Since the 2000s, non-contact atomic force microscopy (nc-AFM) has established itself as a scanning probe method for the topographical, chemical, and electrical mapping of the surface of a sample down to the atomic scale . When used in an ultrahigh-vacuum (UHV) system and at, or close to, liquid helium temperature (4–10 K, LT UHV), the method allows for the direct characterization of individual molecules with intramolecular contrast , opening up the field of studying on-surface reactions or tip-induced chemistry .

The method also makes it possible to quantify the interatomic interaction forces that develop between the tip and the surface acquired in spectroscopic data cube modes with both high sensitivity and high spatial resolution. Recently, the force sensitivity has been pushed forward, and forces as low as 100 fN have been reported on artificial atoms formed by quantum corrals .

In nc-AFM, the probe, whose mechanical behavior may advantageously be compared to that of a one-dimensional simple harmonic oscillator (SHO) of resonance frequency f1 (flexural fundamental eigenmode) and stiffness k1, is sinusoidally excited at f1 by a phase-locked loop (PLL) that also guarantees a constant oscillation amplitude, A1. If the tip is far enough from the surface, that is, at distances where the strength of the tip–surface interatomic forces is negligible with respect to the restoring force induced by the excitation, its resonance frequency remains unchanged, f1. When the tip is in the range of attractive interatomic forces Fint(r), that is, for tip–surface separations r 1 nm, non-linear effects modify the oscillator dynamics, which shifts its resonance frequency down to lower values < f1. The resulting frequency shift Δf = − f1< 0 is tracked by the PLL and used as the input of the Z-controller to form a “topographic image”, which is actually a “constant-Δf” image. Alternatively, the image can also be acquired at constant height, which then forms a local Δf map of the surface. Δf is expressed according to :

where ru(z) = z + A1(1 − cos(u)) is the instantaneous tip–surface position, and z is the shortest distance between the tip and the surface during one oscillation cycle. Thus, if A1 and k1 are properly calibrated, the interaction force may be quantified, however, through non-trivial inversion procedures . The amplitude calibration in nc-AFM using the so-called constant-γ method is well documented and reasonably accurate , even if recently reported methods seem more accurate . Conversely, it seems that the direct stiffness calibration of nc-AFM probes in UHV and at low temperature has never been reported. Furthermore, because the force sensitivity in nc-AFM critically depends on the mechanical stability of both probe and tip, it seems crucial to perform the probe stiffness calibration in situ, that is, within the LT UHV system, by means of a non-destructive method.

In UHV and at room temperature, nc-AFM experiments are mostly carried out with silicon cantilevers, similar to those used during AFM experiments in air or in liquid. Their stiffness rarely exceeds 100 N/m. In UHV and at low temperature, the use of cantilevers is more tedious because of the required in situ optical detection setup. Nc-AFM experiments are then mostly performed with quartz sensors, essentially implemented according to two geometries: quartz tuning fork (QTF) or length-extensional resonator (LER) . The commercial versions of these probes are the qPlus sensor (Scienta-Omicron) and the KolibriSensor (SPECS) , respectively. It is known that these sensors offer several advantages: (i) Their large stiffness (≃1800 N/m for qPlus and ≃540 × 103 N/m for KolibriSensor), much greater than that of silicon cantilevers. It prevents the snap of the tip into contact and enables the use of small oscillation amplitudes (A1 ≃ 50 pm), which render the probe highly sensitive to the short-range regime of interatomic forces. (ii) Their high quality factor (Q, ≃105 in a LT UHV system), which renders the PLL highly sensitive to the frequency tracking. (iii) Their piezoelectric nature, which facilitates the readout of the tip deflection, based on the piezoelectric charge induced by the quartz upon oscillation through a simple I/V, or charge, preamplifier , as compared to the heavy optical detection setup required for silicon cantilevers.

Nowadays, the qPlus sensor is the probe that is most commonly used with LT UHV microscopes. This is why we focus on this type of probe in this work. In this design, only one prong of the QTF is fixed . At the extremity of the free prong, a thin, etched wire (usually W or PtIr), less than a millimeter long, is glued, which forms the tip. The tip is electrically connected to an electrode that collects the tunneling current if scanning tunneling experiments are to be performed along with nc-AFM experiments. The qPlus sensors feature a resonance frequency of f1 ≃ 25 kHz and a most commonly reported stiffness of 1800 N/m . This estimate was first proposed in 2000 , following previous works , and was based on geometric criteria of the sensor that did not consider the influence of the added tip. Thus, this value of the stiffness reported for the early versions of qPlus, which is still used in most of the recent works to perform the frequency shift-to-force conversion (see, e.g., the supplementary material of and ), is not necessarily compatible with that of modern ones. Furthermore, because the detailed geometry of each tip is never the same (regarding, e.g., diameter and length), and because it cannot be glued on the prong with high reproducibility (regarding, e.g, mass of glue and location on the prong), the mechanical properties of each sensor must differ in detail. Therefore, the actual stiffness of each probe must differ and has no reason to match a particular predefined value.

