Mechanical dissipation in silicon flexures

doi:10.1016/j.physleta.2005.10.103

Physics Letters A

Volume 351, Issues 4–5, 6 March 2006, Pages 205-211

https://doi.org/10.1016/j.physleta.2005.10.103 Get rights and content

Abstract

The thermo-mechanical properties of silicon make it of significant interest as a possible material for mirror substrates and suspension elements for future long-baseline gravitational wave detectors. The mechanical dissipation in 92 μm thick $〈 110 〉$ single-crystal silicon cantilevers has been observed over the temperature range 85 K to 300 K, with dissipation approaching levels down to $ϕ = 4.4 \times 10^{−7}$ .

Introduction

Long baseline gravitational wave detectors operate using laser interferometry to sense the differential strain, caused by the passage of gravitational waves, between mirrors suspended as pendulums. These detectors operate over a frequency range between the pendulum modes of the suspensions (typically few Hz) and the lowest internal resonances of the mirrors (few 10's of kHz). One important limit to the displacement sensitivity of current and planned detectors in the frequency range of operation is off-resonance thermal noise in the mirrors and suspensions driven by thermal fluctuations. Thus, low mechanical loss materials, such as silica, sapphire and silicon, are currently used or proposed for detectors at the forefront of this research.

Improved sensitivity at low frequencies (few Hz to few 100 Hz) will require further reduction in the level of thermal noise from the test masses and their suspensions. A possible route for achieving this is through cooling. Fused silica, the most commonly used test mass material, exhibits a broad dissipation peak at around 40 K and therefore is not a promising candidate for cooling [1]. Sapphire and silicon, however, are good candidates. Work is currently being carried out in Japan on developing cooled sapphire test masses and suspension fibers for use in a transmissive Fabry–Perot based interferometer [2], [3], and in Europe and the US research is underway on the use of silicon at low temperatures [4], [5].

At higher frequencies (greater than a few 100 Hz) the performance of current interferometers is not limited by thermal noise from the optics but by photo-electron shot noise, whose significance can be reduced by circulating higher optical powers in the interferometer. However, power absorbed by the test masses and mirror coatings can cause excessive thermally induced deformations of the optics, causing the interferometer to become unstable. The extent of this deformation is proportional to $α / κ$ [6], where α is the linear coefficient of thermal expansion and κ the thermal conductivity of the test mass material. Changing from a transmissive to a reflective topology could eliminate thermal loading from substrate absorption, provided coatings of suitably low transmission are available. Used in such a topology, the high thermal conductivity of a silicon mirror substrate would allow circulating powers approximately seven times higher than could be supported by sapphire for the same induced surface deformation making silicon of significant interest as a test mass substrate from a thermal loading standpoint [4].

At room temperature the thermal noise resulting from thermo-elastic effects in interferometers using crystalline optics has been predicted to be a significant noise source in the frequency band of gravitational wave detection [7]. The level of intrinsic and expected thermo-elastic dissipation in silicon is broadly comparable to sapphire at room temperature [8]. However, on cooling the linear thermal expansion coefficient of silicon becomes zero at two temperatures, $\sim 125 K$ and $\sim 18 K$ [9], and thus around these two temperatures thermoelastic dissipation could be expected to be negligible. It is thus of interest to study the temperature dependence of mechanical dissipation in silicon samples for potential use as suspension elements and mirror blanks. This Letter is restricted to studies of thin silicon flexures.

Studies of dissipation in silicon samples of a variety of geometries and types have been carried out by other authors. In particular, dissipation in silicon flexures has been studied in samples of the type used in atomic force microscopes [10], [11]. However, these cantilevers have dimensions considerably smaller than would be suitable for use in the test mass suspensions of gravitational wave interferometers and thus are in a regime where measured dissipation may be dominated by different sources of dissipation than the dimensions that have been studied here [12].

Section snippets

Experimental procedure

The single-crystal cantilevers tested were fabricated from a silicon wafer by a hydroxide chemical etch. The anisotropic nature of such etching allows the reduction of thickness whilst a masked, thick end can remain as a clamping block to reduce any ‘slip-stick’ losses as the cantilever flexes [13]. The geometry of the cantilevers obtained is shown in Fig. 1. The silicon was boron-doped with a resistivity of 10–20 Ω cm.

The thick end of each cantilever was held in a stainless steel clamp and

Loss as a function of temperature

The measured loss, $ϕ_{measured} (ω)$ , is the sum of dissipation arising from a number of sources, $ϕ_{measured} (ω) = ϕ_{thermoelastic} (ω) + ϕ_{bulk} (ω) + ϕ_{surface} (ω) + ϕ_{clamp} (ω) + ϕ_{gas} (ω) + ϕ_{other} (ω),$ where $ϕ_{thermoelastic} (ω)$ is loss resulting from thermoelastic damping, $ϕ_{bulk} (ω)$ is the bulk (or volume) loss of the material, $ϕ_{surface} (ω)$ is the loss associated with the surface layer, $ϕ_{clamp} (ω)$ is the loss associated with the clamping structure, $ϕ_{gas} (ω)$ is the loss due to damping from residual gas molecules and $ϕ_{other} (ω)$ is

Surface loss

Mechanical loss measurements carried out on silicon samples of sub-micron thickness suggest that the measured loss is dominated by surface losses [18], [21]. These may be due to a combination of the following:

1.
A thin layer of oxidized silicon on the surface [21];
2.
Shallow damage to the crystal structure (atomic lattice) from surface treatment;
3.
Contaminants absorbed on or into the surface from the surroundings or from polishing;
4.
General (or local) surface roughness [22].

Yasumura et al. measured the

Conclusions and future work

Our measurements of the mechanical dissipation of single crystal silicon cantilevers as a function of temperature in general show the dissipation decreasing as temperature decreases. At room temperature the measured dissipation is strongly dependent on the level of thermoelastic dissipation in the sample, however, at lower temperatures other loss mechanisms become dominant. Losses associated with the surface of the samples are expected to be significant, but at a level lower than the measured

Acknowledgements

We would like to thank our colleagues in the GEO 600 project for their interest. We wish to thank V. Mitrofanov and J. Faller for useful discussions. We are grateful for the financial support provided by PPARC and the University of Glasgow in the UK and the NSF in the USA (award No. NSF PHY-0140297).

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Physics Letters A

Mechanical dissipation in silicon flexures

Abstract

Introduction

Section snippets

Experimental procedure

Loss as a function of temperature

Surface loss

Conclusions and future work

Acknowledgements

Phys. Lett. A

Phys. Lett. A

Phys. Lett. A

Phys. Lett. A

Sens. Actuators A

Systems With Small Dissipation

Int. J. Mod. Phys. D

Proc. SPIE

Class. Quantum Grav.

Phys. Rev. A

Phys. Rev. Lett.

Science