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Modeling of an electrostatic torsional actuator: demonstrated with an RF MEMS switch
Sensors and Actuators A 97±98 (2002) 337±346
Modeling of an electrostatic torsional actuator: demonstrated with an RF MEMS switch Robert Sattlera,*, Florian PloÈtzb, Gernot Fattingerb, Gerhard Wachutkaa a
Institute for Physics of Electrotechnology, Munich University of Technology, Arcisstrasse 21, 80290 Munich, Germany b In®neon Technologies AG, 81730 Munich, Germany Received 4 September 2001; received in revised form 22 November 2001; accepted 4 December 2001
Abstract We present an ef®cient methodology for setting up MEMS macromodels which are based on a physical device description and lead to tractable mathematical relations for the device operation. Since design and technology parameters are input parameters of the resulting model, our approach is in particular suited for design studies. In addition to the reduction in degrees of freedom, and hence, the reduced simulation time, macromodels can easily be coupled with the electronic circuitry, and thus the entire device can be simulated on system level. The model also allows for squeeze-®lm damping effects. Its practicality is demonstrated with reference to an electrostatic torsional RF-switch with low actuation voltage intended for the use in mobile communication. Furthermore, simulated and experimental data of the resonance modes are compared for validation and an improved design for faster switching is proposed and optimized. # 2002 Elsevier Science B.V. All rights reserved. Keywords: Switch; Transient; Macromodel; Squeeze-®lm damping
1. Introduction MEMS-based devices for telecommunication applications, such as electromechanical switches, tunable capacitors and mechanical ®lters, have been intensively studied over the last few years. Especially micromechanical RF-switches offer advantages over their electronic solid state counterparts (FETs and PIN diodes) because of their low power consumption, high isolation capability and good linearity. Different types of MEMS switches are being studied by various groups with the designs focusing on two classes of switches. Shunt switches for applications at 10±100 GHz [1,2] and series switches with a low ohmic contact for the lower Gigahertz range. The latter have been realized in different design variants such as bridges, cantilevers [3] and torsional devices [4,5]. Because of their superior electrical performance, RFswitches are an attractive alternative to standard solid-state devices for signal routing, ®lter-path selection and GSM/ UMTS-switching. Depending on the speci®c application, switching times in the microsecond range have to be
*
Corresponding author. Tel.: 49-89-289-23109; fax: 49-89-289-23134. E-mail address: [email protected] (R. Sattler).
achieved. This can be a serious challenge for MEMS switches because of their mechanical nature. 2. Design and fabrication The design of the switch presented in this work is based on a torsion-type actuator with two contacts for dual-mode switching. Mechanical operation of the switch is performed by electrostatic actuation with switching voltages in the range of 10 V. Fig. 1 shows a topview micrograph of the switch device. A tiltable poly plate is suspended by two torsional springs attached at the left and right side of an anchor post located in its center. The center-based suspension enables stress relaxation in the structural polysilicon. The plate is perforated to allow for the sacri®cial layer etching underneath and to reduce air damping effects during operation. A reduced hole density in the transition area from the springs to the mainplate increases the stiffness of the structure. The driving voltage is applied between one of two actuation electrodes positioned underneath both sides of the switch and the mainplate. The electrostatic attraction between the respective electrodes causes a torsion of the mainplate until a pair of signal lines is bridged by a contact bar at one side, while the opposite side is de¯ected upward at the same time, leaving the respective contact open.
0924-4247/02/$ ± see front matter # 2002 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 4 - 4 2 4 7 ( 0 1 ) 0 0 8 5 2 - 4
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Fig. 1. Topview micrograph of a micro-machined RF-switch.
There exist algorithms and software tools which extract compact models by means of curve ®tting from previous FEM simulations. But with this method, design parameters are no longer visible in the compact model. Therefore, with a view to performing design studies, the underlying FEM simulations have to be reiterated. An alternative approach makes use of physically transparent models which are formulated in terms of all relevant design parameters describing geometry, topography, material properties, etc. A systematic methodology of formulating transparent compact models is, among others, provided by a thermodynamic system description in terms of driving forces and resulting ¯ows of the relevant physical quantities [6]. This leads to a full system description as a ``generalized Kirchhof®an network'' which, in turn, is equivalent to a system of ordinary differential-algebraic equations for the node variables that can be solved using a standard analog network simulator. 3.1. Theory In a ®rst-order approximation, the mainplate of the torsional actuator is considered to be a tiltable, rigid body. Hence, the only degree of freedom is the angle of torsion j (see Fig. 3). The differential force acting on a segment of the actuation electrode is equal to that of a differential parallel plate capacitor with width w and in®nitesimal length dr: 1 w dr dF EU 2 2 D r sin j2
Fig. 2. SEM micrograph of torsional switch device.
Micrograph images of the completed device are displayed in Fig. 2. The devices are fabricated in surface micromachining technology using a two-layer poly-Si process with oxide as sacri®cial layer. The ®rst poly-layer is used for the actuation electrodes, leads and anchor post, whereas the second layer forms the tiltable plate. The lower signal line electrodes are fabricated using tungsten, while additive gold electroplating is employed for creating the top-electrodes. The sacri®cial layer is ®nally removed by wet etching in hydro¯uoric acid, for releasing the freely movable mainplate. More details on design and fabrication can be found in [13].
(1)
where E denotes the electric permittivity, U the applied voltage, and D is the gap width between torsion axis and bottom electrodes (see Fig. 3). In the notation of Kirchhof®an network variables, the quantity conjugate to the generalized force (across) variable dj/dt (angular velocity of torsion) is the torque M which plays the role of the generalized ¯ow (through) quantity. Z L Z L 1 r M r dF EU 2 w dr (2) 2 r sin j2 0 0 D Integration over the effective electrode area on the mainplate yields 1 U 2 L2 1 g log 1 g (3) M E 2 w 2 2 D g 1 g
3. Macromodeling A widely employed method to model microsystems is the continuous ®eld approach or an equivalent numerical realization such as, e.g. the ®nite element method (FEM). However, for the analysis of the transient behavior of the device, the numerical effort is often becoming prohibitive. Therefore, the degrees of freedom have to be reduced to a number that is still tractable and yet accurate enough for predictive simulation.
