Abstract
The noise of the frequencyshift signal Δf in noncontact atomic force microscopy (NCAFM) consists of cantilever thermal noise, tip–surfaceinteraction noise and instrumental noise from the detection and signal processing systems. We investigate how the displacementnoise spectral density d^{z} at the input of the frequency demodulator propagates to the frequencyshiftnoise spectral density d^{Δ}^{f} at the demodulator output in dependence of cantilever properties and settings of the signal processing electronics in the limit of a negligible tip–surface interaction and a measurement under ultrahighvacuum conditions. For a quantification of the noise figures, we calibrate the cantilever displacement signal and determine the transfer function of the signalprocessing electronics. From the transfer function and the measured d^{z}, we predict d^{Δ}^{f} for specific filter settings, a given level of detectionsystem noise spectral density d^{z}_{ds} and the cantileverthermalnoise spectral density d^{z}_{th}. We find an excellent agreement between the calculated and measured values for d^{Δ}^{f}. Furthermore, we demonstrate that thermal noise in d^{Δ}^{f}, defining the ultimate limit in NCAFM signal detection, can be kept low by a proper choice of the cantilever whereby its Qfactor should be given most attention. A system with a lownoise signal detection and a suitable cantilever, operated with appropriate filter and feedbackloop settings allows room temperature NCAFM measurements at a low thermalnoise limit with a significant bandwidth.
Introduction
In this contribution, we discuss noise in frequencymodulation noncontact atomic force microscopy (NCAFM) using cantilevers as force sensors and optical beam deflection (OBD) for signal detection. Figure 1 shows a schematic diagram of an NCAFM setup based on OBD to illustrate the signal path and the quantities describing noise. Measured quantities discussed here are often electrical signals that are equivalent to quantities describing the mechanical oscillation of the cantilever. The calibration procedure described in Section 1 of Supporting Information File 1 establishes a relation between the representation in mechanical and electrical units. During NCAFM operation, the cantilever with eigenfrequency f_{0} is excited to oscillation at the resonance frequency f_{r}, which differs from its eigenfrequency by the frequency shift Δf = f_{r} − f_{0} when there is a tip–surface interaction. The mechanical oscillation, i.e., a periodic displacement z(t) of the cantilever with amplitude A, is converted into the oscillation signal V_{z}(t) by the positionsensitive detector (PSD) connected to the preamplifier. The amplitude A of this signal is determined and stabilised to a preset value by the amplitude feedback loop. Signal processing in NCAFM involves the demodulation of the periodic cantileverdisplacement signal V_{z}(t) as well as filtering in the frequency domain to yield the frequency shift Δf(t) carrying the information on the tip–surface interaction [1]. Demodulation is commonly performed by a phaselocked loop (PLL) circuit [2]. As schematically depicted in Figure 1, the amplitude response of the PLL unit can formally be decomposed into the amplitude response G_{demod} of the demodulator and the amplitude response G_{filter} of an inloop or output filter. The characteristics of G_{filter} can be set by the user according to the needs of the experiment.
Noise in NCAFM consists mainly of three contributions: noise arising from the thermal excitation of a cantilever or another force sensor, noise caused by the detection system and signal processing electronics [3,4], and instabilities arising from the interaction of the force microscopy tip with the surface as well as arising from the feedback loops stabilising the cantilever oscillation amplitude and the tip–surface distance [5]. Here, we investigate noise for the case of negligible tip–surface interaction and discuss the cantileverdisplacement thermalnoise spectral density as well as the displacementequivalent noise spectral density introduced by the detection system. This is carried out here in search of the ultimate limits of detection defined by thermal noise, while a systematic study of the tip–sample interaction noise that is present in any NCAFM imaging or spectroscopy experiment will be the subject of forthcoming work. Here, we entirely focus the discussion on cantileverbased NCAFM; however, the concepts, theoretical framework, and experimental strategies for the noise analysis can easily be transferred to systems based on other force sensors and detection schemes.
