Characterizing the free and surface-coupled vibrations of heated-tip atomic force microscope cantilevers

Combining heated-tip atomic force microscopy (HT-AFM) with quantitative methods for determining surface mechanical properties, such as contact resonance force microscopy, creates an avenue for nanoscale thermomechanical property characterization. For nanomechanical methods that employ an atomic force microscope cantilever's vibrational modes, it is essential to understand how the vibrations of the U-shaped HT-AFM cantilever differ from those of a more traditional rectangular lever, for which analytical techniques are better developed. Here we show, with a combination of finite element analysis (FEA) and experiments, that the HT-AFM cantilever exhibits many more readily-excited vibrational modes over typical AFM frequencies compared to a rectangular cantilever. The arms of U-shaped HT-AFM cantilevers exhibit two distinct forms of flexural vibrations that differ depending on whether the two arms are vibrating in-phase or out-of-phase with one another. The in-phase vibrations are qualitatively similar to flexural vibrations in rectangular cantilevers and generally show larger sensitivity to surface stiffness changes than the out-of-phase vibrations. Vibration types can be identified from their frequency and by considering vibration amplitudes in the horizontal and vertical channels of the AFM at different laser spot positions on the cantilever. For identifying contact resonance vibrational modes, we also consider the sensitivity of the resonant frequencies to a change in applied force and hence to tip-sample contact stiffness. Finally, we assess how existing analytical models can be used to accurately predict contact stiffness from contact-resonance HT-AFM results. A simple two-parameter Euler-Bernoulli beam model provided good agreement with FEA for in-phase modes up to a contact stiffness 500 times the cantilever spring constant. By providing insight into cantilever vibrations and exploring the potential of current analysis techniques, our results lay the groundwork for future use of HT-AFM cantilevers for accurate nanomechanical property measurements.