The Pull-In of Symmetrically and Asymmetrically Driven Microstructures and the Use in DC Voltage References

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1 IEEE Instrumentation and Measurement Technology Conference Anchorage, AK, USA, 1-3 May 00 The Pull-In of Symmetrically and Asymmetrically Driven Microstructures and the Use in DC Voltage References L.A. Rocha, E. Cretu* and R.F. Wolffenbuttel Delft University of Technology, ITS/Et, DIMES, Mekelweg 4, 68 CD Delft, The Netherlands *Also at: Melexis, Transportstraat 1, Tessenderloo, Belgium Abstract Micromechanical structures have been designed, fabricated in silicon and tested for use as on-chip voltage reference. Applications are in electrical metrology and in integrated silicon Microsystems. Microbeams of 100µm length, 3µm width and 11µm thickness are electrostatically actuated with a very reproducible pull-in voltage. The structure can be either symmetrically or asymmetrically actuated, resulting in different pull-in voltages, as well as different operational specifications e.g. hysteresis and reproducibility. A two dimensional (-D) energybased analytical model for the static pull-in is derived for both actuation types and compared with measurements. Devices have been designed and fabricated in an epi-poly process. Measurements show a pull-in voltage in the V range for the asymmetric case and V range for the symmetric case, both in agreement with modeling. Keywords MicroElectroMechanical Systems (MEMS), DC voltage reference, pull-in. I. INTRODUCTION The advances in MicroSystem Technology (MST, also referred to as Micro-Electro-Mechanical System -MEMStechnology), has enabled application in many fields, such as: automotive, optical switching for communication, control and analytical chemistry. This technology is gradually also penetrating into mainstream Instrumentation and Measurement (I&M) application areas, such as electrical metrology [1]. Two devices have resulted: the thermal rmsto-dc converter and the electrostatic rms-to-dc converter []. An important device within electrical metrology is the dc reference. Zener references are used as so-called transfer standard, however, the operation of a Zener diode is based on avalanche breakdown and is associated with a high noise level. Recent research has demonstrated that the pull-in voltage of a micromechanical structure can be used as a dc reference with a reduced noise [3,4]. The pull-in voltage of beams of different design has been investigated with the purpose of using the micromechanical structure as an on-chip voltage reference. As the electrostatic force in a vertical field is inversely proportional to the square of the deflection and the restoring spring force of the beam is, in a first approximation, linear with deflection, an unstable system results in case of a deflection beyond a critical value. The pull-in voltage, V pi, is defined as the voltage that is required to obtain this critical deflection. The basic phenomenon is the loss of stability at the equilibrium position, where the elastic forces equilibrate the electrostatic ones. Two different modes of operation are possible: symmetric drive and asymmetric drive. The key performance parameter in metrology is long-term device stability. The microstructure design and the selected electrostatic mode of operation for maximum stability are presented here. II. DEVICE OPERATION As long-term stability is the most important property of the device, the design is based on a single-sided clamped beam fabricated using crystalline material. The structure can be either symmetrically or asymmetrically driven, resulting in different pull-in voltages (Fig. 1). In case of asymmetric actuation, voltages are applied between the beam and bottom right and top left electrodes, while for the symmetric drive, all the four electrodes are used. Figure 1: Single-side clamped structure. A simple qualitative analysis shows that the symmetric drive yields a larger pull-in voltage as compared to driving the same device asymmetrically. This is due to fact that in the asymmetrically operated device, the electrostatic energy is /0/$ IEEE