Fig. 3. Schematic side view of the torsional actuator (L: length of actuation electrode).
R. Sattler et al. / Sensors and Actuators A 97±98 (2002) 337±346
where we have introduced the normalized angle g
sin j j sin j0 j0
and the maximum torsional angle j0 : arcsin
D D L L
The capacitance can be calculated from the electric ®eld distribution E r U= D r sin j according to Z Z L U Q dr (4) E~ E d~ a Ew D r sin j electrode 0 hence C
Q wL E log 1 U D
The total torque is composed of the electrostatic torque, damping, inertia and the mechanical torque exerted by the tethers on which the tiltable mainplate is suspended:
Minertia J j
GI j; ltether
From the linearized Reynolds equation the damping constant for a rectangular plate can be derived as [7]: c A=h3 , where h is the squeeze-®lm thickness and A is a constant depending on the geometry of the plate and the viscosity of the gas. In analogy to the calculation of the electrostatic torque (Eq. (2)), the damping constant in dependence of the angle of torsion can be calculated as Z L A L2 1 g2 jr jdr cA (7) D3 1 g2 2 rj3 L D This is an interesting result, since only the constant parameter A has to be calibrated with reference to experimental data. 3.2. Results
g
If the actuator is driven with charge as control variable, then Q/C has to be substituted for U in Eq. (3), yielding: 1 Q2 1 g M log 1 g (5) 2 Ew log 1 g2 1 g
Mtether kj 2
339
_ Mdamping cj; (6)
where G denotes the shear modulus, I the area moment of inertia, and ltether is the length of the torsional springs, while c is the effective damping constant of the mainplate and J its inertia moment. The parameters for the torsional stiffness of the tethers and the moment of inertia can be calculated from the design parameters.
The equations are coded in SpectreHDL [8] (analog hardware description language similar to VHDL-AMS). This allows us to apply the standard analysis methods as provided by an analog network analyzer. 3.2.1. Direct current analysis 3.2.1.1. Voltage-controlled analysis. Performing a voltagecontrolled dc analysis allows to compute the pull-in hysteresis of the switch. The comparison with the experimental data is displayed in Fig. 4 and shows good agreement. The maximum angle, where the stable equilibrium between the electrostatic and the mechanical torque becomes unstable and pull-in occurs, amounts to g 0:4404 (Figs. 4 and 5). The voltage at the pull-in point results in: r k D3 Upi 0:827 (8) Ew L3
Fig. 4. Pull-in and pull-out characteristics for various geometric parameters. Simulated results (below) in comparison with measured data (above). The results of the simulation are plotted as normalized angle g vs. voltage.
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iterative relaxation scheme for the coupling of the electrostatic and mechanical domain. However, when approaching the pull-in voltage, relaxation methods show a rapidly decreasing convergence rate which is intolerable for the computation of the pull-in voltage. In this case charge can be used as homotopy parameter [11] to follow the displacement versus voltage characteristics even in the unstable branches. This approach requires no extra treatment of the unstable region beyond the snap-down point and reduces the numerical effort considerably. In Fig. 5, a number of operating points beyond the pull-in voltage are depicted, which were computed by charge-controlled homotopy. 3.2.2. Alternative current small signal analysis
Fig. 5. Charge-controlled analysis enables stable electromechanical equilibrium beyond the pull-in voltage.
3.2.1.2. Charge-controlled analysis. Charge control is a well-known means to extend the stable operating area of electromechanical actuators [11]. Pull-in does not occur until a normalized angle of g 0:7106 is attained (Fig. 5), with a corresponding charge of r D (9) Qpi 1:798Ewk L The in¯uence of parasitic charge and the microtechnological realization is discussed in [9,10]. But charge control is not only of practical use. Most simulation tools use an
3.2.2.1. Electrostatic spring softening. For a coupled electromechanical modal analysis and resonance measurements of the microswitch, the effect of electrostatic spring softening has to be included. The action of the nonlinear electrostatic torque contributes with negative sign to that of the linear mechanical torque, which leads to a smaller value of the effective spring constant at the operating point considered, and hence, to a decrease of the resonance frequency. Using our compact model the shift of the resonance frequency versus bias voltage can easily be analyzed (Fig. 6) and conforms well with the measured frequency of 72 kHz [5]. 3.2.3. Transient analysis 3.2.3.1. Dynamic pull-in. Performing a transient analysis we investigated the influence of the actuation voltage on the switching time. Pull-in occurs at a dynamic pull-in voltage U dpi < U pi due to the influence of the inertia. If the switch is operated within the dynamic pull-in regime, then a small
Fig. 6. Small-signal analysis showing the shift in resonance frequency due to spring softening.
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Fig. 7. Pull-in dynamics for various actuation voltages. The static pull-in voltage is U pi 8 V.
change in the voltage has a strong impact on the switching time, as it is illustrated in Fig. 7. 3.2.3.2. Influence of an increased electrode area. For relay applications a large, uniform and smooth contact area is essential; this can only be achieved by exerting a high contact force. In order to obtain a higher electrostatic force, a larger electrode area is necessary. Assuming the electrode to be rigid and perfectly supported by the anchor in the center of rotation, the contact force can be calculated as F
Melectrostatic L
Mtether
cos j0
(10)
been performed for `on/off' and `off/on'-switching cycles. Because of the laser wavelength of 532 nm, all larger displacements lead to ambiguities in the photodetector signal. These can be removed by evaluating the different phase shifts. Fig. 9 shows the vertical displacement as extracted from the measured signal for `on/off'-switching. Three designs (A, B and C) differing in the size and the number of anti-damping holes (area of etch holes is 36, 25 and 16% of the area of the mainplate) have been investigated and compared with the results of the macromodel. For the `on/off'-switching the squeezing of the gas film is negligible. The damping constant A was adjusted to the
Considering Eq. (3) we recognize that the contact force increases linearly with the area of the mainplate. This result is an estimate of the bound from above, as the real actuator is neither rigid nor perfectly supported. So part of the ideal force will be lost in the deformation of the anchor and the bending of the electrodes. But with increasing area also the inertia moment of the switch increases, raising the question how the switching speed scales with electrode area. Fig. 8 shows the pull-in time as a function of the electrode area. The stiffness of the tethers was adapted according to Eq. (8) in order to keep the pull-in voltage at the same value for all areas. As reported by Grif®n et al. [7], the damping constant A in Eq. (7) scales with the square of the area. But this approximation does not consider the many anti-damping holes present in our structure. If, for instance, a linear relationship is assumed, the switching speed would even increase with the electrode area. 3.2.3.3. Influence of etch holes. Transient measurements of the switch contact by means of a laser interferometer have
Fig. 8. Variation of switching time and contact force of a quadratic actuator with increasing mainplate area for an actuation voltage of 2 V above the pull-in voltage.