Under ultrahighvacuum (UHV) conditions, the thermal noise of the cantilever is usually small compared to the noise of the detection system due to the high Qfactor of the cantilever in vacuum [6]. The instrumental noise sources in an optical beam deflection (OBD) setup were recently discussed in detail [3] and it was found that the major noise sources are shot noise arising from the photodetector as well as Johnson noise originating from the resistors in the preamplifier. Further noise is generated in the laser diode that is mainly quantum noise for small output power and modehopping noise for large output power [3]. Back reflections of the laser beam into the laser optical resonator may increase mode hopping. The laser spot on the photodiode may further be disturbed by optical interference, creating timevarying speckle patterns due to temperature fluctuations and mechanical instability. It has been shown, however, that by operating the laser diode with radiofrequency modulation, the contribution of the light source to the total noise can be reduced to a negligible minimum [3].
The issue of noise is intimately related to the requirements of the NCAFM system to process signals varying in time. The detection bandwidth B needed to retrieve the full information present in the Δf(t) signal at the output of the PLL system depends on the spectral components produced during a scanning or spectroscopy experiment. Practically, the maximum usable bandwidth B_{max} is defined by the total displacementnoise spectral density as schematically illustrated in Figure 2. In this figure, we show the displacement spectral density d^{z}(f) present at the input of the frequency demodulator with contributions of the measurement signal and noise (see Figure 1) as a function of the frequency f. This quantity is the root of the onesided power spectral density D^{z}(f), which is derived from the displacement signal V_{z}(t) via a Fourier transform as
where S is the calibration factor converting voltage into displacement as defined in Section 1 of Supporting Information File 1 and the Fourier transform of the displacement signal V_{z} with:
For the case of absent tip–surface interaction, d^{z} is a sharp peak centred at the cantilever eigenfrequency f_{0} (f_{0} = 70 kHz in Figure 2) including noise contributions from and , which will be described in detail below. In the presence of a tip–surface interaction, the resonance peak is shifted by the amount ( = −50 Hz in Figure 2) caused by the timeinvariant part of the interaction. Additionally, sidebands appear that represent spectral components in V_{z}(t) created during scanning or spectroscopy. For simplicity, we assume here a scanning of the tip over the surface with a speed v_{t} where a periodic corrugation (period a_{s}) of the surface Δf^{m} creates a sinusoidal modulation at the frequency f_{m} = v_{t}/a_{s} (f_{m} = 30 Hz in Figure 2), i.e., Δf(t) = + Δf^{m}sin(2πf_{m} + φ). Effectively, this is a frequency modulation of V_{z}(t) with a modulation index Δf^{m}/f_{m} producing an infinite number of higher harmonics with rapidly decreasing power [4]. How many of these side peaks can be detected depends on the modulation index of the signal and the noise characteristics of the measurement system. For the hypothetical measurement illustrated in Figure 2, only two sideband peaks are well above the noise floor. Here, the suitable bandwidth B_{max} is defined by the frequency of the second sideband peak.
The frequency demodulator extracts the frequency shift Δf(t) from the periodic displacement signal V_{z}(t) and, for an arbitrary signal, projects the power in the sidebands of D^{z}(f) into the frequencyshift power spectral density D^{Δ}^{f}(f_{m}), which can be represented as:
The frequency shift Δf(t) varies on a time scale that in an imaging experiment is determined by the spatial periodicity of the scanned structure and the scanning speed, rather than by the period of the cantilever oscillation. Therefore, the spectrum of the frequency shift signal present at the output of the demodulator has significant power only in a limited spectral range of f_{m}. The detection bandwidth B of the demodulator is, therefore, usually restricted to a value of the order of 100 Hz to 1 kHz. As the noise is transformed by the demodulator in a similar way, we define and as the frequencyshiftnoise spectral density and the frequencyshiftnoise power spectral density, respectively, and discuss separate noise contributions and to the frequencyshift signal Δf, as the noise contributions of the thermal cantilever excitation and the detection system yield different spectral characteristics. The detection bandwidth B and, consequently, the noise propagation characteristics depend on the PLL amplitude response G_{PLL} = G_{filter} × G_{demod}, which can usually be influenced by the operator through the filter settings (see Figure 1).
