Micromechanical cantilevers are commonly used for detection of small forces in microelectromechanical sensors (e.g., accelerometers) and in scientific instruments (e.g., atomic force microscopes). A fundamental limit to the detection of small forces is imposed by thermomechanical noise, the mechanical analog of Johnson noise, which is governed by dissipation of mechanical energy. This paper reports on measurements of the mechanical quality factor Q for arrays of silicon-nitride, polysilicon, and single-crystal silicon cantilevers. By studying the dependence of Q on cantilever material, geometry, and surface treatments, significant insight into dissipation mechanisms has been obtained. For submicron-thick cantilevers, Q is found to decrease with decreasing cantilever thickness, indicating surface loss mechanisms. For single-crystal silicon cantilevers, significant increase in room temperature Q is obtained after 700/spl deg/C heat treatment in either N/sub 2/ Or forming gas. At low temperatures, silicon cantilevers exhibit a minimum in Q at approximately 135 K, possibly due to a surface-related relaxation process. Thermoelastic dissipation is not a factor for submicron-thick cantilevers, but is shown to be significant for silicon-nitride cantilevers as thin as 2.3 /spl mu/m.

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Quality factors in micron- and submicron-thick cantilevers

We report measurements of intrinsic dissipation in micron-sized suspended resonators machined from single crystals of galium arsenide and silicon. In these experiments on high-frequency micromechanical resonators, designed to understand intrinsic mechanisms of dissipation, we explore dependence of dissipation on temperature, magnetic field, frequency, and size. In contrast to most of the previous measurements of acoustic attenuation in crystalline and amorphous structures in this frequency range, ours is a resonant measurement; dissipation is measured at the natural frequencies of structural resonance, or modes of the structure associated with flexural and torsional motion. In all our samples we find a weakly temperature dependent dissipation at low temperatures. We compare and contrast our data to various probable mechanisms, including thermoelasticity, clamping, anharmonic mode-coupling, surface anisotropy and defect motion, both in bulk and on surface. The observed parametric dependencies indicate that the internal defect motion is the dominant mechanism of intrinsic dissipation in our samples.

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Intrinsic dissipation in high-frequency micromechanical resonators

The thermo-optic coefficient dn/dT of silicon was measured at 1550 nm in the wide temperature range from 5 K to 300 K. For this purpose an interferometric measurement scheme was applied using the silicon sample as a Fabry-Perot etalon. The high resolution of this setup revealed a thermo-optic coefficient as low as 10−8 K−1 at 5 K. The presented results show an excellent agreement with former measurements above 30 K including a value of dn/dT=1.8×10−4 K−1 at 300 K.

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Thermo-optic coefficient of silicon at 1550 nm and cryogenic temperatures

A simple model of thermoelastic dissipation is proposed for general, free standing microelectromechanical (MEMS) and nanoelectromechanical (NEMS) 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 K) 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

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Thermoelastic loss in microscale oscillators

Thermal noise of optical reference cavities sets a fundamental limit to the
 frequency instability of ultrastable lasers. Using Levin’s formulation of the
 fluctuation-dissipation theorem, we correct the analytical estimate for the spacer
 contribution given by Numata et al. [Phys. Rev. Lett.93, 250602
 (2004)]. For detailed analysis,
 finite-element calculations of the thermal noise focusing on the spacer geometry,
 support structure, and the usage of different materials have been carried out. We
 find that the increased dissipation close to the contact area between spacer and
 mirrors can contribute significantly to the thermal noise. From an estimate of the
 support structure contribution, we give guidelines for a low-noise mounting of the
 cavity. For mixed-material cavities, we show that the thermal expansion can be
 compensated without deteriorating the thermal noise.

Thermal noise in optical cavities revisited

Measurements of the mechanical quality factor Q in a single crystal of silicon vs. temperature have been made. A value of 2 × 109 has been measured at T = 3.5K.

A. W. Hoffman

Papers

Measurements of the mechanical Q of single-crystal silicon at low temperatures