For the 1-DOF case, we can obtain the release voltage by solving the equation,in, and considering the known stopper position as the displacement. For a stopper at a normalized position the following expression for the release voltage is obtained (asymmetric case): (14) Analogously, in the symmetric case the release voltage can be expressed as (15) For the 2-DOF structure, the release voltage can be calculated, by solving (9) in variable, while the state variables (, ) correspond to the equilibrium position given by (16) with and being the stop displacement and stop angle, respectively. The system can be solved numerically, and the results are presented in the next sections. IV. TEST STRUCTURES FABRICATION The epi-poly process was used for the fabrication of the test structures [18], [19]. This process is very suitable for the fabrication of relatively thick and high aspect ratio free-standing beams on top of a silicon wafer. Epitaxial growth at about 700 nm/min is used to yield a thick polysilicon layer on top of a dielectric oxide. After deposition the polysilicon layer is patterned using deep reactive ion etching (DRIE). Microstructures are subsequently released by selectively etching the underlying dielectric sacrificial layer using the DRIE holes as access channel. This process is used for fabrication of thick polysilicon structures, despite the fact that beam characteristics are more likely to be affected by stress and stress gradients in vertical and lateral direction, compared to identical devices fabricated in mono-crystalline silicon. A part of this research is to investigate whether the resulting stability is impaired to an extent that bulk micromachined structures should be used instead. A. Structure 1 Classical 1-DOF The structure used to test the 1-DOFmodel (see Fig. 6)consists offourfoldedbeams,375- longand2.8- wide,connectedto a rigid central bar of length. A set of parallel plate comb driveactuators, with aninitialgapof2 between fixedandmovablestructures, areusedforactuation.themeasurementofthedisplacementisdonebysensingthechangeincapacitanceontwosets of sensing capacitors. Stoppers located on either side of the rigid bar limit the movement after pull-in is reached.

6 ROCHA et al.: ANALYTICAL MODELING OF STATIC PULL-IN 347,, and. For the asymmetric case, the displacement on the structure is computed using the first equation of system (3) at increasing voltages until the pull-in value given by (4). Similarly, the displacement is computed for decreasing voltages starting at the release voltage given by (14). For the symmetric case (6) and (15) yield the pull-in and release voltages respectively. The measured pull-in and release voltages for the 1-DOF structure are presented in Fig. 8. For the asymmetric case, the measured values are in clear agreement with the predicted ones, while for the symmetric drive there is a small deviation. Measurements on the symmetric drive reveal larger deviations from theoretical prediction, which indicates a larger dependency on variations in the dimensions due to tolerances in fabrication. As mentioned before, for the symmetric drive the spring force needs to counterbalance a difference in electrostatic forces, compared to an electrostatic force addition in the asymmetric mode of operation. Both the simulations and the measurements confirm the more abrupt pull-in for the symmetric drive. Fig. 7. Photograph of the single-side clamped structure. B. Structure 2 Inverted Pendulum With Interdigitated Finger Sense Capacitor This device is basically a free-standing lateral beam (200- long, 3- wide and depth of 10.6 ) anchored at one end (the base) only (see Fig. 7). The beam can be deflected by electrostatic actuation in the plane of the wafer using a voltage applied across the parallel-plate capacitors (2- gap). These are composed of two sets of electrodes located alongside the freestanding tip, with counter electrodes mechanically anchored to the substrate. The deflection can be measured using a set of differential sense capacitors located alongside the free-standing tip. Electrically insulated stoppers are placed for limiting the lateral motion. V. CASE STUDIES In this section, the previously derived models are computed for the test structures and compared with experimental results. Finite element modeling is also used and compared with analytical and experimental results (structure 2 only). Preliminary long-term measurement results on both structures are presented and discussed. Pull-in is measured by capacitance readout in all structures. A voltage is applied to the actuator capacitors and pull-in is detected by the sudden change in capacitance. A. Classical 1-DOF Structure The schematic of this structure is presented in Fig. 2. The main dimensions, needed to compute the pull-in and release voltages are:,,, B. 2-DOF Inverted Pendulum The 2-DOF model derived in the previous sections was applied to the second structure. Fig. 3 shows the schematic of the device. The main dimensions and material properties needed to numerical compute the pull-in and release voltages are:,,,,,,,, and. Finite element analysis (FEA) has also been also used to check the analytical model, and to compare with experimental measurements. The predicted pull-in voltage from the analytical model is ; the trajectory of the equilibrium point as a function of the applied voltage is shown in Fig. 9. Fig. 10 presents the results for the 2-D FEA model. Using contact elements in Ansys simulator, the pull-in and release voltages of the structure have been computed. The simulations predict a pull-in voltage of and a release voltage about. The clear agreement with the outcome of the analytical model demonstrates that such physical modeling approaches may be used as well for higher dimensional structures (2 or more degrees of freedom). The pull-in voltage for the symmetric mode of operation was subsequently computed, as well as the release voltages for both operation modes (using the 2-D presented hysteresis model). Fig. 11 shows the predicted behavior of the structure for both actuation modes. Until now, two modes of actuation were differentiated symmetric and asymmetric. Looking to Fig. 3, is clear that an asymmetric mode can be operated in two ways: voltages can be applied onthebottomrightandtop leftelectrodes asymmetricright or the top right and bottom left electrodes can be actuated asymmetric left. These two different actuations for asymmetric drive, together with the symmetric drive (all electrodes are driven) have been applied to the fabricated structures. Fig. 12 presents the measured pull-in voltages for the three types of actuation. Unexpectedly, different values for the pull-in voltage are measured for the two types of asymmetric actuation. This indicates that the structure is not at the position

