Quantum Limit of Quality Factor in Silicon Micro and Nano Mechanical Resonators

Abstract

Micromechanical resonators are promising replacements for quartz crystals for timing and frequency references owing to potential for compactness, integrability with CMOS fabrication processes, low cost, and low power consumption. To be used in high performance reference application, resonators should obtain a high quality factor. The limit of the quality factor achieved by a resonator is set by the material properties, geometry and operating condition. Some recent resonators properly designed for exploiting bulk-acoustic resonance have been demonstrated to operate close to the quantum mechanical limit for the quality factor and frequency product (Q-f). Here, we describe the physics that gives rise to the quantum limit to the Q-f product, explain design strategies for minimizing other dissipation sources, and present new results from several different resonators that approach the limit.

A 10-GHz film-thickness-mode cavity optomechanical resonator

“…On the other hand, the thickness mode exhibits both excellent transmission and a highest fÁQ product of 1.9 Â 10 13 Hz, which is close to the theoretical limit of 2.5 Â 10 13 Hz reported in Ref. 16.…”

“…A high mechanical Q of 1830 (in air) at 10.4 GHz is demonstrated using a 550-nm-thick micro-disk, leading to an fÁQ product (frequency-quality factor product) of 1.9 Â 10 13 Hz, which approaches the material limit of AlN. 16 This thickness mode-based piezo-optomechanical resonator can be expected to serve as a core element in high-frequency microwave applications. 17,18 It also provides a promising platform for efficient coherent signal conversion among electrical, optical, and mechanical domains.…”

“…For high frequency resonators, a more useful figure of merit is the fÁQ product, which is fundamentally limited by quantum phonon scattering in the material known as the Akhiezer effect. 16 The fÁQ products of the first eight radial-contour modes as well as the fundamental thickness mode are plotted in Fig. 2(c).…”

Gold thin film viscoelastic losses of a length extension mode resonator

“…We demonstrated with finite element analysis that the specific design of the LEM resonator family allows Q anchor >10 8 . The Q.f product for quartz is limited by Akhiezer losses this maximum value for MHz range frequencies is given in [2]: Q.f=3.2x10 13 . Assuming viscous fluid damping (1/Q air ) is negligible since the resonator is held under vacuum and so is the thermoelastic damping (1/Q TED ) since the resonator is in a length extension mode it appears that the maximum value Q tot for a 1.3 MHz LEM resonator is well above one million allowing applications in devices such as MEMS oscillators.…”

Strong vibration nonlinearity in semiconductor-based nanomechanical systems

“…Much attention have been recently attracting Si-based nano-and micromechanical systems, see [15,16] and references therein. In such systems there was observed an unexpectedly large change of the amplitude dependence of the vibration frequency with the varying electron density [17,18].…”