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Fig. 9. Measured transient behavior for `on/off'-switching compared to simulation for three differently perforated switches with fitted damping constants c.
measured data for the three designs (Fig. 9). Then the same damping constants were used to model the `off/on'switching. Fig. 10 shows that the measurement can be reproduced quite well, but only if the improved damping model (Eq. (7)) is used. Due to the compression of the gas in the gap the damping constant is not constant any more, but depends on the angle of torsion, which is a new and worthwhile result. 3.2.4. System simulation Due to technical constraints the available supply voltage is in many cases too low to operate the switch. This problem can be solved by means of a charge pump. A cascade of several capacitors and diodes is driven by an alternating voltage generated from the dc supply voltage. At each stage, the voltage is increased, but the available output current is limited by the corresponding demand of charge per cycle (I max U0 C1 f ). Therefore, a charge pump can be described as a voltage source with a virtual source impedance, leading to a voltage drop but without any resistive losses. Also the
Fig. 10. Measured transient behavior for `off/on'-switching compared to simulation for three different perforated switches. The result for a model using a uniform (angle-independent) damping constant is shown for comparison.
ripple increases, if the load current is high compared to the available capacitors. Because charge pumps are sensitive to the behavior of the load they drive, it is important for the circuit designer to have a reliable simulation model for the interaction of these two components. In the case studied in this work, the charge pump has only one stage, and resistors (R1, R2, RL) are included to represent the losses in a non-ideal circuit. The torsional actuator is represented as a variable capacitance. As the model of the actuator is coded in an analog hardware description language, it can easily be included in the circuitry of the charge pump (Fig. 11). The in¯uence of the pump frequency on the behavior of the system is presented in Fig. 12. The exponential increase of the output voltage indicates a proper operation of the charge pump. But during the pull-in the capacitance of the switch increases, and if the pump frequency is not suf®ciently high, then the output voltage of the charge pump drops too much. This can delay or even prevent the switching operation.
Fig. 11. Schematic of a simple charge pump combined with the macromodel of the switch.
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have seen from the results of the pull-in hystereses and the transient behavior. But if the stiffness of the switch is importantÐas it is important for the mode shapes or the maximum achievable contact forceÐthen a ®eld solver is the best choice, even though it implies much higher computational expenses. 4.1. Resonance modes The resonance modes of the mainplate have been studied using the FEM simulator Memcad [12]. The discretization of the (simpli®ed) device structure by an FEM mesh is shown in Fig. 13. The darker parts in the mesh indicate a higher density of the material due to a lower perforation of the structure. The experimental mode analysis is exactly re¯ected in the simulation procedure: For a given dc bias voltage, an ac voltage is superimposed and applied to one of the actuation electrodes, then the harmonic response at each operating point is determined with a modi®ed Mach±Zehnder laser-interferometer. The comparison of simulated and measured mode shapes together with the extracted resonance frequencies is presented in Fig. 14. One should note that, by applying a voltage to one of the actuation electrodes, not all possible modes can be excited. For example, lateral displacements can never be excited nor detected. Hence, the simulated values of those modes are omitted in the list presented. As damping is not considered in the simulation, the calculated resonance frequencies are a little higher compared to measurement. An exception is the ®rst mode. In the switching mode the resonance frequency is increased due to squeeze-®lm spring forces. Fig. 12. Time-dependence of the output voltage U2 of the charge pump and the pull-in characteristics for various pump frequencies.
4. Continuous field modeling In the compact model the actuator is considered to be rigid. This simpli®cation is justi®ed for many cases, as we
4.2. Improved switching time The mayor contribution to the moment of inertia of the switch is the contact metalization. To reduce the switching time we removed the contact bar from the moving structure. Instead of moving down the contact bar to bridge the signal lines as shown in the left part of Fig. 15, the seesaw-type
Fig. 13. Meshing of the torsion plate for FEM-modal analysis.
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Fig. 14. (a) Simulated and (b) measured resonance modes of the torsional plate.
Fig. 15. Schematic illustration of downward (left) and upward (right) switching (not to scale).
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Fig. 16. SEM micrograph of improved switch design.
Fig. 17. Optimization of the gap between mainplate and upper contact for a maximum contact force.
Fig. 18. Vertical displacement of the switch for various actuation voltages in comparison with measurements.
