Non-contact atomic force microscopy (nc-AFM) offers a unique experimental framework for topographical imaging of surfaces with atomic and/or sub-molecular resolution. The technique also permits to perform frequency shift spectroscopy to quantitatively evaluate the tip–sample interaction forces and potentials above individual atoms or molecules. The stiffness of the probe, k, is then required to perform the frequency shift-to-force conversion. However, this quantity is generally known with little precision. An accurate stiffness calibration is therefore mandatory if accurate force measurements are targeted. In nc-AFM, the probe may either be a silicon cantilever, a quartz tuning fork (QTF), or a length extensional resonator (LER). When used in ultrahigh vacuum (UHV) and at low temperature, the technique mostly employs QTFs, based on the so-called qPlus design, which actually covers different types of sensors in terms of size and design of the electrodes. They all have in common a QTF featuring a metallic tip glued at the free end of one of its prongs. In this study, we report the stiffness calibration of a particular type of qPlus sensor in UHV and at 9.8 K by means of thermal noise measurements. The stiffness calibration of such high-k sensors, featuring high quality factors (Q) as well, requires to master both the acquisition parameters and the data post-processing. Our approach relies both on numerical simulations and experimental results. A thorough analysis of the thermal noise power spectral density of the qPlus fluctuations leads to an estimated stiffness of the first flexural eigenmode of ≃2000 N/m, with a maximum uncertainty of 10%, whereas the static stiffness of the sensor without tip is expected to be ≃3300 N/m. The former value must not be considered as being representative of a generic value for any qPlus, as our study stresses the influence of the tip on the estimated stiffness and points towards the need for the individual calibration of these probes. Although the framework focuses on a particular kind of sensor, it may be adapted to any high-k, high-Q nc-AFM probe used under similar conditions, such as silicon cantilevers and LERs.
IntroductionSince 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.
Four Supporting Information files support our framework. They contain important results from the literature and are organized to help the reader to follow our developments. Supporting Information File 1 reminds the most salient results of the Euler–Bernoulli model and how it sustains the point-mass SHO equivalence. Supporting Information File 2 reminds fundamental elements of signal processing applied to discrete time signals, which include the power spectral density (PSD), a key tool for the stiffness calibration. Supporting Information File 3 reminds the expression of the thermal noise PSD of a SHO in thermal equilibrium within a thermostat. The PSD of the stochastic thermal force giving rise to the fluctuations of the SHO is derived as well, which is used in the numerical simulations. Supporting Information File 4 discusses the relevance of a digital antialiasing filter on the measured thermal noise PSD.
Stiffness Calibration Methods: A Brief ReviewThis section reminds some salient results about stiffness calibration methods reported in the literature, which forms the context of the present study.
In the following, unless specified otherwise, the word “probe” either means a silicon cantilever or the free prong of a qPlus sensor. The discussion is restricted to probes with a rectangular cross section (length l, thickness t, width w are such that l ≫ t and l ≫ w) treated in the Euler–Bernoulli model of the embedded beam, extensively detailed, for example, in (cf. also Supporting Information File 1). The displacement of the probe is assumed to occur vertically (along the z axis, as defined in Figure 1) and to be small with respect to all its dimensions (elastic deformation only). The word “deflection” means the displacement that takes place at the free end of the probe with respect to its equilibrium position z = 0. Torsional effects are not accounted for, which is justified in the section “Framework to the stiffness calibration” (cf. subsection “Experimental context”).
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 methodsGeometric 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:
where E, w, t, and l are Young’s modulus, width, thickness, and length of the probe, respectively. Cleveland et al. early exploited this concept to determine the stiffness of soft levers . But these authors, as well as Sader et al. , also proposed a calibration method of the static stiffness based on the measurement of the unloaded resonant frequency of the probe flexural fundamental eigenmode (f1), whose mass mprobe is to be estimated then. In this framework, the probe is assumed to behave as an equivalent SHO; then, the static stiffness is derived from f1 according to: . The quantity
is the effective mass of the fundamental eigenmode of the probe; mprobe = ρwtl is calculated from the density ρ, thickness t, and plan view dimensions (length l and width w) of the probe. The quantity μe,1 is the probe’s normalized effective mass of the fundamental eigenmode, taking the value μe,1 ≃ 0.2427 for l/w > 5 (cf. also Supporting Information File 1). Because the experimental determination of the length of the probe is prone to less error than that of its thickness (l ≫ t), Cleveland et al. and Sader et al. removed the thickness dependence from Equation 2 and came to an equivalent expression for
:
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 noise methodsThermal 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.
