Silicon cantilevers locally heated from 300 K up to the melting point: Temperature profile measurement from their resonances frequency shift

Abstract

When heated, micro-resonators present a shift of their resonance frequencies. We study specifically silicon cantilevers heated locally by laser absorption and evaluate theoretically and experimentally their temperature profile and its interplay with the mechanical resonances. We present an enhanced version of our earlier model [Sandoval et al., J. Appl. Phys. 117, 234503 (2015)], including both elasticity and geometry temperature dependency, showing that the latter can account for 20% of the observed shift for the first flexural mode. The temperature profile description takes into account thermal clamping conditions, radiation at high temperature, and lower conductivity than bulk silicon due to phonon confinement. Thanks to space–power equivalence in the heat equation, scanning the heating point along the cantilever directly reveals the temperature profile. Finally, frequency shift measurement can be used to infer the temperature field with a few percent precision.

Thermo-optical bistability in silicon micro-cantilevers

“…When heated, the cantilever mechanical resonance frequencies shift owing to a change in the stiffness due to the temperature dependency of Young's modulus and thermal expansion. Knowing the thermo-mechanical properties of silicon, the cantilever temperature can be deduced by tracking the frequency shift [17]. The experimental setup uses a single laser beam (Fig.…”

“…At high P 0 , corresponding to high temperatures, the silicon extinction coefficient κ is large enough to kill the interferences, the cantilever becomes non-transparent. The temperature elevation deduced from the frequency shift [17] measured for the first two mechanical modes is displayed in Fig. 8-b.…”

“…When the cantilever is in vacuum and a laser beam is focused at its free end (Fig. 1), the heat generated by absorption can only dissipate via conduction along cantilever length through the tiny crosssection, resulting in significant temperature rise [16,17,33,34]. In addition, owing to the relatively weak absorption in silicon, the light experiences multiple reflections within cantilever thickness which acts as a Fabry-Pérot cavity.…”

“…The fraction AP 0 of light absorbed by the cantilever will result in a rise of its temperature. In vacuum, neglecting the thermal radiation, the only dissipation process is thermal conduction along the cantilever [17]. The stationary temperature profile can be estimated using the Fourier law…”

“…This actuation is a photo-thermal effect: the light absorbed locally heats and bends the cantilever. Indeed, using moderate laser powers (a few mW) on AFM silicon cantilevers, one can reach huge temperatures [16,17] (up to silicon melting at 1410°C) owing to the relatively large absorption of visible light by silicon.…”