## Heterodyne Force Microscopy

### Nondestructive Subsurface Characterization on the Nanoscale

- Heterodyne Force Microscopy - Nondestructive Subsurface Characterization on the Nanoscale
- Fig. 1: (a) A HFM setup, in which the cantilever (sample) is excited at frequency ωt (ωs) with amplitude At (As). The (subsurface) information is contained in the amplitude Adiff and phase φdiff of the cantilever motion at the difference frequency ωdiff = |ωs-ωt|. (b) The amplitude A sub(t) (top) and the deflection δ (bottom) measured on Si wafer as a function of the height of the cantilever‘s base zb (that determines tip-sample distance z). Note that the x-axis is not linear in z due to the cantilever‘s deflection. zb = 0 defines the border between the attractive and the repulsive regime, at which the effective force on the cantilever is zero. It is impossible to indicate the position, at which the cantilever touches the surface for the first time, due to the missing information on the tip-sample interaction in the experiment. Note that At stays almost constant even deep in the repulsive part of the tip-sample interaction: it reduces only to 99.7% of its free amplitude.
- Fig. 2: The amplitude of the difference frequency in the experiment (a) and in the numerical calculation (b) as a function of the height of the cantilever‘s base zb (that determines the tip-sample distance z). (b) Here, we added As + At to zb for visibility. As is 0.1 nm, 0.5 nm, 1 nm, 2 nm, 5 nm, and 10 nm and is color coded from black to light gray. At is fixed and equal to 1 nm. The black arrows indicate the approach and retract curve. (a) and (b) both show a local maximum before the mixing amplitude reaches an even higher plateau (Aplateau) in the repulsive part of the tip-sample interaction.
- Fig. 3: (a) The sum of the two ultrasonic excitations appears to oscillate at the difference frequency with amplitude Abeating, as indicated with the dashed red lines (left panel). Obviously, the Fourier Transform (right panel) only shows the two original frequencies. This effect is called beating. (b) The comparison between the theoretical values Abeating and the values of the plateau‘s, Aplateau, obtained from numerical calculations in figure 2b.

It has always been a desire in microscopy to non-destructively image below a surface. The introduction of ultrasound in a scanning probe microscope has shown the possibility to image subsurface nanoparticles on a nanometer scale. This technique is called **Heterodyne Force Microscopy** (HFM). Despite the successful application of HFM, the measurements initially remained qualitative. **Recent developments unravelled the dynamics in these HFM measurements and paved the way for quantitative experiments**.

**HFM Basics**

In HFM both the sample and the cantilever are excited with ultrasonic sound waves at slightly different frequencies ωs and ωt, which are both in the order of a few MHz (fig. 1a) [1]. These frequencies are in general far above the first resonance frequency of AFM cantilevers. The nonlinear tip-sample interaction, which consists of an attractive Van-der-Waals and a repulsive Hertzian contact force, mixes these ultrasonic excitations and generates a heterodyne, low-frequency signal at their difference frequency ω_{diff} = |ωs-ωt|, which is usually chosen well below the first resonance frequency of the cantilever such that the cantilever really oscillates also at the difference frequency. It is exactly the amplitude A_{diff} and phase φ_{diff} of this oscillation, which contains the subsurface information [2-4].

**Cantilever Dynamics**

To develop a microscope that enables quantitative imaging of subsurface structures at the nanoscale, detailed knowledge of the cantilever dynamics in close vicinity to a vibrating surface as well as of the propagation of the ultrasonic waves through the sample is a prerequisite requirement. We showed that the ultrasonic wave propagating through the sample is Rayleigh scattered by subsurface nanoparticles [5]. The resulting amplitude contrast is negligible, but the generated phase contrast should be detectable, as it is in the order of a few millidegrees. One major question at this time was: how does the cantilever pick up the ultrasonic wave that propagated through the sample. Recent experimental and numerical work provided deep insight into this issue, as we discuss in this paper.

We used a standard tapping mode cantilever with a spring constant of 2 N/m (f_{res} = 73.4 kHz). Let us now first discuss the ultrasonic cantilever motion and then the generation of the signal at the difference frequency.