As discussed early by Butt and Jaschke and explicitly measured by others , the rigorous analysis of the thermal fluctuations is to be performed in terms of modal decomposition of the probe deflections over its eigenmodes. Then, the total deflection of the probe’s free end due to thermal fluctuations results from the superposition of the deflections of equivalent SHOs embodying the eigenmodes of the probe, which are assumed to be independent . Because of their high quality factors, the former assumption is particularly valid for qPlus sensors (cf. also Supporting Information File 1). Thus:
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), .
Combining Equation 8, Equation 9, and Equation 10 yields:
Equation 11 is similar to Equation 7 if the summation is performed over all the eigenmodes of the probe (, cf. ).
From an experimental point of view, the number of accessible eigenmodes (m) is limited because the detection bandwidth of the fluctuations is restricted. Then, the relative error introduced in the estimated static stiffness is
, which may be estimated. For instance, restricting the detection bandwidth to the fundamental eigenmode of an unloaded probe (α1 = 1.875) sets m = 1 and
In other words, the modal stiffness k1 of the fundamental eigenmode of an unloaded probe exceeds the static stiffness ks by 3%, or equivalently, 97% of the thermal fluctuations are due to the probe’s fundamental eigenmode.
Equation 9 states that the measurement of the thermal fluctuations of the deflection of the n-th eigenmode over an arbitrary long time interval might allow us to derive the corresponding modal stiffness. But this is not feasible in practice because of measurement noise, which usually exceeds thermal noise. There are several origins to measurement noise. For qPlus sensors, two main noise sources may be considered: the preamplifier, which converts the piezoelectric current of the QTF into a scalable voltage signal, and the subsequent analog/digital converter (ADC), which converts the analog signal into a digital signal to be processed by the digital control unit of the microscope. It is difficult to quantify the strength of those sources with respect to that of the thermal noise based on the time trace of the fluctuations as it gives no idea on how the noise is spectrally spread within the system. This is why the analysis of noisy signals is rather performed from their PSD. The PSD Sz(f) of an analog signal z(t) featuring thermal fluctuations with measurement noise is defined according to:
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:
Thus, if the measurement noise PSD (f) is quantified, the quantity
depicting the thermal fluctuations of the probe can be estimated from the measurement of Sz(f) (Equation 15) and, thus, also the probe stiffness (Equation 7).
Because the Fourier transform is intrinsically two-sided (f ∈ ℝ), the integration in Equation 12, Equation 13, and Equation 15 spreads from −∞ to +∞. The two-sided representation of the DFT forms a strict, self-consistent, mathematical background; however, in the case of the PSD, its one-sided representation is preferred (f ∈ [0;+∞[). In addition, the observables that are measured from stochastic signals usually relate to rms values (e.g., Vrms). It is therefore preferable to express their corresponding PSD from the rms value of their Fourier transform. Supporting Information File 2 explicitly details the connection between the two-sided expression of the PSD and that of the one-sided rms PSD (cf. Supporting Information File 2, Equations S14 and S15). In the following, we will only use the spectral expression/representation of the one-sided rms PSD of the signal zth(t), which is defined for f ≥ 0 only. Unlike in Supporting Information File 2, the “rms” superscript will be systematically omitted in the notations in order to lighten them, but it is maintained in the units.
A large part of the thermal fluctuations stems from the probe’s fundamental eigenmode. Thus, it is interesting to compare (f) to the formal expression of the one-sided rms tn-PSD of an equivalent SHO (resonance frequency f1, quality factor Q1, and stiffness k1), which is established in Supporting Information File 3 (cf. Equation S8). This quantity is written as:
The function exhibits a resonance for f = f1 (u = 1), and then:
Equation 15, Equation 16, Equation 17, and Equation 18 form the analytical framework for the analysis of the thermal fluctuations, which ultimately allows for the probe stiffness calibration from Equation 7. They are used according to three methodological approaches, which all are reported in the literature:
Method 1:These methods all require a good estimate of the measurement noise PSD (f), otherwise the estimated stiffness will be uncertain (cf. hereafter).