2 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, due to asymmetries in the structure. The balancing in the electrostatic domain causes the pull-in to take place at a higher voltage level. The higher pull-in voltage at given device dimensions could be an advantage. However, the dependence on device asymmetries and imperfections, and thus on device fabrication, is likely to favor the asymmetric drive. For a more conclusive analysis analytical modeling is required. III. D PULL-IN MODEL The modeling is based on the energy of the system (the coupling between mechanical and electrical energy). Two parameters (w 1 and ϕ 1 ) fully determine the configuration of the electromechanical system (Fig. 1 and Fig. ). in variable V, and the state variables (w 1, ϕ 1 ) correspond to the equilibrium position determined from U U w 0 ( ; ) = 1 xeq V ( w, ; ) = 1 1 V U eq ϕ eq (3) x 0 ϕ 1 III.1 Asymmetric Drive On the asymmetric drive two of the four electrodes are actuated. The mechanical energy is described by U elastic l x 1 ϕ EI 1 y w1 w = 1 ( w = 1, ϕ 1) EI y dx ϕ1 3ϕ 1 + (4) 0 x lx lx lx where E is the Youngs Modulus, I y is Moment of Inertia and l x is the length of the beam, and the electrical energy (taking w, ϕ ) (0,0) as zero-level energy) is described by ( 1 1 = U electric 1 ( w1, ϕ 1; V ) = V [ C( w1, ϕ1) + C( w1, ϕ1) C0 ] (5) Figure : Identification of the state variables used in the model. The total energy of the system can be written as U w, ϕ ; V ) = U ( w, ϕ ) + U ( w, ϕ ; ) (1) ( 1 1 elastic 1 1 electric 1 1 V where U elastic w 1, ϕ ) represents the mechanical energy and ( 1 U electric ( w 1, ϕ 1; V ) represents the electrical energy stored in the capacitor. The pull-in voltage can be found analytical by solving the determinant equation A local continuation method was implemented in Mathematica TM for tracing the equilibrium point coordinates at increasing voltage. The approach used to solve the problem is based on sweeping of the voltage, from the initial value V 0 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, x1 = { wneq, [ k 1], ϕneq, [ k 1]}. W n denotes the normalized tip deflection wn = w1 /( Ly + ty + dy ). This makes it possible to trace the evolution of the equilibrium point as function of V. For the computed values of w1, eq, ϕ 1, eq, the eigenvalues of the associated Hessian at that point can be computed. If any of these has a negative value, then the equilibrium point is unstable, and drawn as a separate line. First, the analytical model was used and the results were subsequently verified using finite element modeling (FEM). Figure 3 presents the predicted pull-in voltage from both analytical modeling and finite element simulations. The predicted pull-in voltage from the analytical model is V pi =9.6V, in agreement with the FEM results. U ( w1, ϕ1; V ) x = 0 ()

3 III.3 Hysteresis Model An important aspect of movable microelectromechanical systems with a stop position is mechanical hysteresis [5]. In our case, this phenomenon is important, as it seriously complicates the design of a feedback system required to control the structure. It should be emphasized that the hysteresis in such a MEMS device is not due to a parasitic or practical device limitation. Rather is inherent to the basic device operation and not to e.g. stiction. Figure 3: Variation of the equilibrium point with applied voltage. III. Symmetric Drive For the symmetric type of actuation, an approximate analytical expression for the pull-in can be derived. The mechanical energy for the symmetric drive is the same as the one for the asymmetric drive and described by (4). The Electric energy is described by 1 Uelectric ( w1, ϕ1; V) = V ( 1, 1) 4 0 Ctotal w ϕ C C ( w, ϕ ) = Cw (, ϕ ) + C( w, ϕ ) + Cw (, ϕ ) + C( w, ϕ ) total (6) The hysteresis phenomenon can be demonstrated using the one degree of freedom case of a parallel plate structure. Computing both the electrical and mechanical force for such a case we can see the evolution of the equilibrium position with increasing voltage Fig. 4. Pull-in occurs when the mechanical force no long equilibrates the electrical one (V=V 4 ). The last equilibrium position occurs at x n =1/3. After pull-in the structure will stop at the designed stopper position (in this example we considered ½ of the initial gap), where the electric force equilibrates the sum of the mechanical force with the reaction force of the stopper. A characteristic of the symmetric drive is that the structure remains in an equilibrium position ( w 1, ϕ1) = (0,0), 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 () using ( w 1, ϕ1) = (0,0), yields the following analytical expression for the pull-in voltage of the symmetric drive V pi 3 d 0n EI yt = (7) C l ( 3d + t (3 + ( 3 + t ) t )) 0 x 0n where d 0n = d 0 /( L y + t y + d y ) denotes the normalized inter-plate gap distance, and t = t y /( L y + t y + d y ) the normalized zero-voltage overlapping length between the movable and fixed plates. Such an analytical expression is very important for the designer, because the influence of the various dimensions on the pull-in voltage can be analyzed. For the dimensions of the structure used: l x =100µm, d 0 =µm, L y =98µm, d y =8µm and t y =50µm, the pull-in for the symmetric drive is V pi =1.6V according to (7). Figure 4: Explaining the hysteresis