7 348 JOURNAL OF MICROELECTROMECHANICAL SYSTEMS, VOL. 13, NO. 2, APRIL 2004 Fig. 8. Computed and measured pull-in and release voltages for 1-DOF structure for a) asymmetric and b) symmetric drive. Fig. 9. Variation of equilibrium point with applied voltage. as we expected, but has a initial zero-voltage equilibrium position. The causes for this behavior are related to technological processing steps. Residual stress and (vertical and lateral) gradients are usually present in a structural

8 ROCHA et al.: ANALYTICAL MODELING OF STATIC PULL-IN 349 Fig. 10. Results of FEM simulation on an asymmetrically operated device. Fig. 11. Results of the 2-DOF model for a device asymmetrically and symmetrically driven. layer fabricated in a surface micromachining process. The fact that this structure is clamped at one side only ensures that the residual stress does not affect the pull-in of the beam. However, a stress gradient would displace the structure from the initial zero position. 1) The Effect of Stress Gradients on the Pull-In Voltage: Among the mechanical properties specified for the epi-poly structural layer is a vertical stress gradient of [20] (i.e., in the direction normal to the plane of the wafer). This stress gradient cannot be the cause of the different left-right pull-in voltages measured, because such a gradient would cause a bending of the beam away from the substrate plane, causing no change on the initial zero position. A possible cause for the observed values can be the in-plane ( lateral) stress gradient (i.e., parallel to the plane of the wafer). Information on a lateral gradient stress is not available, but its presence seems plausible. Reinterpretation of the measurements performed (in several 2-DOF devices) suggests a value of about 0.2. Using beam stress theory [21], the displacement and angle caused by such a value of stress gradient in the beam are calculated. Values of have been computed for a 0.2 stress gradient. If the mechanical energy caused by the in-plane stress gradient is in-

9 350 JOURNAL OF MICROELECTROMECHANICAL SYSTEMS, VOL. 13, NO. 2, APRIL 2004 Fig. 12. Experimental results for a device a) asymmetrically right, b) asymmetrically left, and c) symmetrically driven.