The stiffness calibration of silicon cantilevers at room temperature in air, liquid, or UHV by means of destructive and non-destructive methods has been discussed quite extensively in the literature , leading to a set of a dozen distinct approaches. A “global calibration initiative” has even been launched by Sader . Conversely, much less references are available for qPlus sensors , and, among these, none of them deals with the direct stiffness calibration of the probe in a LT UHV system.

The goal of the present work is to propose a framework based on thermal noise measurement to calibrate the stiffness of qPlus sensors operated in a LT UHV system. The concept was introduced by Hutter and Bechhoefer, and Butt et al. in 1993 , and was further improved by Butt and Jaschke in 1995 . It is based on the measurement of the spectrum of the fluctuations of the free end of the probe excited by thermal noise. The peak of the thermal noise spectrum at the resonance frequency of the probe may be related to its stiffness if the mechanical behavior of the probe can be modeled as that of an equivalent SHO. Our framework combines experimental measurements performed with qPlus sensors in UHV at 9.8 K and numerical simulations of the thermal fluctuations of a SHO under equivalent conditions. The numerical results permit to refine the experimental strategy, which allows us to achieve an uncertainty of 10% maximum in the calibration.

This work is inspired by, and based on, results from the literature, but extends the scope of AFM probes stiffness calibration through thermal noise measurement to very stiff (k > 1000 N/m) and large-Q (Q > 105) probes, such as qPlus sensors. To this end, many theoretical and practical elements are detailed, which usually are either not clearly stated or little discussed in the literature because they are not salient with softer probes, but they become crucial with very rigid and high-Q probes in LT UHV.

Finally, we want to stress that although this work treats the particular case of qPlus sensors, the framework can be adapted to any other kind of nc-AFM probes used in LT UHV, including other types of QTFs, silicon cantilevers, and LERs, with the KolibriSensor among the latter.

The paper is organized as follows. The section “Stiffness calibration methods: a brief review” briefly introduces the bibliographic context of the stiffness calibration methods, restricted to the main of our requirements. Section “Framework to the stiffness calibration” details the concepts of stiffness calibration based on thermal noise measurement, along with our assumptions. Section “Numerical simulations” details the numerical approach to the stiffness calibration. Section “Experimental results” presents the experimental results, which are discussed in section “Discussion” before the Conclusion.

This section reminds some salient results about stiffness calibration methods reported in the literature, which forms the context of the present study.

Figure 1: SEM images of the type of qPlus sensor used in this work. All dimensions are estimated with a relative uncertainty of 5%. a- Side view showing the complete geometry of the qPlus sensor. The circuitry of the electrodes is well identifiable. b- Front view of the qPlus free prong showing the tip glued to the right-hand side. c- Magnification of the dotted rectangle shown in b-.

Since we focus on the stiffness calibration in UHV, we also restrict the context to cases where the hydrodynamic function of the fluid surrounding the probe , if described in the model, plays no role.

Following Burnham’s classification , we essentially focus on two categories of non-destructive calibration methods, referred to as “geometric” and “thermal” methods.

Geometric methods permit to calculate the stiffness of the probe from its dimensions and the mechanical properties of its constitutive material. When the load is applied at the free end of the probe, its static stiffness is given by:

As discussed by Burnham et al. , for rectangular probes with a stiffness of ≃1 N/m, Cleveland/Sader’s calibration methods agree within 17% of the manufacturer’s nominal value.

In 2012, Lübbe et al. extended Cleveland/Sader’s approach to the Euler–Bernoulli model and derived an expression giving the static stiffness of the probe from the resonance frequency of any of its flexural eigenmodes :

where αn is a term occurring in functions that define the Euler–Bernoulli model used to describe the probe oscillation. αn is the solution of a so-called dispersion relation, written as :

leading to α1 = 1.875, α2 = 4.694, α3 = 7.864, …, αn ≃ (n − 1/2)π.

The latter formalisms consider a homogeneous probe without influence of the added mass due to the presence of the tip at its free end (unloaded case). This added mass changes the eigenmodes geometry, though. This results in a change of the value of the constant αn of each eigenmode. Lozano et al. , Lübbe et al. , and Yamada et al. have addressed the issue of the tip mass correction in the Euler–Bernoulli model. To this end, an extended probe oscillation model is used , which leads to a new equation for αn (loaded case), now written in order to not confuse it with the solution of the unloaded case:

where μ = mtip/mprobe is the ratio between the tip mass and the probe mass. Hence, μ must now be established before obtaining the value of .

In their work, Lübbe et al. point out that the direct stiffness calibration from the probe dimensions yields values with an uncertainty of ±25% as the result critically depends on the probe thickness, which is difficult to determine experimentally. But the uncertainty is reduced to ±7% when the measured fundamental eigenfrequency is included in the calculation and a tip mass correction is applied.

Thermal methods are based on the measurement of the probe’s thermal fluctuations when it is in thermal equilibrium within a thermal bath . The influence of thermal noise on a system has first been investigated by Nyquist and Johnson in 1928 with electric resistors . Their seminal work has later been formalized by the linear response theory and the fluctuation–dissipation theorem (FDT) , establishing a connection between fluctuations about equilibrium and the response of a system to external forces upon its susceptibility (or response function).