4 When we reduce the applied voltage we would like to return to a stable position right after below the pull-in voltage, but that is not the case. Rather, after pull-in the electric force increases as the gap decreases ( Fel ( V4 ;0.5) on Fig. 4). Because the electric force is now larger than it was when it collapsed, a lower voltage is needed to make that the electric force gets the same value as the mechanical force ( Fel ( V3;0.5) on Fig. 4). When the mechanical force equilibrates the electric force (the reaction of the stopper becomes zero), we have two clear solutions. Because the higher displacement solution is on an unstable position, the structure returns to the stable zero position. The hysteresis depends on the stop position. Designing a structure with a stop position closer to the deflection at pullin results on a smaller value of the hysteresis. Therefore if we can calculate the hysteresis and the pull-in value from device dimensions, we fully describe analytically the static behavior of the structure. stop position. For our D model this can be done by solving (3) in variable V and the state variables (w 1, ϕ 1 ) correspond to the equilibrium position given by s = w + ( t + s )sin( ϕ ) (8) y 1 x x 1 The system was solved numerically for both symmetric and asymmetric cases (s y =µm, t x =100µm and s x =10µm), and the predicted behavior is presented on figure 5. IV. DEVICE FABRICATION A modification on surface micromachining, a so-called epipoly process [6,7], was used for the fabrication of 11µm thick single-side clamped 100 µm long free-standing structures with electrode structures at the tip. A fabricated pull-in device is shown in figure 6. Figure 6: Fabricated microstructure. Figure 5: Predicted capacitance changes for a) asymmetric and b) symmetric drive. The goal of the hysteresis model is to find the voltage V hyst, for which the mechanical force equals the electric one at the The device is basically a free-standing lateral beam anchored at one end (the base) only. The beam can be deflected by electrostatic actuation in the plane of the wafer using a voltage applied across parallel plate capacitors composed of two sets of electrodes located alongside the free-standing tip, with counter electrodes anchored to the substrate. The deflection can be measured using the differential sense capacitor located directly on top of the substrate and aligned with the square-shape electrode at the

5 tip of the beam. These buried polysilicon electrodes are electrically isolated from the substrate and placed symmetrically on either side of a guard electrode directly underneath the axial direction of the beam. Finally, there are electrically isolated stoppers to limit the lateral motion. The electrodes beneath the movable structure are used for capacitive detection of the pull-in voltage. The readout circuit is based on a capacitor bridge with two active arms. The complication in the readout is the af resolution requirement in the presence of pf parasitic capacitors. For symmetric actuation, the measured value was about 11 V, and showed a relatively large sample-to-sample deviation. The larger deviation of measurements to theoretical prediction, and the larger spreading in pull-in values, indicates a larger dependency on variations in the dimensions due to tolerances caused by the fabrication techniques. As mentioned, this is due to the fact that in symmetric drive, a difference in electrostatic forces is balanced by the spring force, whereas an electrostatic force is directly balanced in the asymmetric mode of operation. V. EXPERIMENTAL RESULTS The first results are presented on figure 7 for both cases. As expected, the asymmetric drive presents a parabolic change on the capacitance until the pull-in is reached, and the symmetric case presents a much sharper transition at pull-in. The measured pull-in voltage, for the asymmetric actuation, was about 9.1V, which is near the predicted value