10 ROCHA et al.: ANALYTICAL MODELING OF STATIC PULL-IN 351 TABLE I EXPERIMENTAL DATA AND NUMERICALLY COMPUTED VALUES FOR THE PULL-IN AND RELEASE VOLTAGES (STRUCTURE 2), WHICH INCLUDES THE EFFECT OF STRESS GRADIENTS IN THE DEVICES Fig. 13. Stability test at constant temperature. cluded in the model, an additional term must be introduced in the expression of elastic energy. Using (17) we can compute again the new pull-in values for all three actuation cases (considering either a positive or negative ). Table I presents the results obtained. There is a good agreement between the predicted and experimental values. The small deviations (less than 5%) are due to process tolerances and uncertainties regarding the value of the Young s Modulus. Taking into account the bending caused by a stress gradient in the model leads to a better agreement with the observed experimental results (differences between asymmetric right and asymmetric left). This is especially true for the symmetric case, where the incorrect assumption of a stable position (where the electrostatic forces would cancel each other), has the strongest impact. The initial, zero-voltage bending caused by the stress gradient induces an asymmetry in the configuration which lowers the pull-in value, compared to the ideal case. Even if the analytical 2-DOF model is quite simple, it adequately describes the relevant aspects of the pull-in of micromechanical structures (including the bistable regime). C. Long-Term Measurements After validation of the analytical models, some of the devices have been used for long-term stability measurements. Long-term stability of pull-in voltage is essential for application in metrology, or for material characterization. The time stability of the pull-in voltage has been measured using the 2-DOF structure. Pull-in has been measured over 26 days at constant temperature and the result is presented in Fig. 13. Other devices (1-DOF structure) have been used to measure stability with temperature. The measurements were performed after stabilization during a 2-week burn-in period. Fig. 14 shows one of those measurements. A temperature coefficient of about is observed. Two interesting aspects are observed from Figs. 13 and 14. The first one is the drift during the first 8 days (see Fig. 13) and stabilization afterwards, while the second characteristic is the dependence of the pull-in voltage on temperature (see Fig. 14). The initial drift is expected to be due to either charging of a dielectric layer between the electrodes or to mechanical stress relaxation. To identify the actual mechanism another measurement on a 2-DOF structure was performed (see Fig. 15). After an operation break (no voltage is applied) the initial voltage value reappears, thus eliminating the hypothesis of mechanical stress relaxation. The temperature dependence of the pull-in voltage is caused by temperature sensitivity of the geometrical dimensions and of the modulus of elasticity. These dependencies are quantitatively analyzed in what follows. 1) Surface Charging: During fabrication of the devices, a Teflon-like film is deposited on the sidewalls [22]. This polymer, used as a passivation layer and deposited during plasma etching, is not removed at the very end of processing steps and is very likely a cause of the 17 mv drift observed over the first 8 days. The electrostatic force between electrodes

11 352 JOURNAL OF MICROELECTROMECHANICAL SYSTEMS, VOL. 13, NO. 2, APRIL 2004 Fig. 14. Stability test pull-in voltage versus temperature. Fig. 15. Stability test at constant temperature and operation breaks. considering the presence of trapped charges on the electrode surface has been derived in [13] and can be written as can be expressed as (not considering quadratic terms). If now, this temperature dependent spring is introduced in (4) and we take the derivative to temperature (T), the following expression is found: (18) where is an offset voltage leading to a shift of the parabolic force versus voltage curve due to trapped charges, is the thickness of the polymer layer, is the polymer permittivity, and is the charge density. Values found in literature for Teflon-like layers report a and [23]. Typical values for the thickness of the polymer layer are in the range of few nm. A realistic value, such as, yields, in agreement with the observed drift. 2) Effect of Thermal Expansion on the Pull-In Voltage: The source of temperature dependence of the pull-in voltage appears to be the Young s Modulus temperature dependence plus the thermal expansion of the polysilicon. These material characteristics change the mechanical spring of the system, leading to changes in the voltage required to yield critical deflection (pull-in voltage). If we disregard the effect of thermal expansion on the capacitance, remains as the temperature dependent part and (19) From (19) it can be concluded that the pull-in thermal coefficient is not linear, and that it increases with temperature (note that ). For the epi-poly process used to fabricate the devices [20], a polysilicon with thermal expansion coefficient is specified, while the Young s modulus thermal coefficient is. For the 1-DOF device presented, and considering a constant temperature coefficient ( in (19)), atcof results, in reasonable agreement with the measured value of. The difference between measured and computed values could be attributed to the uncertainties in he exact values of the thermal expansion and Young s modulus thermal coefficients. The temperature coefficient depends on both the characteristics of the poly-layer used and the details of the suspension of the structure.