structure shown in the right part features a downward tilting of one side of the mainplate in conjunction with an upward movement of the opposite side in order to close the contacts from below (Fig. 15, right side). This allows the omission of the contact bar on the tiltable poly-plate. In this way, the mass of the switch plate is reduced, resulting in faster switching times for this type of a design. The contact area of the device with upward switching is shown in the SEM micrograph of Fig. 16. However, now the gap between the switch and the upper contact must be optimized (Fig. 17). On the side where the mainplate is pulled down, it must almost be in contact with the ground electrode to get the maximum electrostatic torque. Thus, if the gap between the switch and the upper contact is too small, then the switch cannot come
close to the ground electrode, but if the gap is too large, then the switch touches the ground electrode and reduces the contact force. Measurements of the switch displacement for various actuation voltages are compared with FEM simulations in Fig. 18. 5. Conclusion A surface micromachined switch for dual mode operation has been studied by experiments and numerical simulation. A compact model based on a physical device description has been formulated, which provides not only important insights in the static and dynamic behavior of the device, but also
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allows performing parameter studies at acceptable computational cost. The in¯uence of the actuation voltage, the electrode area and the etch holes on the device operation was investigated and compared with corresponding experiments. Once the damping constants are calibrated, the model can be used for predictive simulations. Furthermore our compact model can easily be incorporated in a full system model by attaching it to the electric circuitry needed for the voltage generation and device control. The pump frequency of a charge pump turned out to be a crucial parameter for a proper device operation. The resonance frequencies of the switch have been measured by laser interferometry and were found to be in good agreement with previously simulated results. An improved microswitch structure has been proposed where pull-down and bridging of the signal contacts occur on opposite sides of the mainplate. The design was optimized by contact simulations and compared with experimental data. Acknowledgements The authors would like to express their special thanks to S. Hoffmann for the circuit diagram of the charge pump. Also acknowledged are A. Meckes, K.-G. Oppermann, H. Kapels, M. Franosch and T. Bever for their technical support as well as M. Handtmann for valuable discussions. References [1] C.L. Goldsmith, Y. Zhimin, S. Eshelman, D. Denniston, Performance of low-loss RF MEMS capacitive switches, IEEE Microwave Guided Wave Lett. 8 (8) (1998) 269±271. [2] S.P. Pacheco, L.P.B. Katehi, C.T. Nguyen, Design of low actuation voltage RF MEMS switch, in: Proceedings of the International Microwave Symposium 2000, Boston, 2000, pp. 165±168. [3] J.J. Yao, M.F. Chang, A surface micromachined miniature switch for telecommunications applications with signal frequencies from dc up to 4 GHz, in: Proceedings of the Eurosensors IX, 1995, pp. 384±387. [4] D. Hah, E. Yoon, S. Hong, A low voltage actuated micromachined microwave switch using torsion springs and leverage, in: Proceedings of the International Microwave Symposium 2000, Boston, 2000, pp. 157±160. [5] F. PloÈtz, S. Michaelis, R. Aigner, H.-J. Timme, J. Binder, R. NoeÂ, A low-voltage torsional actuator for application in RF-microswitches, in: Proceedings of the Eurosensors XIV, Copenhagen, Denmark, August 2000, pp. 297±300. [6] G. Wachutka, Tailored modeling: a way to the virtual transducer fab, Sens. Actuators A 46/47 (1995) 603±612. [7] W.S. Griffin, H.H. Richardson, S. Yamanami, A study of squeezefilm damping, J. Basic Eng. (1966) 451±456. [8] http://www.cadence.com. [9] R.N. Guardia, R. Aigner, W. Nessler, M. Handtmann, L.M. CastanÄer, Control positioning of torsional electrostatic actuators by current driving, in: Proceedings of the ASDAM, Smolenice Castle, Slovakia, October 2000, pp. 91±94. [10] R.N. Guardia, A. Dehe, R. Aigner, L. CastanÄer, New current drive method to extend the stable operation range of electrostatic actuators:
experimental verification, in: Proceedings of the Transducers'01, Munich, Germany, June 2001, pp. 760±763. [11] E.-R. KoÈnig, G. Wachutka, Analysis of unstable behaviour occuring in electromechanical microdevices, in: Proceedings of the MSM, Puerto Rico, USA, April 1999, pp. 330±333. [12] http://www.memcad.com. [13] F. PloÈtz, S. Michaelis, R. Aigner, H.-J. Timme, J. Binder, R. NoeÂ, A low-voltage torsional actuator for application in RF-microswitches, Sens. Actuators A 92 (1±3) (2001) 312±317.
Biographies Robert Sattler was born in Memmingen, Germany, in 1970. He studied physics at the University of New South Wales, Sydney, Australia, and the University of Regensburg, where he received the Diplom-Physiker degree in 1998. Since then he is working towards his PhD degree at the Institute for Physics of Electrotechnology at Munich University of Technology. His research activities are focused on modeling of coupled effects in microsystems. Florian PloÈtz was born in Regensburg, Germany, in 1973. He studied physics at the University of Colorado at Boulder, USA, and the University of Regensburg, where he received the Diplom-Physiker degree in 1998. Since then he is working towards his PhD degree in electrical engineering at the University of Paderborn and is with the MEMS group of Infineon in Munich, formerly Siemens Corporate Technology. His research interests are in the area of high-frequency microsystems. Gernot Fattinger was born in Linz, Austria, in 1975. He studied semiconductor and solid state physics at the University of Linz before he joined the MEMS group at Infineon Technologies, formerly Siemens Corporate Technology, in 1999. There he accomplished his diploma thesis about interferometric measurement methods for RF-MEMS and received the Diplom-Ing degree in 2000. Currently he is working towards his PhD degree in physics, studying RF microelelctromechanical devices. Gerhard Wachutka received DSc degree from the Ludwig-MaximiliansUniversitaÈt, Munich, Geramny, in 1985. From 1985 to 1988, he was with Siemens Corporate Research & Development, Munich, where he headed a modeling group active in the development of modern high-power semiconductor devices. In 1989, he joined the Fritz-Haber-Institute of the Max-Planck-Society, Berlin, Germany, where he worked in the field of theoretical solid-state physics. From 1990 to 1994, he was head of the microtransducers modeling and characterization group of the Physical Electronics Laboratory at the Swiss Federal Institute of Technology (ETH), Zurich. There he also directed the microtransducers modeling module of the Swiss Federal Priority Program M2S2 (micromechanics on silicon in Switzerland). Since spring 1994, he has been heading the Institute for Physics of Electrotechnology at the Munich University of Technology, where his research activities are focused on the design, modeling, characterization, and diagnosis of the fabrication and operation of semiconductor microdevices and microsystems. He has authored or coauthored more than 180 publications in scientific or technical journals. He is consultant of research institutes in industry and university, and he serves as reviewer for various scientific journals and other institutions. Among his many educational activities, he has set-up and taught courses funded by European Community training programs such as UETP, EUROFORM, and EUROPRACTICE. Professor Wachutka is member of the IEEE, the American Electrochemical Society, the American Materials Research Society, the ESD Association, the VDE Association for Electrical, Electronic and Information Technologies, the German Physical Society, the American Physical Society, and the AMA Society for Sensorics.
