150 JOURNAL OF MICROELECTROMECHANICAL SYSTEMS, VOL. 20, NO. 1, FEBRUARY 2011 Electromechanical Sensing of Charge Retention on Floating Electrodes David Elata, Member, IEEE, Vitaly Leus, J. Provine, Arnon Hirshberg, Student Member, IEEE, and Roger T. Howe, Fellow, IEEE Abstract—This paper considers the electromechanical response of electrostatic actuators that are driven by both voltage and charge. The model system is an electrostatic actuator in which the suspended electrode is subjected to a driving voltage and the fixed electrode, which is electrostatically floating, is loaded by charge. The response of the system is analyzed using energy methods, and it is shown that the system has two distinct pull-in voltages. It is also shown that the amplitude of charge on the floating electrode is proportional to the average of these two pull-in voltages. Test- actuators were designed, fabricated, and characterized, and their measured response validates the theoretical predictions. A nondis- ruptive measurement of charge is proposed and demonstrated which enables to monitor charge decay over time. [2010-0052] Index Terms—Electrostatic actuators, floating electrode, pull-in. I. I NTRODUCTION T HE RESPONSE of electrostatic actuators to a given volt- age can be enhanced and linearized by adding a suf- ficiently large dc bias [1]. Biasing also enables the use of electrostatic transducers as sensors. Voltage sources for biasing (e.g., using charge pumping) may be unnecessary if actuators and sensors are preloaded with a fixed charge. In some devices, charge is stored in electrets, which are dielectric elements loaded with fixed injected charge [2], [3]. Precharged electret transducers may be advantageous in autonomous devices which operate within a highly constrained energy budget. However, charge may also be stored in floating conducting electrodes. For example, charged floating electrodes have been recently proposed for memory storage [4]. In that study, it was shown that charge retention can be extended if the floating electrode is encapsulated. To assess the reliability of charged floating electrodes, in encapsulated actuators and sensors, a simple and nondisruptive method for measuring charge is essential. Indirect measurement of charge by an nMOSFET has been demonstrated [4], but this measuring method requires imbedding electronic circuitry in the device. Manuscript received March 1, 2010; revised September 6, 2010; accepted September 23, 2010. Date of publication December 3, 2010; date of current version February 2, 2011. Subject Editor N. de Rooij. D. Elata, V. Leus, A Hirshberg are with the Technion—Israel Institute of Technology, Haifa 32000, Israel (e-mail: elata@technion.ac.il; levitaly@ tx.technion.ac.il; harnon@tx.technion.ac.il). J. Provine and R. T. Howe are with Stanford University, Stanford, CA 94305 USA (e-mail: jprovine@stanford.edu; rthowe@stanford.edu). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/JMEMS.2010.2090499 The main purpose of this paper is to demonstrate that charge on a floating electrode can be measured nondisruptively by characterizing the electromechanical response of an electrosta- tic actuator that is affected by this charge. This demonstra- tion includes both a theoretical prediction and experimental verification. The theoretical prediction is derived by using en- ergy methods to analyze the electromechanical response of the system. A second purpose of this paper is to offer new insight and elucidate the correct way of using energy methods to analyze the response of transducers that are affected by both voltage and charge. Particularly, we emphasize the correct way of considering the energy of the voltage source in the analysis. Transducers that are affected by both voltage and charge have been considered before. Though electrets are clearly ben- eficial, dielectric charging may be detrimental in devices that use dielectric layers for isolation. Dielectric charging [5]–[10] affects the electromechanical response of microelectromechan- ical actuators. For example, proper operation of RF-MEMS capacitive switches requires a predictable and stable pull-in voltage (e.g., [11]), but this voltage is strongly affected by dielectric charging [8]. Moreover, a total lockdown of capac- itive switches was shown to be caused by spatial distribution of injected charges [10]. A rigorous analysis of the effect of injected charges on the electromechanical response of electrostatic actuators was presented by Rottenberg et al. [10]. In that investigation, electrostatic forces due to applied voltage and injected charge were derived and used to solve the equilibrium equation of the system. The alternative approach of using energy methods which is applied in this paper is more general. For example, though this paper only considers the static response of transducers that are affected by both voltage and charge, energy methods can be used to directly compute critical parameters of the dynamic response, without having to perform time integration of momentum equations [14]. Although this paper considers the effect of charge on a floating electrode, the same modeling approach is applicable and relevant to systems in which charge is fixed in a dielectric layer. This will be demonstrated in a future paper. In the following section, the total potential of an electrostatic actuator that is simultaneously driven by voltage and charge is formulated. In Section III, the static response of the actuator is derived from the total potential, and it is shown that the actuator has two distinct pull-in voltages. In Section IV, we emphasize the correct way of accounting for the energy of the voltage 1057-7157/$26.00 © 2010 IEEE