Thermal energy and probe fluctuations are linked by the equipartition theorem, which states that the energy transferred from a thermal bath to a dynamic system equals kBT/2 for each of its degrees of freedom, kB being the Boltzmann constant and T the temperature of the thermostat. Here, as discussed in , the probe is described as an equivalent point-mass SHO that features stochastic deflections of its free end along the vertical z axis over time due to thermal noise, forming a signal zth(t). The equipartition theorem is written as:

where ⟨⟩ represents a virtually infinite time averaging. The quantity

is the power of the probe fluctuations (mean quadratic deflections) induced by thermal noise over time. For a qPlus sensor of stiffness ks ≃ 1800 N/m at T = 9.8 K, the rms deflection induced by thermal noise is ≃ 270 fm.

where is the power of the thermal fluctuations of the n-th flexural eigenmode (resonance frequency fn, quality factor Qn, and stiffness kn), and the summation represents all eigenmodes of the probe. Under the assumption of thermal equilibrium, the thermal noise-induced deflection of each eigenmode follows the equipartition theorem, such that:

Upon proper normalization of the solution functions of the Euler–Bernoulli model, the modal stiffness kn of an unloaded probe may be connected to the static one ks (or equivalently ) according to :

Thus, for the fundamental eigenmode of an unloaded probe (α1 = 1.875), .

It is also usually assumed that thermal noise and measurement noise are uncorrelated. Thus, noting the power of the measurement noise:

the power of the measured thermal noise probe deflections is such that:

The power of the thermal fluctuations without measurement noise is ultimately given by integration of a quantity we name the thermal noise PSD (tn-PSD) (f), according to:

Technically, the PSD is defined from the Fourier transform (f) of the time trace of the signal z(t) forming the Fourier pair z(t) (f), according to:

The function exhibits a resonance for f = f1 (u = 1), and then:

Our experimental results are interpreted with the help of numerical simulations, but experimental and numerical approaches rely both on the same methodology.

Our framework to the stiffness calibration consists in processing the time trace of the qPlus thermal fluctuations to extract the quantity . To this end, the time signal of the thermal fluctuations including, or not (in the case of numerical simulations), measurement noise, is acquired over a windowing duration Tw. This process is repeated to form a statistic set of M time traces of the fluctuations (Mexp ≥ 500, Mnum ≥ 60). Then, the properly normalized one-sided rms PSD spectrum of each trace is calculated. The M rms PSD spectra are ultimately averaged resulting in a single final spectrum.

The value of the stiffness is then deduced according to the methods 1–3 described before. However, as discussed in detail by Cole or Sader et al. , in the case of noisy signals like the tn-PSD, the use of non-linear least-squares fits, such as those required in method 2, is problematic. Their convergence and the accuracy of the fit coefficients may depend on type of noise, type of fit functional, minimization algorithm, and number of coefficients to fit along with their boundary conditions and may lead to erroneous results. Because some of these difficulties were faced when processing our data, we do not use method 2 for the experimental stiffness calibration and restrict the analysis to methods 1 and 3.

The experimental results presented in this work have been acquired with a closed-cycle UHV SPM Infinity microscope from Scienta-Omicron, operated at 9.8 K. We use commercial qPlus sensors purchased from Scienta-Omicron.

As with many other results dealing with that topic, a central assumption of this work is that the mechanical behavior of the qPlus sensor may be described by that of an equivalent SHO of resonance frequency f1, quality factor Q1, amplitude at the resonance A1, and stiffness k1. The relevance of that approximation is verified by recording the resonance curve around f1 and checking to which extent the measured amplitude A(f) and phase φ(f) can be fitted by the SHO model for large quality factors (cf. section “Experimental results”, subsection “Equivalent SHO”):

Doing so, f1 and Q1 will be determined accurately. For the sake of the forthcoming discussions, we use typical orders of magnitude for qPlus sensors operated in LT UHV, namely f1 ≃ 25 kHz and Q1 ≃ 2 × 105.

The concept of PSD applied to the measurement of thermal fluctuations was introduced by assuming a continuous, that is, analog time signal. But on the experimental level, the thermal fluctuations are meant to be processed by the Nanonis MIMEA control unit, which is based on a digital FPGA architecture, such that the signal is ultimately discrete in time and of finite duration Tw. Let us assume the signal to be sampled with a period Ts, that is, a sampling frequency fs = 1/Ts, yielding a buffer of N samples. The windowing duration is then:

Therefore, the problem is to determine a correct value for δf that must satisfy δf ≪ . With our parameters, δf ≪ 60 mHz. We arbitrarily set δf = 20 mHz, that is, Tw = 50 s. According to the definition of the quality factor, the SHO bandwidth is wf = f1/Q1, that is, here wf ≃ 125 mHz. Thus, it is essential to have at least ≃7 samples within the wf bandwidth around f1 in the tn-PSD.

On the hardware level, the analog signal of the qPlus deflections is sampled at the maximum rate imposed by the MIMEA control unit, fs,max = 40 MHz, resulting in a Shannon–Nyquist frequency of 20 MHz. At that frequency, because of the analog antialiasing filter, the digital signal provided by the ADC is