International audienceThis work introduces a MEMS oscillator composed exclusively of mechanical switches as logic components. The electromechanical model of the system is developed and the conditions for a periodic response are established

Barniol, N.

Juillard, Jérôme

Munoz-Gamarra, J.L.

HAL-CentraleSupelec

Analysis of a MEMS-based ring oscillatorJ.L. Munoz-Gamarra, N. Barniol, Je´roˆme JuillardTo cite this version:J.L. Munoz-Gamarra, N. Barniol, Je´roˆme Juillard. Analysis of a MEMS-based ring oscillator.ISCAS’12, May 2012, Seoul, North Korea. pp.2103-2106, 2012. <hal-00725674>HAL Id: hal-00725674https://hal-supelec.archives-ouvertes.fr/hal-00725674Submitted on 27 Aug 2012HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.L’archive ouverte pluridisciplinaire HAL, estdestine´e au de´poˆt et a` la diffusion de documentsscientifiques de niveau recherche, publie´s ou non,e´manant des e´tablissements d’enseignement et derecherche franc¸ais ou e´trangers, des laboratoirespublics ou prive´s.Analysis of a MEMS-based ring oscillator  J. L. Muñoz-Gamarra, N. Barniol Depart. Electronic Eng. Universidad Autonoma de Barcelona 089193 Bellaterra, SPAIN joseluis.munoz.gamarra@uab.es J. Juillard E3S, SUPELEC Gif-sur-Yvette, FRANCE jerome.juillard@supelec.fr  Abstract— This work introduces a MEMS oscillator composed exclusively of mechanical switches as logic components. The electromechanical model of the system is developed and the conditions for a periodic response are established. I. INTRODUCTION  Microelectromechanical systems (MEMS) have been proposed as an alternative to standard CMOS-based  switches due to their ideally zero leakage current and abrupt switching behaviour [1, 2].. In [3], the basis for several MEMS-based logic components are laid and several implementations of inverters or NAND gates are proposed. The parallelism between CMOS and MEMS switches has also been extended to fields which are traditionally dominated by the CMOS technology, such as logic circuits [4] or memory cells [5]. With logic circuits in mind, one may wonder whether a purely MEMS-based synchronous logic can be achieved, i.e. can a digital clock signal be generated using only MEMS switches? And can this clock signal be used to drive other MEMS-based logic components? The aim of this paper is to show how such a classical concept as the CMOS ring oscillator (a loop consisting of an odd number of inverters in series) may be extended into the MEMS domain and in which respect we may answer positively to the first of these two questions. In section II, the electromechanical model of a MEMS cantilever switch is presented. In section III, we show under what hypotheses a MEMS-based ring oscillator can be modelled as an autonomous impact oscillator [6, 7] and what the conditions for the existence of a periodic regime are. In section IV, the results obtained in section III under some simplifying assumptions and those obtained from the simulation of the device are compared and some general design guidelines are discussed. II. ELECTROMECHANICAL MODEL OF THE SWITCH Fig. 1 represents the studied out of plane cantilever switch configuration. When a sufficient voltage is applied across gate and source, the flexible part of the source bends as a result of electrostatic forces until its tip touches the drain and an electrical contact is established. The simplest MEMS inverter consists of two switches, PMEMS and NMEMS, respectively with source voltages ddV  and ssV , having a common drain and a common gate (Fig. 2). With correct design [2-5], the drain (output) voltage is ddV  when the gate (input) voltage is ssV  and vice-versa. Putting an odd number of such inverters in series and closing the loop results in an intrinsically unstable system, known as a ring oscillator, a simplified model of which is established in the next sub-sections.  L LLGhb G SOURCE GATE DRAIN Figure 1.  Schematic of a MEMS cantilever switch and notations. A. Electrical model In a ring oscillator, the switches are either in the “free mode”, such as NMEMS in Fig. 2 which is moving toward or away from the drain, or in the contact mode, such as PMEMS in Fig. 2. In the contact mode, a resistive path between source and drain is established. As long as mechanical contact is maintained, DV  is governed by:  SDDonon VVdtdVCR   where onR  is the resistance of the path between source and drain and onC  is the total capacitance when contact is established. onR  can be assimilated to the contact resistance ( coR  in Fig 2) between source and drain, which is typically several orders of magnitude greater than the intrinsic resistance of the drain or of the source ( R  in Fig.2). If ononon CR  is small compared to the mechanical contact duration, DV  goes to SV . Typical values for the contact resistance in MEMS relays (taken from [8]) are in the 1-10k range (depending on the contact force and area) while the gate-source capacitance is in the 0.1-10fF range. In the free mode, provided the change in SGPC  or SGNC  is not too large compared to LC , the gate/drain voltages can be considered as constant. Note that the behaviour of our oscillator is considerably different from that of the one proposed in [7], where the mechanical and electrical time constants have the same order of magnitude. It is assumed throughout the rest of the paper that ssV =0. VDD PMEMS NMEMSCLVG=0 VRCSGP CSGNCSDNRCO Figure 2.  