Finite element method (FEM) modeling has been applied successfully to calibrate the stiffness of both silicon cantilevers and qPlus sensors . For qPlus sensors, however, FEM does not offer a generic approach. Indeed, as presented in the Introduction, the fact whether the sensors are custom-made or commercial, the tip shape (nature, diameter, and length of the wire), the precise location where it is glued on the free prong, along with the nature and quantity of glue used to hold the wire, imply a large range of geometric parameters, which ultimately influence the resulting stiffness of the probe. Falter et al. pointed out this issue , and the authors outlined the urge of stiffness calibration for each sensor. The main conclusion of their FEM modeling shows quantitative agreement with the beam formula (Equation 2) if the beam origin is shifted to the position of zero stress onset inside the tuning fork base; however, there was a systematic overestimation of the experimental stiffness due to the tip gluing geometry.
In the 2000s, Rychen et al. proposed an approach to the calibration of the modal stiffness of QTFs used below 4.2 K and at 5 mbar . The method is based on the measurement of the admittance of the piezoelectric current produced by the fork upon oscillation. The authors fitted that quantity with a Butterworth–Van Dyke-type electrical equivalence, and they put in relation the fitted electrical parameters with those of an equivalent mechanical SHO. This approach is valuable as it is performed in situ (however here not in UHV) and is non-destructive. However, it requires the precise knowledge of the piezoelectric constant of the quartz, and, with current qPlus designs, it was shown that the Butterworth–Van Dyke equivalent circuit failed at describing all their features .
Framework to the Stiffness Calibration MethodologyOur 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.
Experimental contextThe 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.
Scanning electron microscopy (SEM) pictures of one of these probes are shown in Figure 1. SEM analysis was performed with a Zeiss GeminiSEM 500 ultrahigh-resolution FESEM at 15 kV. Secondary electron detection was used for imaging. At 15 kV, the resolution is 0.6 nm. Energy-dispersive X-ray spectroscopy (EDS) chemical analyses have been performed too, for which an EDAX Octane Silicon Dri Detector (129 eV energy resolution for manganese) coupled to the SEM was used at 15 kV. A large side view (cf. Figure 1a) shows the overall probe geometry. The qPlus sensors we use feature QTFs whose prong geometry is asymmetric. The sensor is glued by means of an insulating epoxy glue (white areas) on a massive metallic holder, not visible in the figure, that is mechanically clamped into the scanning piezo featuring a ring electrode for mechanical excitation. The QTF surface features a set of three metallic electrodes evaporated on it. Their chemical composition has been characterized by EDS as consisting of a ≃200 nm thick layer of Au on a thinner chromium layer to favor the adhesion and wetting of Au. The massive electrode is for grounding. The two thinner ones, running along the free prong, are for the piezoelectric current and tunneling current readouts. The free prong is l = (2045 ± 100) μm long. The tip, indicated at the end of the free prong, consists of a W wire that is 50 μm in diameter, better visible in Figure 1b. It is glued with a conducting epoxy, visible in the pictures, on the side of the free prong, near its end. With this asymmetry, oscillations due to the first torsional eigenmode of the qPlus sensor might occur and perturb the detection of the thermal motion due to the first bending mode. However with regular QTFs, the first torsional eigenmode is expected to be above, or near, our sampling frequency of ≃155 kHz (cf. subsection “Acquisition parameters of the thermal fluctuations as a discrete time signal”). The torsional resonance is, therefore, far away from our considered frequency range, and torsional effects should not influence, to a large extent, our measurements. The W wire has been cut at the top end and etched at the bottom end to form the tip. One can estimate its height as that of an effective cylinder, as indicated in the picture, h = (574 ± 30) μm. However, this quantity is only representative of this particular qPlus sensor and is expected to vary from one sensor to another. The free prong has a width w = (132 ± 7) μm and a thickness t = (222 ± 12) μm, as measured from Figure 1c, which is the magnification of the dotted rectangle shown in Figure 1b. From these geometric quantities, the estimated static stiffness of the probe (cf. Equation 2) is ks = (3322 ± 1270) N/m. To get this value, we have considered Young’s modulus and density of quartz, E = (78.7 ± 1.6) GPa and ρ = (2.65 ± 0.06) × 103 kg·m−3, respectively. It is also reminded that Equation 2 does not include the contribution of the tip.
