# Stiffness calibration of qPlus sensors at low temperature through thermal noise measurements

Laurent Nony, Sylvain Clair, Daniel Uehli, Aitziber Herrero, Jean-Marc Themlin, Franck Para, Alessandro Pioda, Christian Loppacher

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Laurent Nony, Sylvain Clair, Daniel Uehli, Aitziber Herrero, Jean-Marc Themlin, et al.. Stiffness calibration of qPlus sensors at low temperature through thermal noise measurements. 3èmes journées du GDR Nanosciences en Champ proche Sous Ultravide, Nov 2023, Nancy, France. hal-05288334

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Submitted on 29 Sep 2025

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# Stiffness calibration of qPlus sensors at low temperature through thermal noise measurements

L.Nony¹, S.Clair¹, D.Uehli², A.Herrero², J.-M. Themlin¹, F. Para¹
A.Pioda² and C.Loppacher¹

1-Aix-Marseille University, CNRS, IM2NP, UMR 7334, Marseille, France
2-SPECS Zurich GmbH

qPlus sensors are the most representative probes for combined non-contact AFM-STM experiments in UHV and at low temperature and have shown an extreme force sensitivity (1). They rely on a quartz tuning fork one prong of which, holding the tip, is free to oscillate. However, it is not the tip-surface interaction force $F_{int}$ that is primarily probed in nc-AFM, but the frequency shift $\Delta f$, of the resonance frequency of the tip ($f_1$) in interaction with the surface, according to (2):

$$ \Delta f = \frac{f_1}{2\pi k_1 A_1} \int_0^{2\pi} F_{int}(r_u) \cos(u) du, $$

where $r_u$ is the instantaneous tip-surface position. Here, $A_1$ and $k_1$ are the oscillation amplitude and stiffness of the probe, respectively. Thus, if $A_1$ and $k_1$ are well-calibrated, $F_{int}$ may be quantified. The amplitude calibration is usually performed through the so-called constant-$\gamma$ method (3). Conversely, it seems that the direct stiffness calibration of qPlus in UHV and at low temperature has never been reported.

In this work, we report the stiffness calibration of qPlus sensors in situ at 9.8 K through thermal noise measurements. Because the probe features a large nominal stiffness (>2 kN/m), along with a high-Q factor (>10⁵), we have developed a thorough methodological framework which, with the help of numerical simulations, allows us to calibrate the stiffness of the probe with an uncertainty of less than 10% (cf. fig. 1).

<figure><img src="image_2.png" /><figcaption>Figure 1: a-Numerical thermal noise power spectrum density (tn-PSD) of an oscillator of resonance frequency $f_1=25$ kHz, stiffness $k_1=1800$ N/m, Q-factor $Q_1=10^5$ (black curve) at 9.8 K and comparison with the corresponding theoretical tn-PSD (red curve). b-Experimental tn-PSD of a qPlus with $f_1=29183$ Hz, $Q_1=145000$ at 9.8 K and comparison with the theoretical tn-PSD with $k_1=2700$ N/m.</figcaption></figure>

## REFERENCES

1. F.J. Giessibl, *Rev. Sci. Instrum.* **90**, 011101 (2019).

2. F.J. Giessibl, «Advances in atomic force microscopy», *Rev. Mod. Phys.* **75**, 949 (2003).

3. F.J. Giessibl and H. Bielefeldt, *Phys. Rev. B* **61**, 9968 (2000).