MEMS inverter and its electrical model when VG=0V. B. Mechanical Model The beams in the ring oscillator can be modelled with the Euler-Bernoulli equation, in which we account for squeezed-film damping as the dominant damping phenomenon and electrostatic actuation:       1)1(,2)1(,122230223334443wFwVGbtwbhtwwGbxwLbhEceff where  1,0x  is the normalized coordinate along the length of the beam ( 1x  corresponding to the tip), displacement  txw ,  is normalized with respect to gap G , V  is the voltage applied to the beam, cF  is the contact force applied at the beam tip and   ,  is the characteristic function of the actuation electrode (equal to 1 if   ,x  and 0 otherwise). E  and   are the material’s Young’s modulus and mass density, eff  the effective viscosity of the surrounding medium and 0  the permittivity of vacuum. The solution of (2) can be approximated as:           xWtaxWtatxw 2211,   where   211 21 UUW    212 21 UUW   and  xU1  and  xU 2  are the first two eigenmodes of the cantilever, with eigenvalues 41  and 42 , normalized so that     111 21 UU . Consequently,   111 W  and   012 W . Mechanical contact between source and drain then takes place when 1a , regardless of 2a . We use the Galerkin method to derive the following nonlinear mechanical model of the beam:       afafaaaBKa ce  2V  awhere  21 , aaT a  and         424142414241424142192 LEhK  b  2121  jiijBB      dxwWWhGbB jieffij 3321a  c   21  iifef    dxwWhGf ii 23012 d   otherwise   ,a  if   0 111 0fc    aBaK cc  eNote that the contact force (5-e) is not necessarily conservative. Using (1) and (5), a MEMS-based ring oscillator with an arbitrary number of stages can be simulated and several of its characteristics can be explored (influence of contact stiffness and damping, influence of supply voltage on the existence of periodic solutions, etc.). However, the model developed in this section, in spite of its apparent simplicity, is difficult to simulate because of (i) the existence of discontinuities, and of (ii) the coexistence of several phenomena with very different time scales (ideally, the transition from a high to a low state should be very short compared to the mechanical contact duration, and even shorter compared to the period of oscillation). In the following section, we propose a semi-analytical tool based on simplifying assumptions which will allow us to predict the existence of periodic regimes without resorting to computationally-intensive simulation tools. III. SEMI-ANALYTICAL LIMIT CYCLE PREDICTION The following set of assumptions is made. First of all, only small amplitude motion of the beam is considered, so that    0BaB   and    0faf ee   in (5-a). This allows us to recast (5) in the following state-space form:   zbbAzz ce  2V  with   aaz    0f0bee and   0BKI0A 22  It is also assumed that mechanical contact is instantaneous and governed by:       cc tt Szz 100000000100001S  where ct  designates the instant at which contact occurs. Equation (8) implies that, when a contact takes place, the state of the system is unchanged, except for the first modal velocity coefficient, governed by:       cc tata 11    Coefficient 1  then represents the mechanical losses during contact. Between impacts, a beam is then governed by:  2VebAzz   Finally, 2V  in (10) can take either of two values, ideally equal to 02 offV  and 22 ddon VV  , provided the electric time constant is small1. The commutation from one value to the other takes place when the output of the previous inverter in the ring oscillator loop changes.  From this set of hypotheses, the existence of simple limit cycles can be predicted. In particular, let us assume that a limit cycle with period T , half-period 2/T  and delay   can take place in the system, meaning that:     TTtVTttVtVonon,2/,2/,,0,0,(11) where the origin of time is taken at the moment when the mechanical contact occurs. Note that   depends on the number of stages in the ring oscillator: 0  for a one-stage oscillator, 3/T  for a three-stage oscillator, etc. The analytical solution of (10-11) is straightforward, provided the beam does not come into contact with the drain for  Tt ,0 . Under this hypothesis, we find:                                                           1 Otherwise, the load capacitance is only partially charged or discharged. This, however, does not affect very much the rest of the analysis.    2240 onTTTTf VezeeeIzezAAAA     where   Tf zz ,   00 zz , ee bAz 1  and Me  designates the matrix exponential of M . If the motion is periodic, we should have      0zz T , so that, using (8), (12) becomes:      2241120onTTTTonVVeAAAA zeeeIeSz    For a given value of  , 2onV  should then be adjusted so that:    00001 z  Since   and 2onV  are positive, this is only possible if, for instance,       010001 0 z  Furthermore, at time 0 , the beam should be bouncing away from the drain (and not “into it”), i.e.:      010100 0 z  Finally, (13) is valid only if there are no spurious contacts for  Tt ,0 . There is no analytical way of verifying this, although the analytical expression of  tz  exists. One should then tabulate  itz  for  Tti ,0  and verify that, i , the first component of  itz  (corresponding to  ita1 ) is smaller than  . If this condition is met, and (15) and (16) as well, then a limit cycle with period T , half-period 2/T , delay   and no supplementary impacts between 0 and T  may take place in the system. By setting   and sweeping a range of values of T , it is then possible to plot curves giving T  versus onV . Note that the procedure described above can be extended to the case when several impacts