1 Nonlinear Dynamics in Nanomechanical Oscillators Stav Zaitsev, Ronen Almog, Oleg Shtempluck, and Eyal Buks Abstract— In the present work we investigate nonlinear dy- namics in a nanomechanical doubly clamped beam made of PdAu fabricated using bulk nanomachining and e-beam lithography. The beam is driven into nonlinear regime of oscillations and the response is measured by an electron beam displacement detector. In one set of experiments we study the impact of nonlinear damping on the dynamics in the bistable regime of operation. For data analysis we introduce a nonlinear damping term to Duffing equation. The experiment shows conclusively that accounting for nonlinear damping effects is needed for correct modeling of the dynamics. In another set of experiments we study intermodulation mechanical gain near the onset of bistability . As predicted by a theoretical analysis, we find high intermodulation gain when the system is operated close to a bifurcation. I. I NTRODUCTION The field of micro-machining is forcing a profound redefi- nition of the nature and attributes of electronic devices. This technology allows fabrication of a variety of on-chip fully integrated sensors and actuators with a rapidly growing range of applications. In many cases it is highly desirable to shrink the size of mechanical elements down to the nano-scale [1], [2]. This allows increasing the speed of operation by increasing the frequencies of mechanical resonances and enhancing their sensitivity as sensors. Moreover, as devices become smaller their power consumption goes down and their cost can be sig- nificantly lower. Some key applications of NEMS technology include magnetic resonance force microscopy (MRFM) [3], [4] and mass-sensing [5]. Further miniaturization is also motivated by the quest for mesoscopic quantum effects in mechanical systems [6], [7], [8]. Nonlinear effects are of great importance for nanomechan- ical devices. The relatively small applied forces needed for driving a nanomechanical oscillator into a nonlinear regime is usually easily accessible. Thus, a variety of useful applications such as frequency synchronization, frequency mixing and conversion, and parametric amplification, can be implemented by applying modest driving forces. Moreover, monitoring the displacement of a nanomechanical oscillator oscillating in the linear regime may be difficult when a displacement detector with high sensitivity is not available. Thus, in many cases the nonlinear regime is the only useful regime of operation. How- ever, to optimize the properties of NEMS devices operating in the nonlinear regime it is important to study the underlying physics. In the present work we study damping and intermodulation gain in a nanomechanical oscillator. We find that correct This work was supported by the German Israel Foundation under grant 1-2038.1114.07, the Israel Science Foundation under grant 1380021, the Deborah Foundation and Poznanski Foundation. The authors are with the Department of Electrical Engineering and Microelectronics Research Center, Technion, Haifa 32000, Israel (e-mail: eyal@ee.technion.ac.il). Fig. 1. The device consists of a narrow doubly clamped beam (length 200 m, width 0.25 m and thickness 0.2 m) and wide electrode. The excitation force is applied as voltage between the beam and the electrode. modeling of the response of the system in the nonlinear regime is possible only when nonlinear damping is taken into account. Moreover, we characterized the nonlinear response by studying intermodulation, and find that high gain is achieved when operating close to a bifurcation. II. EXPERIMENTAL SETUP For the experiments we employ nanomechanical oscillators in the form of doubly clamped beams made of PdAu (see Fig. 1 ). The bulk nano-machining process used for sample fabrica- tion is similar to the one described in [9], [10]. Measurements of mechanical properties are done in-situ a scanning electron microscope, where the imaging system of the microscope is employed for displacement detection [10]. A driving force is applied to the beam by applying a voltage to the nearby electrode. With a relatively modest driving force the system is driven into the regime of nonlinear oscillations [10], [11]. III. NONLINEAR DAMPING A key property of devices based on mechanical oscillators is the rate of damping. For example, in many cases the sensitivity of NEMS sensors is limited by thermal fluctuation which is related to damping via the fluctuation dissipation theorem. In general, a variety of different physical mechanisms can contribute to damping, including bulk and surface defects, thermoelastic damping, nonlinear coupling to other modes, phonon-electron coupling, clamping loss, etc.. Identifying ex- perimentally the contributing mechanisms in a given system can be highly challenging, as the dependence on a variety of parameters has to be examined systematically. Nanomechanical systems suffer from low quality factors Q relative to their macroscopic counterparts [2]. This behavior suggests that damping in nanomechanical devices is dominated by surface properties, since the relative number of atoms Proceedings of the 2005 International Conference on MEMS, NANO and Smart Systems (ICMENS’05) 0-7695-2398-6/05 $20.00 © 2005 IEEE