**Ultrasonic Cantilever Motion**

In standard tapping-mode-operation of an **Atomic Force Microscope**, the cantilever is excited at its first resonance frequency and one uses the decrease in vibrational amplitude to regulate the tip-sample distance [6]. It was, therefore, totally surprising that the ultrasonic amplitude of the tip, A_{t}, remains not only constant during the approach, but also when indenting deeply into the sample [7]. The reason is the extremely high spring constant of the cantilever (k ~ 1000 N/m) at MHz frequencies, which has to be compared to the bending mode spring constant of only 2 N/m. As a consequence, the cantilever bends when getting in full contact with the sample, but simultaneously leaves the high frequency motion unaltered. A constant ultrasonic amplitude A_{t} at MHz frequencies was never observed before and is fully surprising, as one intuitively would expect other effects, like tip-sample damping, to reduce the amplitude A_{t }significantly. Figure 1b experimentally shows that A_{t} indeed remains (almost) constant while deeply indenting a hard Silicon wafer with a Silicon tip. In full contact, 99.7% of the free amplitude is left while the static deflection is as large as δ = 10 nm. Although the tip-sample interaction starts to reduce A_{t }when the cantilever comes into contact, this reduction is indeed fully negligible. This shows that the amplitude A_{t} can be considered as constant in regular HFM experiments.

**Difference Frequency Generation**

The signal at the difference frequency must the generated via a nonlinear process. Therefore, based on standard textbook equations, one expects that the amplitude of the difference frequency is proportional to the second derivative of the tip-sample interaction. This would result in two local maxima: one in the attractive Van-der-Waals regime and one in the repulsive regime. In contrast, only a single local maximum is observed in figure 2a and b. The vertically dashed lines in figure 2a show that these local maxima are in the attractive regime of the tip-sample interaction (δ < 0). Instead of a second maximum, we see a "plateau" in the repulsive part of the tip-sample interaction. Although totally counterintuitive, the height of this plateau perfectly fits the amplitude of, which is a linear effect. As depicted in figure 3a, the sum of the ultrasonic excitations appears to oscillate at the difference frequency with an amplitude Abeating. However, as the Fourier Transform only contains the two original ultrasonic frequencies, the cantilever excitation at the difference frequency should be absent and we should not measure any signal. Nevertheless, the table in figure 3b shows that the beating amplitude Abeating perfectly fits the height of the plateaus observed in figure 2b. This shows that beating dominates the difference frequency generation in HFM [8] and, in the meantime we have developed a complete theory for this [9].

**Conclusion**

The combined results of the experiments and the numerical calculations discussed above give us a quantitative understanding of both the ultrasonic cantilever motion and the generation of the difference frequency signal. Both aspects show totally unexpected results: the ultrasonic vibration amplitude of the cantilever stays constant while indenting the sample and the amplitude of the heterodyne difference frequency is dominated by a linear effect called beating.

**Outlook**

These results set the stage to a quantitative understanding of reported and future HFM measurements, as this is still missing. A study that takes into account both the altered tip-sample interaction above buried nanoparticles and the cantilever dynamics might explain the reported subsurface contrasts. G.J. Verbiest performed all this research during his PhD time at the University of Leiden under the supervision of M.J. Rost, who initiated this research. Although G.J. Verbiest has moved to the RWTH Aachen University, where he investigates the electromechanical coupling in graphene, he is still strongly involved in the ongoing research in Leiden.

**Acknowledgement**

The research described in this paper has been performed under and financed by the NIMIC [10] consortium under project 4.4.

**References**

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[2] Shekhawat G.S. and Dravid V.P.: Science 310, 89 (2005)

[3] Tetard L *et al.*: Appl. Phys. Lett. 93, 133113 (2008)

[4] Kimura, K. *et al*.: Ultramicroscopy 133, 41 (2013)

[5] Verbiest G.J. *et al*.: Nanotechnology 23, 145704 (2012)

[6] Garcıa R. and San Paulo A.: Phys. Rev. B. 60, 074961 (1999)

[7] Verbiest G.J. *et al*.: Nanotechnology 24, 365701 (2013)

[8] Verbiest G.J. *et al.*: Ultramicroscopy 135, 113 (2013)

[9] Verbiest G.J. and Rost M.J.: Nature Communications submitted

[10] www.realnano.nl

**Authors
Dr. Gerard Verbiest**

RWTH Aachen

2nd Institute of Physics A

Aachen, Germany

**Dr. Marcel Rost**

Leiden University

Kamerlingh Onnes Laboratory

Leiden, The Netherlands