from Mathematica model (V pi =9.6 V). The difference is mainly due to the simplifying assumptions used, the process deviations and uncertainties in the value of the Young s modulus used in the numerical computation. Figure 8: Measured behavior of the structure for a) asymmetric and b) symmetric drive. Figure 8 presents the total behavior of a sample (with increasing and decreasing voltage) for the two actuations being studied. When we compare the results with those obtained from the modeling (Fig. 5), a reasonable agreement is observed. For the asymmetric case we have a measured hysteresis gap of about 1V, which is in agreement with the model. The small shift in position is mainly due to the simplifications in the model. Figure 7: Experimental results. a) asymmetric and b) symmetric actuation. For the symmetric case, the release voltage is in close agreement with the model. However, the pull-in takes place at a voltage level lower than expected, due to the high sensitivity to tolerances mentioned before. After pull-in, the

6 structure goes to a position where the electric force is in equilibrium with the sum of the mechanical force and the reaction force on the stopper. Because of this new equilibrium situation, the small imperfections, that do give a significant contribution to the pull-in, do not affect the release voltage. Actually, the release voltage is closer to the model value as compared to the asymmetric case, because the non-idealities (stray capacitances, fringe fields) that are not considered in the model tend to cancel out in the symmetric device. VI. CONCLUSIONS For proper operation of this structure as a voltage reference, the pull-in should be as abrupt as possible and the effect should be reproducible. As shown, the asymmetric drive performs much better with respect to pull-in voltage reproducibility, while the symmetric drive has a better defined threshold. The key performance parameter that leads to the selection of either type is strongly application dependent. For use as a dc reference, where stability and device-to-device reproducibility is crucial, the asymmetric case seems to offer superior operational performance as compared to the symmetric one. [3] A.S. Oja, J. Kyäräinen, H. Seppä and T. Lampola, "A micromechanical DC-voltage reference", Conference digest of CPEM000, Sydney, Australia, May , pp [4] E. Cretu, L.A. Rocha and R.F.Wolffenbuttel, "Using the Pull-in Voltage as Voltage Reference", in the Proceedings Transducer01, Munich, Germany, June , Vol. 1, pp [5] J.R. Gilbert, G.K. Ananthasuresh, S.D. Senturia "3D modeling of contact problems and hysteresis in coupled electro-mechanics", Micro Electro Mechanical Systems, 1996, MEMS '96, Proceedings, pp [6] [7] M. Offenberg, F.Lärner, B. Elsner, H.Münzel and W.Rierhmüller, Novel process for an integrated accelerometer, Proc. Transducers 95, Stockholm, Sweden, June 5-9, Vol.1, pp [8] J. Wibbeler, G. Pfeifer and M. Hietshold, Parasitic charging of dielectric surfaces in capacitive MEMS, Sensors and Actuators, Vol. A71 (1998) Although the symmetric drive seems not to be the best choice to operate the device, it can be used as good indicator of the tolerances and imperfections in a surface micromachining process. Its actuation characteristics may also be helpful in obtaining a better insight in the dependence on fundamental design parameters and influence of tolerances and parasitic effects on the pull-in voltage. The design and fabrication of a pull-in microstructure for application in metrology has been described in this paper. The feasibility has been demonstrated. First stability measurements on the asymmetric drive indicate a pull-in voltage change of 1mV after the first 10 days of continuous operation, and stabilization afterwards. The initial drift is expected to be due to charging of a dielectric layer between the electrodes [8]. We suspect a polymer to be deposited on the sidewall during plasma etching to be responsible. Measurements are scheduled to verify long-term stability and to investigate whether mechanical-thermal noise is a dominant performance limiting factor. REFERENCES [1] R.F. Wolffenbuttel and C.J. van Mullem, "Microtechnology and Microsystems in Measurement Applications", IEEE I & M magazine, Vol. 4, N 3 September 001, pp [] B.P. van Drieënhuizen and R.F. Wolffenbuttel, "Integrated electrostatic RMS-to-DC converter using IC-compatible surface micromachining", in Proc. Transducer 95, Stockholm, Sweden, June , pp