Modeling of an electrostatic torsional actuator: demonstrated with an RF MEMS switch Robert Sattlera,*, Florian PloÈtzb, Gernot Fattingerb, Gerhard Wachutkaa a
Institute for Physics of Electrotechnology, Munich University of Technology, Arcisstrasse 21, 80290 Munich, Germany b In®neon Technologies AG, 81730 Munich, Germany Received 4 September 2001; received in revised form 22 November 2001; accepted 4 December 2001
Abstract We present an ef®cient methodology for setting up MEMS macromodels which are based on a physical device description and lead to tractable mathematical relations for the device operation. Since design and technology parameters are input parameters of the resulting model, our approach is in particular suited for design studies. In addition to the reduction in degrees of freedom, and hence, the reduced simulation time, macromodels can easily be coupled with the electronic circuitry, and thus the entire device can be simulated on system level. The model also allows for squeeze-®lm damping effects. Its practicality is demonstrated with reference to an electrostatic torsional RF-switch with low actuation voltage intended for the use in mobile communication. Furthermore, simulated and experimental data of the resonance modes are compared for validation and an improved design for faster switching is proposed and optimized. # 2002 Elsevier Science B.V. All rights reserved. Keywords: Switch; Transient; Macromodel; Squeeze-®lm damping
1. Introduction MEMS-based devices for telecommunication applications, such as electromechanical switches, tunable capacitors and mechanical ®lters, have been intensively studied over the last few years. Especially micromechanical RF-switches offer advantages over their electronic solid state counterparts (FETs and PIN diodes) because of their low power consumption, high isolation capability and good linearity. Different types of MEMS switches are being studied by various groups with the designs focusing on two classes of switches. Shunt switches for applications at 10±100 GHz [1,2] and series switches with a low ohmic contact for the lower Gigahertz range. The latter have been realized in different design variants such as bridges, cantilevers [3] and torsional devices [4,5]. Because of their superior electrical performance, RFswitches are an attractive alternative to standard solid-state devices for signal routing, ®lter-path selection and GSM/ UMTS-switching. Depending on the speci®c application, switching times in the microsecond range have to be
*
Corresponding author. Tel.: 49-89-289-23109; fax: 49-89-289-23134. E-mail address: [email protected] (R. Sattler).
achieved. This can be a serious challenge for MEMS switches because of their mechanical nature. 2. Design and fabrication The design of the switch presented in this work is based on a torsion-type actuator with two contacts for dual-mode switching. Mechanical operation of the switch is performed by electrostatic actuation with switching voltages in the range of 10 V. Fig. 1 shows a topview micrograph of the switch device. A tiltable poly plate is suspended by two torsional springs attached at the left and right side of an anchor post located in its center. The center-based suspension enables stress relaxation in the structural polysilicon. The plate is perforated to allow for the sacri®cial layer etching underneath and to reduce air damping effects during operation. A reduced hole density in the transition area from the springs to the mainplate increases the stiffness of the structure. The driving voltage is applied between one of two actuation electrodes positioned underneath both sides of the switch and the mainplate. The electrostatic attraction between the respective electrodes causes a torsion of the mainplate until a pair of signal lines is bridged by a contact bar at one side, while the opposite side is de¯ected upward at the same time, leaving the respective contact open.
0924-4247/02/$ ± see front matter # 2002 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 4 - 4 2 4 7 ( 0 1 ) 0 0 8 5 2 - 4
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Fig. 1. Topview micrograph of a micro-machined RF-switch.
There exist algorithms and software tools which extract compact models by means of curve ®tting from previous FEM simulations. But with this method, design parameters are no longer visible in the compact model. Therefore, with a view to performing design studies, the underlying FEM simulations have to be reiterated. An alternative approach makes use of physically transparent models which are formulated in terms of all relevant design parameters describing geometry, topography, material properties, etc. A systematic methodology of formulating transparent compact models is, among others, provided by a thermodynamic system description in terms of driving forces and resulting ¯ows of the relevant physical quantities [6]. This leads to a full system description as a ``generalized Kirchhof®an network'' which, in turn, is equivalent to a system of ordinary differential-algebraic equations for the node variables that can be solved using a standard analog network simulator. 3.1. Theory In a ®rst-order approximation, the mainplate of the torsional actuator is considered to be a tiltable, rigid body. Hence, the only degree of freedom is the angle of torsion j (see Fig. 3). The differential force acting on a segment of the actuation electrode is equal to that of a differential parallel plate capacitor with width w and in®nitesimal length dr: 1 w dr dF EU 2 2 D r sin j2
Fig. 2. SEM micrograph of torsional switch device.
Micrograph images of the completed device are displayed in Fig. 2. The devices are fabricated in surface micromachining technology using a two-layer poly-Si process with oxide as sacri®cial layer. The ®rst poly-layer is used for the actuation electrodes, leads and anchor post, whereas the second layer forms the tiltable plate. The lower signal line electrodes are fabricated using tungsten, while additive gold electroplating is employed for creating the top-electrodes. The sacri®cial layer is ®nally removed by wet etching in hydro¯uoric acid, for releasing the freely movable mainplate. More details on design and fabrication can be found in [13].
(1)
where E denotes the electric permittivity, U the applied voltage, and D is the gap width between torsion axis and bottom electrodes (see Fig. 3). In the notation of Kirchhof®an network variables, the quantity conjugate to the generalized force (across) variable dj/dt (angular velocity of torsion) is the torque M which plays the role of the generalized ¯ow (through) quantity. Z L Z L 1 r M r dF EU 2 w dr (2) 2 r sin j2 0 0 D Integration over the effective electrode area on the mainplate yields 1 U 2 L2 1 g log 1 g (3) M E 2 w 2 2 D g 1 g
3. Macromodeling A widely employed method to model microsystems is the continuous ®eld approach or an equivalent numerical realization such as, e.g. the ®nite element method (FEM). However, for the analysis of the transient behavior of the device, the numerical effort is often becoming prohibitive. Therefore, the degrees of freedom have to be reduced to a number that is still tractable and yet accurate enough for predictive simulation.