The qPlus sensor is assumed to be in thermal equilibrium at T = 9.8 K. The temperature is measured within the head of the microscope by a Si diode and readout by a Lakeshore 335 Controller. The microscope being in closed-cycle has been thermalized at that temperature for several weeks. A HQA-15M-10T charge preamplifier from Femto collects the piezoelectric current generated by the qPlus sensor and sends the preamplified signal into a Nanonis OC4 oscillation control unit for digital conversion and a Nanonis Mimea SPM control system from SPECS for processing this signal. An analog low-pass filter in the OC4 acts as an antialiasing filter before digital conversion by an ADC. It is implemented as a third-order overall Butterworth filter with a Sallen–Key topology and with a fixed cut-off frequency of 5 MHz, meaning that dampening is 60 dB per decade. There is no additional filter in the analog part, and there is no antialiasing filter in the digital domain. Before thermal noise measurements, the qPlus deflections are accurately calibrated into metric units with a custom-made script implemented in the Nanonis MIMEA control unit that performs the constant-γ calibration procedure of the oscillation amplitude in nc-AFM mode . The accuracy of the calibration is cross-checked in STM mode, which guarantees an accuracy of 5% in the amplitude calibration.
A dedicated software data acquisition module implemented by SPECS into the Nanonis MIMEA control software is used to acquire the Mexp time traces of the signal z(t) featuring the thermal fluctuations of the qPlus sensor and to process them accordingly to yield the averaged rms PSD spectrum of the thermal fluctuations (cf. subsection “Acquisition parameters of the thermal fluctuations”).
Because Mexp ≃ 500 time traces are acquired, lasting Tw ≃ 50 s each (cf. also subsection “Acquisition parameters of the thermal fluctuations”), the total acquisition lasts a couple of hours. In order to lower the parasitic noise level, these measurements are carried out overnight. The protocol might seem long and demanding, but it is the properties of the qPlus sensor that constrain one to drastic acquisition parameters. In 2013, Lübbe et al. had already come to the same conclusion in the case of silicon cantilevers in UHV at room temperature . To overcome the drawback of the acquisition duration with high-Q probes, they introduced a quick and efficient alternative method based on the spectral analysis of the frequency shift detected by the PLL. If their concept was transposable to the case of qPlus sensors without loss of accuracy in the calibration, which is not established so far, it would be an advantage over the current method.
During the thermal noise measurement (Sz(f)), the qPlus sensor is located far from the sample such that no interaction force may develop between tip and surface. All inputs to the microscope are grounded (e.g., high voltage lines of the X, Y, and Z scanner, coarse motor, and bias). The tunneling current readout is also grounded. It is also made sure that no parasitic external noise source (mechanical or electrical) adds to the measurement. The measurement noise PSD ((f)) is recorded under similar conditions, except that the input of the charge amplifier is not connected and let open (qPlus sensor not connected). The acquisition parameters of Sz(f) and
(f) are detailed hereafter.
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.
Acquisition parameters of the thermal fluctuations as a discrete time signalThe 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:
In UHV and at low temperature, the qPlus sensor’s thermal fluctuations stem from stochastic phonons of the quartz in thermal equilibrium with the thermostat at the energy kBT. The phonons excite the qPlus sensor and statistically repeat over time with a frequency spectrum yielding a thermal noise rms PSD described by that of an equivalent SHO (Equation 17). The spectrum conceals the mechanical properties of the SHO and exhibits a resonance at f1. Thus, stochastic phonons with frequencies at, or close to, the SHO resonance frequency, produce long-standing oscillations, particularly if the SHO’s quality factor is large, which is the case with qPlus sensors. It is therefore mandatory to acquire the thermal fluctuations over a duration window Tw that is much larger than the intrinsic equilibration time of the SHO defined as τ1 = 2Q1/f1, hence, Tw ≫ τ1. With the orders of magnitude that were chosen above, τ1 ≃ 16 s.
Furthermore, setting Tw implicitly means setting δf, that is, the frequency resolution δf of the rms PSD spectrum:
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
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