12 ROCHA et al.: ANALYTICAL MODELING OF STATIC PULL-IN 353 VI. CONCLUSION Three main conclusions with respect to the modeling can be drawn from the presented work. First, the good agreement between analytical model and results on all relevant aspects (including the preliminary results of long-term measurements) lead to the conclusion that analytical models can be used to characterize the electromechanical behavior of the structure with acceptable accuracy. Significant gains in design cycle time are achieved as compared to FEM on the same computer. Second, relatively simple test structures could be used to extract the needed information on material properties, such as stress gradient, temperature coefficient and charge densities. Literature data has been adequate in quantitatively explaining the phenomena observed. Third, the hysteresis is intrinsic to device operation, but is predictable for a given drive mode. The analytical modeling has provided the insight leading to a number of significant conclusions on basic device performance, not previously drawn from FEM. In electrostatic actuators and servo-operated capacitive accelerometers the displacement range should be maximum. Even if the symmetric drive offers a higher pull-in voltage, Fig. 9 clearly indicates that the usable displacement is very limited due to the more sudden pull-in. For this reason the asymmetric drive with the more gradual displacement-to-voltage characteristic is to be preferred in this application. For proper operation of a pull-in structure as a voltage reference, the pull-in should be as abrupt as possible and the effect should be reproducible. The symmetric drive has a better-defined threshold with an abrupt pull-in. However, for use as a dc voltage reference, where stability and device-to-device reproducibility is crucial, the asymmetric drive offers superior operational performance as compared to the symmetric one. Therefore, also in this application area, the asymmetric drive should be selected. Due to the dependence on process tolerances, the symmetric drive can be used as a good process monitor. Its displacement vs. voltage characteristic may also be helpful in obtaining a better insight into the dependence on fundamental design parameters and influence of tolerances and parasitic effects on the pull-in voltage. The asymmetric drive can be used as a versatile test structure for measuring in-plane stress gradients in the structural layer in a surface micromachining process, by alternating between applying the voltage to the upper left-lower right and lower leftupper right electrodes. The temperature coefficient of the pull-in voltage is significant, but is accurately predictable and thus can be corrected or compensated for [24]. The major source of drift is relaxation of charge in a dielectric layer, which can be significantly reduced by proper surface treatment. REFERENCES [1] C. S. Liu, A. Barzilai, O. Ajakaiye, H. K. Rockstad, and T. W. Kenny, Performance enhancements for micromachined tunneling accelerometer, in Proc. Transducers99, 1999, pp [2] L. Que, J. S. Park, and Y. B. Gianchandani, Bent-beam electrothermal actuators Part I: Single beam and cascaded devices, J. Microelectromech. Syst., vol. 10, pp , June [3] S. T. Cho, K. Najafi, and K. D. Wise, Internal stress compensation and scaling in ultra-sensitive silicon pressure sensors, IEE Tr. ED, vol. 39, pp , [4] P. M. Osterberg and S. D. Senturia, M-TEST: a test chip for MEMS material property measurement using electrostatically actuated test structures, J. Microelectromech. Syst., vol. 6, pp , June [5] H. 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Eurosensors XVII, 2003, pp Luis Alexandre Rocha was born in Guimarães, Portugal, in In 1995, he began to study electronic engineering at the University of Minho, Portugal, where he graduated in Since February 2001, he has been pursuing the Ph.D. degree at the Department for Microelectronics, Faculty of Electrical Engineering Mathematics and Computer Science of the Delft University of Technology, Delft, The Netherlands. The topic of his research includes the study and design of MEMS for application in microinstruments.

13 354 JOURNAL OF MICROELECTROMECHANICAL SYSTEMS, VOL. 13, NO. 2, APRIL 2004 Edmond Cretu was born in Romania in He received the M.Sc. degree in electrical engineering from the politechnic University of Bucharest in 1989 and the Ph.D. degree from Delft University of Technology, in He was Researcher in Romanian Academy and Associate Assistant at the Faculty of Electrical Engineering of Politehnica University of Bucharest. Since March 2000, he has worked for Melexis Belgium, as Senior Designer and Project Manager in the field of inertial systems, with emphasis on MEMS-based gyroscope systems. Reinoud F. Wolffenbuttel received the M.Sc. degree in 1984 and the Ph.D. degree in 1988, both from the Delft University of Technology, Delft, The Netherlands. Between 1986 and 1993, he has been an Assistant Professor and since 1993, he has been an Associate Professor at the Department of Microelectronics, Faculty of Information Technology and Systems of the Delft University of Technology and is involved in instrumentation and measurement in general and on-chip functional integration of microelectronic circuits and silicon sensor, fabrication compatibility issues and micromachining in silicon and microsystems, in particular. He was a Visitor at the University of Michigan, Ann Arbor, in 1992, 1999, and 2001, Tohoku University, Sendai, Japan, in 1995 and EPFL Lausanne, Switzerland in Dr. Wolffenbuttel is the recipient of a 1997 NWO pioneer award. He has served as General Chairman of the Dutch National Sensor Conference in 1996, Eurosensors in 1999, and the MicroMechanics Europe Workshop in 2003.