Fig. 3. Schematic side view of the torsional actuator (L: length of actuation electrode).
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where we have introduced the normalized angle g
sin j j sin j0 j0
and the maximum torsional angle j0 : arcsin
D D L L
The capacitance can be calculated from the electric ®eld distribution E r U= D r sin j according to Z Z L U Q dr (4) E~ E d~ a Ew D r sin j electrode 0 hence C
Q wL E log 1 U D
The total torque is composed of the electrostatic torque, damping, inertia and the mechanical torque exerted by the tethers on which the tiltable mainplate is suspended:
Minertia J j
GI j; ltether
From the linearized Reynolds equation the damping constant for a rectangular plate can be derived as [7]: c A=h3 , where h is the squeeze-®lm thickness and A is a constant depending on the geometry of the plate and the viscosity of the gas. In analogy to the calculation of the electrostatic torque (Eq. (2)), the damping constant in dependence of the angle of torsion can be calculated as Z L A L2 1 g2 jr jdr cA (7) D3 1 g2 2 rj3 L D This is an interesting result, since only the constant parameter A has to be calibrated with reference to experimental data. 3.2. Results
g
If the actuator is driven with charge as control variable, then Q/C has to be substituted for U in Eq. (3), yielding: 1 Q2 1 g M log 1 g (5) 2 Ew log 1 g2 1 g
Mtether kj 2
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_ Mdamping cj; (6)
where G denotes the shear modulus, I the area moment of inertia, and ltether is the length of the torsional springs, while c is the effective damping constant of the mainplate and J its inertia moment. The parameters for the torsional stiffness of the tethers and the moment of inertia can be calculated from the design parameters.
The equations are coded in SpectreHDL [8] (analog hardware description language similar to VHDL-AMS). This allows us to apply the standard analysis methods as provided by an analog network analyzer. 3.2.1. Direct current analysis 3.2.1.1. Voltage-controlled analysis. Performing a voltagecontrolled dc analysis allows to compute the pull-in hysteresis of the switch. The comparison with the experimental data is displayed in Fig. 4 and shows good agreement. The maximum angle, where the stable equilibrium between the electrostatic and the mechanical torque becomes unstable and pull-in occurs, amounts to g 0:4404 (Figs. 4 and 5). The voltage at the pull-in point results in: r k D3 Upi 0:827 (8) Ew L3
Fig. 4. Pull-in and pull-out characteristics for various geometric parameters. Simulated results (below) in comparison with measured data (above). The results of the simulation are plotted as normalized angle g vs. voltage.
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iterative relaxation scheme for the coupling of the electrostatic and mechanical domain. However, when approaching the pull-in voltage, relaxation methods show a rapidly decreasing convergence rate which is intolerable for the computation of the pull-in voltage. In this case charge can be used as homotopy parameter [11] to follow the displacement versus voltage characteristics even in the unstable branches. This approach requires no extra treatment of the unstable region beyond the snap-down point and reduces the numerical effort considerably. In Fig. 5, a number of operating points beyond the pull-in voltage are depicted, which were computed by charge-controlled homotopy. 3.2.2. Alternative current small signal analysis
Fig. 5. Charge-controlled analysis enables stable electromechanical equilibrium beyond the pull-in voltage.
3.2.1.2. Charge-controlled analysis. Charge control is a well-known means to extend the stable operating area of electromechanical actuators [11]. Pull-in does not occur until a normalized angle of g 0:7106 is attained (Fig. 5), with a corresponding charge of r D (9) Qpi 1:798Ewk L The in¯uence of parasitic charge and the microtechnological realization is discussed in [9,10]. But charge control is not only of practical use. Most simulation tools use an
3.2.2.1. Electrostatic spring softening. For a coupled electromechanical modal analysis and resonance measurements of the microswitch, the effect of electrostatic spring softening has to be included. The action of the nonlinear electrostatic torque contributes with negative sign to that of the linear mechanical torque, which leads to a smaller value of the effective spring constant at the operating point considered, and hence, to a decrease of the resonance frequency. Using our compact model the shift of the resonance frequency versus bias voltage can easily be analyzed (Fig. 6) and conforms well with the measured frequency of 72 kHz [5]. 3.2.3. Transient analysis 3.2.3.1. Dynamic pull-in. Performing a transient analysis we investigated the influence of the actuation voltage on the switching time. Pull-in occurs at a dynamic pull-in voltage U dpi < U pi due to the influence of the inertia. If the switch is operated within the dynamic pull-in regime, then a small
Fig. 6. Small-signal analysis showing the shift in resonance frequency due to spring softening.
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Fig. 7. Pull-in dynamics for various actuation voltages. The static pull-in voltage is U pi 8 V.
change in the voltage has a strong impact on the switching time, as it is illustrated in Fig. 7. 3.2.3.2. Influence of an increased electrode area. For relay applications a large, uniform and smooth contact area is essential; this can only be achieved by exerting a high contact force. In order to obtain a higher electrostatic force, a larger electrode area is necessary. Assuming the electrode to be rigid and perfectly supported by the anchor in the center of rotation, the contact force can be calculated as F
Melectrostatic L
Mtether
cos j0
(10)
been performed for `on/off' and `off/on'-switching cycles. Because of the laser wavelength of 532 nm, all larger displacements lead to ambiguities in the photodetector signal. These can be removed by evaluating the different phase shifts. Fig. 9 shows the vertical displacement as extracted from the measured signal for `on/off'-switching. Three designs (A, B and C) differing in the size and the number of anti-damping holes (area of etch holes is 36, 25 and 16% of the area of the mainplate) have been investigated and compared with the results of the macromodel. For the `on/off'-switching the squeezing of the gas film is negligible. The damping constant A was adjusted to the
Considering Eq. (3) we recognize that the contact force increases linearly with the area of the mainplate. This result is an estimate of the bound from above, as the real actuator is neither rigid nor perfectly supported. So part of the ideal force will be lost in the deformation of the anchor and the bending of the electrodes. But with increasing area also the inertia moment of the switch increases, raising the question how the switching speed scales with electrode area. Fig. 8 shows the pull-in time as a function of the electrode area. The stiffness of the tethers was adapted according to Eq. (8) in order to keep the pull-in voltage at the same value for all areas. As reported by Grif®n et al. [7], the damping constant A in Eq. (7) scales with the square of the area. But this approximation does not consider the many anti-damping holes present in our structure. If, for instance, a linear relationship is assumed, the switching speed would even increase with the electrode area. 3.2.3.3. Influence of etch holes. Transient measurements of the switch contact by means of a laser interferometer have
Fig. 8. Variation of switching time and contact force of a quadratic actuator with increasing mainplate area for an actuation voltage of 2 V above the pull-in voltage.
