Intermodal coupling spectroscopy of mechanical modes in microcantilevers

Abstract

Atomic force microscopy (AFM) is highly regarded as a lens peering into the next discoveries of nanotechnology. Fundamental research in atomic interactions, molecular reactions, and biological cell behaviour are key focal points, demanding a continuous increase in resolution and sensitivity. While renowned fields such as optomechanics have marched towards outstanding signal-to-noise ratios, these improvements have yet to find a practical way to AFM. As a solution, we investigate here a mechanism in which individual mechanical eigenmodes of a microcantilever couple to one another, mimicking optomechanical techniques to reduce thermal noise. We have a look at the most commonly used modes in AFM, starting with the first two flexural modes of cantilevers and asses the impact of an amplified coupling between them. In the following, we expand our investigation to the sea of eigenmodes available in the same structure and find a maximum coupling of 9.38 × 10³ Hz/nm between two torsional modes. Through such findings we aim to expand the field of multifrequency AFM with innumerable possibilities leading to improved signal-to-noise ratios, all accessible with no additional hardware.

Keywords: atomic force microscopy; intermodal coupling; nonlinear mechanics; optomechanics; sideband cooling.

Figure 1

(a) Schematic drawing of the experimental setup. The cantilever is glued to the macrosized piezo driver. The LDV can either send data to the MSA to determine the eigenmode shapes or to the lock-in amplifier for higher bandwidth measurements. The latter also synthesizes the signal applied to the piezo driver. (b) Schematic of the signals used. Three signals are in effect at all times: the red sideband pump ω_p, an off-set red sideband pump ω_h ensuring even heating across the sample and a small one, compared to the previous, sweeping over the sense mode. (c) Comparison between a two-signal measurement (left) and a three-signal measurement (right) ensuring thermal stabilisation. The second flexural mode is coupled with the fifth flexural mode. The sum of heating signal and pump is constant. The stabilisation signal was 3 kHz higher than the red sideband pump, which was set at 3176.9 kHz. Due to their large frequency distance from the observed mode compared to its linewidth, ω_p and ω_h were not included in the graph.

Figure 2

MSA measurements showing the difference in modeshapes between the third order torsional modes investigated in the main text. (a) is T3’ with a node much closer to the added mass of the tip. (b) is T3, with nodes closer to their expected positions. Inset: FEM simulation of T3 eigenmode.

Figure 3

(a) Measurements of the first mode coupled with the second. Increasing the pump amplitude presents both a shift in the frequency and a reduction in effective temperature. Inset: Effective temperature and Q factor as functions of the pump amplitude. (b) Data of the second mode under different pump settings. Mode shapes under increasing amplitude of the pump. (c) Estimation of the coupling strength from data in (b). Slight deviations from the linear fit are caused by the approximation used. (d) Colormap of second mode for different frequency offsets of the pump at fixed amplitude. f_AS refers to the anti-Stokes pump frequency.

Figure 4

Map of the observed modes under anti-Stokes pumps. On the columns we have the sense mode, while the rows designate the mode it is coupled to, from bottom left. The greyed out graphs are setups where no discernible coupling is present. The red ones follow the expectation of the optomechanical Hamiltonian. The yellow ones exhibit nonlinear behaviour not described by the aforementioned Hamiltonian. Blue have a significant frequency shift, on the same order of magnitude as the linewidth of the sense mode, unexplained by cantilever expanding under heating.

Figure 5

(a) Graph for mode combination F2–T3, which has a regime transition. Inset: Coupling rates determined from linewidth changes or eigenmode separation against half the linewidth of cavity mode T3. (b) Matrix showing the coupling rates of all mode combinations. Contoured squares represent combinations between flexural modes only. (c) Same data as in (b) presented in a one-dimensional perspective. Blue points are calculated from data sets with the sense mode lower in frequency than the cavity mode, while red are the opposite. Greyed out points have no discernible coupling.