A simple model of thermoelastic dissipation is proposed for general, free standing microelectromechanical ͑MEMS͒ and nanoelectromechanical ͑NEMS͒ oscillators. The theory defines a flexural modal participation factor, the fraction of potential energy stored in flexure, and approximates the internal friction by assuming the energy loss to occur solely via classical thermoelastic dissipation of this component of the motion. The theory is compared to the measured internal friction of a high Q mode of a single-crystal silicon double paddle oscillator. The loss at high temperature ͑above 150 K͒ is found to be in good agreement with the theoretical prediction. The importance of this dissipation mechanism as a function of scale is briefly discussed. We find that the relative importance of this mechanism scales with the size of the structure, and that for nanoscale structures it is less important than intrinsic phonon-phonon scattering. © 2002 American Institute of Physics. ͓DOI: 10.1063/1.1449534͔Obtaining high Q performance in Si based oscillators in microelectromechanical systems ͑MEMS͒ and nanoelectromechanical systems ͑NEMS͒ is desirable for applications that include charge detectors, 1 radio-frequency filters, 2 magnetic resonance force microscopes, 3 and torque magnetometers. 4 Achieving this goal has proven to be difficult however, and to date, the achieved Qs in MEMS/NEMS oscillators have been orders of magnitude smaller than expected from considerations of fundamental loss mechanisms.5 Explanations of the observed behavior include attachment losses, 6,7 bulk defects, 4 losses associated with the electrical contacts, 8 and surface effects. 9 Unfortunately, untangling the various potential mechanisms is not a straightforward exercise because of the difficulty of controlling and measuring the properties of microscale and nanoscale devices.The MEMS oscillators are typically free standing, planar samples vibrating predominantly either in flexure or torsion. In the case of a simple beam in flexure, thermoelastic ͑TE͒ dissipation is known to be a significant loss mechanism near room temperature. 10 The loss mechanisms in other oscillators involving torsional vibration, or in general possessing a nontrivial mode shape, are however still unknown. We have developed a simple model which predicts the internal friction arising from TE dissipation in these types of oscillators. We have found the internal friction can be quite significant even for nominally torsional vibration modes which one might conclude have no TE loss. Analysis of the experimental results for a macroscopic single-crystal silicon oscillator yields good agreement with the prediction.TE dissipation arises from thermal currents generated by compression/decompression in elastic media. Typically, the TE dissipation arising from longitudinal waves in solids is very small 11 owing to the large distance between the compressed and rarefied regions. In contrast, the compression associated with flexural vibration can give rise to TE dissipation orders of magnitude l...

Two-dimensional arrays of coupled nanomechanical plate-type resonators were fabricated in single crystal silicon using e-beam lithography. Collective modes were studied using a double laser setup with independent positioning of the point laser drive and interferometric motion detector. The formation of a wide acoustic band has been demonstrated. Localization due to disorder (mistune) was identified as a parameter that limits the propagation of the elastic waves. We show that all 400 resonators in our 20×20 array participate in the extended modes and estimate group velocity and density of states. Applications utilizing the resonator arrays for radio frequency signal processing are discussed.

We have performed laser-Doppler vibrometry measurements of the vibration of a double-paddle oscillator. Seven modes with principally out-of-plane motion have been identified. Their resonance frequencies and mode shapes are in excellent agreement with three-dimensional finite element simulations. We have found that the second antisymmetric torsional mode has exceptionally good vibration isolation of its mode shape. This explains its extremely small low temperature internal friction below 10 K (2ϫ10 Ϫ8 3,4 Oscillators of similar design have been used in a variety of applications. 5,6 However, extending these designs to microelectromechanical systems ͑MEMS͒, a subject of recent interest, has not been as successful. The commonly achieved low Q in MEMS oscillators is a key obstacle limiting the applications of such technology, particularly in the area of rf electronics.7 In order to improve our understanding of the loss mechanisms that are at work in silicon based oscillators, and in particular to achieve higher Q versions of the DPOs and MEMS oscillators in general, we have investigated the dynamics and loss mechanisms of a DPO.In this letter we report laser Doppler vibrometry ͑LDV͒ measurements of the lowest seven out-of-plane modes of a DPO and the corresponding finite element modeling. Direct LDV measurements have enabled us to establish a detailed finite element method ͑FEM͒ 8 predictive capability for the frequencies and mode shapes of the DPO. A detailed prediction of the Q is not possible because of the coupling of the oscillator to its external environment. Nevertheless, an empirical relationship between the mode shapes and internal friction is found which provides insight into the damping mechanisms and may be a useful guide to the development of high Q oscillators using FEM tools.The silicon oscillator was made of a 300 m thick, floatzone refined, double-side polished, ͗100͘ oriented, and N-doped silicon substrate with ϳ2 k⍀ cm resistivity. Its long axis was along the ͗110͘ orientation. The oscillator was epoxied to an Invar block inside a vacuum chamber ͑1 mTorr͒. The oscillator was driven capacitively by an electrode mounted under one of the wings. A second identical electrode under the opposite wing was used as a detector to provide a continuous feedback signal for the drive electronics. This setup enabled stable operation of the oscillator at resonance. Capacitive coupling was achieved by depositing a 400 Å gold film onto the side of the DPO which faced the electrodes. The head and neck were not coated in order to minimize the energy loss caused by metal films 9 to the second antisymmetric torsion mode. For details of the oscillator setup, see Ref. 10.With the oscillator driven in a selected resonance its outof-plane surface velocity was measured with a LDV ( ϭ780 nm, 7 mW͒.11 Optical access for the laser was made through an optical window in the chamber. The sensitivity of the instrument for reflection from the polished bare silicon surface corresponds to a detection threshold of 0.1-1 Å over the freq...

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The dynamics of single-crystal silicon ϳ100 m size rectangular paddle oscillators at room temperature have been studied using a recently developed high-resolution scanning laser vibrometer. The dynamic mechanical behavior is determined by scans of the entire device, providing both amplitude and phase spatial maps of the vibratory response. These reveal more than 16 normal modes below 500 kHz. In addition to simple translation and torsional motion, flexural modes of the paddle plate are observed. Quality factors ranging from 1ϫ10 3 to 2ϫ10 4 are measured and are found to be significantly lower than those expected from well-known intrinsic absorption mechanisms. The measurements reveal that there exists significant modification of the expected eigenfrequencies and mode shapes. It is speculated that this is caused by excessive undercutting of the support structure, and that the resulting energy flow into the support leads to increased oscillator loss. Indeed, some correlation is found between observed loss and energy levels resident in the supports. At frequencies where there is relatively little support motion, three-dimensional finite-element modeling accurately predicts the paddle modal behavior.