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Fig. 9. Measured transient behavior for `on/off'-switching compared to simulation for three differently perforated switches with fitted damping constants c.
measured data for the three designs (Fig. 9). Then the same damping constants were used to model the `off/on'switching. Fig. 10 shows that the measurement can be reproduced quite well, but only if the improved damping model (Eq. (7)) is used. Due to the compression of the gas in the gap the damping constant is not constant any more, but depends on the angle of torsion, which is a new and worthwhile result. 3.2.4. System simulation Due to technical constraints the available supply voltage is in many cases too low to operate the switch. This problem can be solved by means of a charge pump. A cascade of several capacitors and diodes is driven by an alternating voltage generated from the dc supply voltage. At each stage, the voltage is increased, but the available output current is limited by the corresponding demand of charge per cycle (I max U0 C1 f ). Therefore, a charge pump can be described as a voltage source with a virtual source impedance, leading to a voltage drop but without any resistive losses. Also the
Fig. 10. Measured transient behavior for `off/on'-switching compared to simulation for three different perforated switches. The result for a model using a uniform (angle-independent) damping constant is shown for comparison.
ripple increases, if the load current is high compared to the available capacitors. Because charge pumps are sensitive to the behavior of the load they drive, it is important for the circuit designer to have a reliable simulation model for the interaction of these two components. In the case studied in this work, the charge pump has only one stage, and resistors (R1, R2, RL) are included to represent the losses in a non-ideal circuit. The torsional actuator is represented as a variable capacitance. As the model of the actuator is coded in an analog hardware description language, it can easily be included in the circuitry of the charge pump (Fig. 11). The in¯uence of the pump frequency on the behavior of the system is presented in Fig. 12. The exponential increase of the output voltage indicates a proper operation of the charge pump. But during the pull-in the capacitance of the switch increases, and if the pump frequency is not suf®ciently high, then the output voltage of the charge pump drops too much. This can delay or even prevent the switching operation.
Fig. 11. Schematic of a simple charge pump combined with the macromodel of the switch.
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have seen from the results of the pull-in hystereses and the transient behavior. But if the stiffness of the switch is importantÐas it is important for the mode shapes or the maximum achievable contact forceÐthen a ®eld solver is the best choice, even though it implies much higher computational expenses. 4.1. Resonance modes The resonance modes of the mainplate have been studied using the FEM simulator Memcad [12]. The discretization of the (simpli®ed) device structure by an FEM mesh is shown in Fig. 13. The darker parts in the mesh indicate a higher density of the material due to a lower perforation of the structure. The experimental mode analysis is exactly re¯ected in the simulation procedure: For a given dc bias voltage, an ac voltage is superimposed and applied to one of the actuation electrodes, then the harmonic response at each operating point is determined with a modi®ed Mach±Zehnder laser-interferometer. The comparison of simulated and measured mode shapes together with the extracted resonance frequencies is presented in Fig. 14. One should note that, by applying a voltage to one of the actuation electrodes, not all possible modes can be excited. For example, lateral displacements can never be excited nor detected. Hence, the simulated values of those modes are omitted in the list presented. As damping is not considered in the simulation, the calculated resonance frequencies are a little higher compared to measurement. An exception is the ®rst mode. In the switching mode the resonance frequency is increased due to squeeze-®lm spring forces. Fig. 12. Time-dependence of the output voltage U2 of the charge pump and the pull-in characteristics for various pump frequencies.
4. Continuous field modeling In the compact model the actuator is considered to be rigid. This simpli®cation is justi®ed for many cases, as we
4.2. Improved switching time The mayor contribution to the moment of inertia of the switch is the contact metalization. To reduce the switching time we removed the contact bar from the moving structure. Instead of moving down the contact bar to bridge the signal lines as shown in the left part of Fig. 15, the seesaw-type
Fig. 13. Meshing of the torsion plate for FEM-modal analysis.
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Fig. 14. (a) Simulated and (b) measured resonance modes of the torsional plate.
Fig. 15. Schematic illustration of downward (left) and upward (right) switching (not to scale).
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Fig. 16. SEM micrograph of improved switch design.
Fig. 17. Optimization of the gap between mainplate and upper contact for a maximum contact force.
Fig. 18. Vertical displacement of the switch for various actuation voltages in comparison with measurements.