take place during a period. However, this requires solving a nonlinear set of equations and quickly becomes impractical when the number of impacts is more than a few. IV. SIMULATION RESULTS We have represented in Fig. 3 the starting transient of  ta1 , obtained by simulating (1) and (5) with Matlab/Simulink. The device is a 1-stage MEMS ring oscillator consisting of two identical beams arranged as in Fig. 2, with shorted gate and drain, with the following geometry:  =0.1,  =0.9,  =0.15, L =25µm, b =1µm, h =0.25µm and G =0.3µm(1/fo=1.88μs, Qsqueezed =15.2, Vsnap=2.07V). The material is assumed to be polysilicon and the surrounding medium air at ambient pressure. The actuation voltage is Vdd =1.4V and we set onR =10k and onC =10fF. In these conditions, the oscillator reaches a periodic regime after a short time, with a period of T =1.15×10-6s. We also show in Fig.4 the steady-state behaviour of  ta1  as predicted by the method presented in section III. There is an excellent match between the simulated response and the predicted one.  0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1x 10-5-0.25-0.2-0.15-0.1-0.0500.050.10.150.2Tip motion a 1(t)Starting transient of the oscillator  6.56 6.58 6.6 6.62 6.64 6.66 6.68 6.7 6.72 6.74x 10-5-0.25-0.2-0.15-0.1-0.0500.050.10.150.2Time (s)Tip motion a 1(t)Steady-state regime of the oscillator  Tip motion of 1st beamTip motion of 2nd beamPredicted limit cycleTip motion of 1st beamTip motion of 2nd beam Figure 3.  Simulated transient response of a1(t) (normalized beam tip position) for the two beams composing the switch (top) and simulated and predicted steady-state response (bottom). 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.40.70.80.911.11.21.31.41.5x 10-6 Predicted vs. simulated oscillation periodDrive voltage Vdd (V)Oscillation period T (s) Figure 4.  Comparison of predicted “T versus Von” curve (black line) and results obtained by transient simulation (green line), starting from Vdd =2.5V. In Fig.4, we compare the results obtained by slowly sweeping the actuation voltage in the transient nonlinear model to the “T  versus onV ” curve obtained with our semi-analytical method - for the transient simulations, T  is in fact the “apparent” period, i.e. the time between two successive rising edges of  tV . As expected [6, 7], depending on the value of onV , the simulated behaviour is not always periodic, even though our semi-analytical method predicts one or more possible periodic limit cycles. For example, transient simulation shows that the system exhibits chaotic behaviour when onV  is between 1.5V and 1.75V, whereas our semi-analytical method predicts the existence of one or two periodic solutions. This illustrates the fact that the stability of the predicted limit cycles is not guaranteed a priori and should be the object of careful study. This situation is even more pronounced for structures with small damping factors (larger Q).  V. CONCLUSION We have shown that under certain conditions, one may contrive to generate a periodic square signal using an odd number of MEMS inverters in a ring oscillator configuration. The governing equations of such a system were established and a semi-analytical method for predicting the existence of simple limit cycles was proposed. Comparison of the results obtained by our method and by simulation incite us to develop tools to study the stability of these limit cycles, for example following [9]. Qualitatively, we have found that too large a quality factor is detrimental to the stability of the system. Also, a possible reason for non-periodic behaviour is that the two natural resonance frequencies of the system (those of the first two eigenmodes of the beam) are not harmonically related. The beam profile could then be carefully designed, for example to tune the beam for periodic behaviour in a given voltage range. Several questions remain open, e.g. heating, friction and wear, etc. which are the subject of ongoing work.  VI. REFERENCES [1] M. L. Roukes, et al., "Low Voltage Nanoelectromechanical Switches Based on Silicon Carbide Nanowires," Nano Letters, vol. 10, pp. 2891-2896, 2010. [2] N. Abele, et al., "Suspended-gate MOSFET: bringing new MEMS functionality into solid-state MOS transistor," IEEE IEDM, 2005, pp. 1075-1077. [3] K. Akarvardar, et al., "Design considerations for complementary nanoelectromechanical logic gates," IEEE IEDM, 2007, pp. 299-302. [4] D. A. Czaplewski, et al., "A nanomechanical switch for integration with CMOS logic," Journal of Micromechanics and Microengineering, vol. 19, 2009. [5] S. Chong, et al., "Nanoelectromechanical (NEM) relays integrated with CMOS SRAM for improved stability and low leakage," IEEE/ACM ICCAD, 2009. [6] S. W. Shaw, "Forced Vibrations of a Beam with One-Sided Amplitude Constraint - Theory and Experiment," Journal of Sound and Vibration, vol. 99, pp. 199-212, 1985. [7] R. Puers, et al., "The autonomous impact resonator: A new operation principle for a silicon resonant strain gauge," International Conference on Solid-State Sensors and Actuators, 1997, pp. 1105-1108 [8] F. Chen, et al., "Integrated circuit design with NEM relays," IEEE/ACM ICCAD, 2008 [9] S. W. Shaw and P. J. Holmes, "A periodically forced piecewise-linear oscillator," Journal of Sound and Vibration, vol. 90, pp. 129-155, 1983. 

Analysis of a MEMS-based ring oscillator

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Analysis of a MEMS-based ring oscillator

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