Abstract
We report the first study on the thermal behavior of the stiffness of individual carbon nanotubes, which is achieved by measuring the resonance frequency of their fundamental mechanical bending modes. We observe a reduction of the Young's modulus over a large temperature range with a slope -(173±65) ppm/K in its relative shift. These findings are reproduced by two different theoretical models based on the thermal dynamics of the lattice. These results reveal how the measured fundamental bending modes depend on the phonons in the nanotube via the Young's modulus. An alternative description based on the coupling between the measured mechanical modes and the phonon thermal bath in the Akhiezer limit is discussed.
Highlights
We use the single clamped resonator layout, where one end of the nanotube is attached to a silicon chip and the other end is free
The restoring force is given solely by the bending rigidity. This enables us to probe the Young’s √modulus Y by measuring the resonance frequency, ω0 ∝ Y [32]. Such a resonance-based methodology is employed in thermoelasticity studies on larger scale systems [11,12,13, 33]
We engineer a platinum particle at the free end of the nanotube, so that the resonator can be measured by scattering optomechanical spectroscopy (Fig. 1a) [31]
Summary
The carbon nanotubes were grown on silicon substrates via chemical vapor deposition. A Zeiss Auriga scanning electron microscope (SEM) was used to select suitable nanotube cantilevers. 6 (a) and (c) presumably consists of hydrocarbons adsorbed during their exposure to air and the particle growth [2] Using such HRTEM images, we determined the number of walls and the associated diameters for six different devices ranging from single wall to seven wall nanotubes. The effect of the added particle at the free end of the CNT on the standard deviation equation can be obtained using the equipartition theorem: σn. The numerical coefficient in Eq 9 is function of the number of modes considered in the summation N and depends on the influence of the added mass The standard deviation of the cantilever is primarily given by that of the fundamental eigenmode independently of the particle mass at the free end
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