Resonance frequency shift of strongly heated micro-cantilevers

Abstract

In optical detection setups to measure the deflection of micro-cantilevers, part of the sensing light is absorbed, heating the mechanical probe. We present experimental evidences of a frequency shift of the resonant modes of a cantilever when the light power of the optical measurement set-up is increased. This frequency shift is a signature of the temperature rise and presents a dependence on the mode number. An analytical model is derived to take into account the temperature profile along the cantilever; it shows that the frequency shifts are given by an average of the profile weighted by the local curvature for each resonant mode. We apply this framework to measurements in vacuum and demonstrate that huge temperatures can be reached with moderate light intensities: a 1000 °C with little more than 10 mW. We finally present some insight into the physical phenomena when the cantilever is in air instead of vacuum.

Mesoscopic virial equation for nonequilibrium statistical mechanics

“…Indeed, recent experiments with heatconducting metals show intriguing deviations from equipartition, related to enhancements of low-frequency vibrational modes that may become even 'hotter' than the highest boundary temperature [3]. Similar deviations from equipartition are observed for strongly heated cantilevers [4] and Brownian particles [5,6]. These are some out of many manifestations of nontrivial effects characterizing systems driven far from thermodynamic equilibrium.…”

Low thermal fluctuations in a system heated out of equilibrium

“…The sensing beam heats the cantilever and a steady temperature profile driven by the absorbed light power is reached, setting the system in a non-equilibrium steadystate. In a first approximation, the temperature T grows linearly with x, the coordinate along the cantilever length [27].…”

“…In vacuum, the main way out for the heat is by conduction [28] along the cantilever length towards the clamped base, which is thermalized at room temperature thanks to its macroscopic size. A huge thermal gradient can easily be reached: a 500 K difference along its 500 µm length with only P = 10 mW of light power [27]. We measure the deflection with a differential interferome-ter [26], whose sensing beam also acts as the heating source.…”

“…Tracking those frequency shifts with light power, we estimate the cantilever temperature profile, and thus its average temperature T avg (more details are available in appendix B and in ref. [27] Although from a thermal point of view, the cantilever can be described within the framework of local equilibrium with a temperature profile T (x), the fluctuations of the deflection of the cantilever are a collective phenomenon, that cannot be described only by the local temperature at the endpoint of the cantilever; these fluctuations rather need to be described using the normal modes of the cantilever. The present experimental system then appears as an ideal toy model to test the behavior of the thermal fluctuations in a Non-Equilibrium Steady State (NESS): the system is large enough to have a strongly non-uniform temperature profile, and small enough to have measurable mechanical fluctuations, its degrees of freedom (i.e., the normal modes) are easily identified and uncoupled, and a one-dimensional description (the Euler-Bernoulli model for a mechanical clamped-free beam, see below) captures the main physical phenomena .…”

“…We plot in Fig. 3 the effective temperatures of the first 3 flexural modes measured on two different cantilevers as a function of the impinging light power P , and compare it to their average temperature T avg (measured by the resonance frequency shift [27]). The behavior of the two cantilevers is strikingly different.…”