structure shown in the right part features a downward tilting of one side of the mainplate in conjunction with an upward movement of the opposite side in order to close the contacts from below (Fig. 15, right side). This allows the omission of the contact bar on the tiltable poly-plate. In this way, the mass of the switch plate is reduced, resulting in faster switching times for this type of a design. The contact area of the device with upward switching is shown in the SEM micrograph of Fig. 16. However, now the gap between the switch and the upper contact must be optimized (Fig. 17). On the side where the mainplate is pulled down, it must almost be in contact with the ground electrode to get the maximum electrostatic torque. Thus, if the gap between the switch and the upper contact is too small, then the switch cannot come
close to the ground electrode, but if the gap is too large, then the switch touches the ground electrode and reduces the contact force. Measurements of the switch displacement for various actuation voltages are compared with FEM simulations in Fig. 18. 5. Conclusion A surface micromachined switch for dual mode operation has been studied by experiments and numerical simulation. A compact model based on a physical device description has been formulated, which provides not only important insights in the static and dynamic behavior of the device, but also
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allows performing parameter studies at acceptable computational cost. The in¯uence of the actuation voltage, the electrode area and the etch holes on the device operation was investigated and compared with corresponding experiments. Once the damping constants are calibrated, the model can be used for predictive simulations. Furthermore our compact model can easily be incorporated in a full system model by attaching it to the electric circuitry needed for the voltage generation and device control. The pump frequency of a charge pump turned out to be a crucial parameter for a proper device operation. The resonance frequencies of the switch have been measured by laser interferometry and were found to be in good agreement with previously simulated results. An improved microswitch structure has been proposed where pull-down and bridging of the signal contacts occur on opposite sides of the mainplate. The design was optimized by contact simulations and compared with experimental data. Acknowledgements The authors would like to express their special thanks to S. Hoffmann for the circuit diagram of the charge pump. Also acknowledged are A. Meckes, K.-G. Oppermann, H. Kapels, M. Franosch and T. Bever for their technical support as well as M. Handtmann for valuable discussions. References [1] C.L. Goldsmith, Y. Zhimin, S. Eshelman, D. Denniston, Performance of low-loss RF MEMS capacitive switches, IEEE Microwave Guided Wave Lett. 8 (8) (1998) 269±271. [2] S.P. Pacheco, L.P.B. Katehi, C.T. Nguyen, Design of low actuation voltage RF MEMS switch, in: Proceedings of the International Microwave Symposium 2000, Boston, 2000, pp. 165±168. [3] J.J. Yao, M.F. Chang, A surface micromachined miniature switch for telecommunications applications with signal frequencies from dc up to 4 GHz, in: Proceedings of the Eurosensors IX, 1995, pp. 384±387. [4] D. Hah, E. Yoon, S. Hong, A low voltage actuated micromachined microwave switch using torsion springs and leverage, in: Proceedings of the International Microwave Symposium 2000, Boston, 2000, pp. 157±160. [5] F. PloÈtz, S. Michaelis, R. Aigner, H.-J. Timme, J. Binder, R. NoeÂ, A low-voltage torsional actuator for application in RF-microswitches, in: Proceedings of the Eurosensors XIV, Copenhagen, Denmark, August 2000, pp. 297±300. [6] G. Wachutka, Tailored modeling: a way to the virtual transducer fab, Sens. Actuators A 46/47 (1995) 603±612. [7] W.S. Griffin, H.H. Richardson, S. Yamanami, A study of squeezefilm damping, J. Basic Eng. (1966) 451±456. [8] http://www.cadence.com. [9] R.N. Guardia, R. Aigner, W. Nessler, M. Handtmann, L.M. CastanÄer, Control positioning of torsional electrostatic actuators by current driving, in: Proceedings of the ASDAM, Smolenice Castle, Slovakia, October 2000, pp. 91±94. [10] R.N. Guardia, A. Dehe, R. Aigner, L. CastanÄer, New current drive method to extend the stable operation range of electrostatic actuators:
experimental verification, in: Proceedings of the Transducers'01, Munich, Germany, June 2001, pp. 760±763. [11] E.-R. KoÈnig, G. Wachutka, Analysis of unstable behaviour occuring in electromechanical microdevices, in: Proceedings of the MSM, Puerto Rico, USA, April 1999, pp. 330±333. [12] http://www.memcad.com. [13] F. PloÈtz, S. Michaelis, R. Aigner, H.-J. Timme, J. Binder, R. NoeÂ, A low-voltage torsional actuator for application in RF-microswitches, Sens. Actuators A 92 (1±3) (2001) 312±317.
Biographies Robert Sattler was born in Memmingen, Germany, in 1970. He studied physics at the University of New South Wales, Sydney, Australia, and the University of Regensburg, where he received the Diplom-Physiker degree in 1998. Since then he is working towards his PhD degree at the Institute for Physics of Electrotechnology at Munich University of Technology. His research activities are focused on modeling of coupled effects in microsystems. Florian PloÈtz was born in Regensburg, Germany, in 1973. He studied physics at the University of Colorado at Boulder, USA, and the University of Regensburg, where he received the Diplom-Physiker degree in 1998. Since then he is working towards his PhD degree in electrical engineering at the University of Paderborn and is with the MEMS group of Infineon in Munich, formerly Siemens Corporate Technology. His research interests are in the area of high-frequency microsystems. Gernot Fattinger was born in Linz, Austria, in 1975. He studied semiconductor and solid state physics at the University of Linz before he joined the MEMS group at Infineon Technologies, formerly Siemens Corporate Technology, in 1999. There he accomplished his diploma thesis about interferometric measurement methods for RF-MEMS and received the Diplom-Ing degree in 2000. Currently he is working towards his PhD degree in physics, studying RF microelelctromechanical devices. Gerhard Wachutka received DSc degree from the Ludwig-MaximiliansUniversitaÈt, Munich, Geramny, in 1985. From 1985 to 1988, he was with Siemens Corporate Research & Development, Munich, where he headed a modeling group active in the development of modern high-power semiconductor devices. In 1989, he joined the Fritz-Haber-Institute of the Max-Planck-Society, Berlin, Germany, where he worked in the field of theoretical solid-state physics. From 1990 to 1994, he was head of the microtransducers modeling and characterization group of the Physical Electronics Laboratory at the Swiss Federal Institute of Technology (ETH), Zurich. There he also directed the microtransducers modeling module of the Swiss Federal Priority Program M2S2 (micromechanics on silicon in Switzerland). Since spring 1994, he has been heading the Institute for Physics of Electrotechnology at the Munich University of Technology, where his research activities are focused on the design, modeling, characterization, and diagnosis of the fabrication and operation of semiconductor microdevices and microsystems. He has authored or coauthored more than 180 publications in scientific or technical journals. He is consultant of research institutes in industry and university, and he serves as reviewer for various scientific journals and other institutions. Among his many educational activities, he has set-up and taught courses funded by European Community training programs such as UETP, EUROFORM, and EUROPRACTICE. Professor Wachutka is member of the IEEE, the American Electrochemical Society, the American Materials Research Society, the ESD Association, the VDE Association for Electrical, Electronic and Information Technologies, the German Physical Society, the American Physical Society, and the AMA Society for Sensorics.