To understand the influence of various experimental parameters and the settings of the PLL filter on , which is the most relevant noise figure in the NCAFM experiment, we derive noise models based on system parameters. Hypotheses and conclusions are tested against the reality of NCAFM experiments, by comparing the noise figures and filter settings for three NCAFM systems based on the OBD scheme and comparing experimental results to the predicted settings for noiseoptimised operation. We find that by the correct choice of the cantilever, by using optimised detection electronics and by appropriate PLL filter settings, the frequencyshift signal Δf can be detected at a low thermalnoise limit over a bandwidth B that is more than 100 Hz for room temperature operation under UHV conditions. The dependence of the thermal limit and other noise figures on relevant experimental parameters is discussed in detail.
Displacement noise
Here, we discuss the displacement noise superimposed on the displacement signal V_{z}(t) in the case of negligible tip–surface interaction. Usually, the signal V_{z}(t) is a noisy sinusoidally oscillating voltage and the noise can be described in the frequency domain by the displacementnoise spectral density . This is the square root of the displacementnoise power spectral density , which is proportional to the unwanted energy per frequency interval stored in the oscillating system.
A cantilever that is not deliberately excited but in equilibrium with a thermal bath at temperature T exhibits random fluctuations resulting in measurable noise in the cantilever displacement signal. This noise can be predicted by a model outlined in Section 2 of Supporting Information File 1. Furthermore, all electrical and optical components that are part of the detection system produce noise, superimposed on the displacement signal. Therefore, the power spectral density of the total displacement signal noise can be described as
where and represent the thermal and the detectionsystem contributions. The quantity as derived in Section 2 of Supporting Information File 1 can be represented as:
Here, is calculated only for the fundamental cantilever oscillation mode with eigenfrequency f_{0}, stiffness k_{0} and Qfactor Q_{0} as the contribution of higher harmonics to the total noise power spectral density is small; the fundamental mode contains 97% or more of the total power extracted by the cantilever from the thermal bath. For the investigation of noise at higher harmonics, f_{0}, k_{0} and Q_{0} would have to be replaced by the respective modal values f_{n}, k_{n} and Q_{n} (see Section 2 of Supporting Information File 1). The noise spectral density as defined in Equation 1 is displayed in Figure 3a for typical experimental parameters exhibiting a sharp peak at f_{0} = 70 kHz (note the logarithmic scale of the ordinate) sticking far out of the whitenoise floor when using lownoise detection electronics. The sharp peak in results from the cantilever resonance. Especially high Qfactor cantilevers strongly amplify the white spectral power of thermal excitation only in a narrow range of frequencies around f_{0} according to Equation 2. The detectionsystem noise represented by is governed by the quality of the optical and electronic components used in the detection system. In contrast to thermal noise, which is a fixed quantity for a given cantilever and temperature, the detectionsystem noise floor can be reduced by technical improvements of the detection system [3,7,8].
Frequencyshift noise
The frequency demodulator of the NCAFM system extracts the cantilever response to the tip–surface interaction from the sidebands of the cantileveroscillation frequency spectrum (see Figure 2) and yields the signal power spectral density present in the sidebands, i.e., the displacement power spectral density D^{z}(f) is transformed to the frequencyshift power spectral density D^{Δ}^{f}(f_{m}) by the demodulation process. The noise contribution in this spectrum is the most relevant noise figure in NCAFM measurements and can be calculated from the demodulator input noise by applying the appropriate demodulator transfer function and an approximation to obtain a simple yet accurate expression for the thermaldisplacement power spectral density . As the frequency noise is represented as a function of the modulation frequency f_{m}, it is desirable to represent the displacement noise as a function of f_{0} ± f_{m}. For , we use the following approximation [1] instead of the precise result from Equation 2:
This expression is a very good approximation for modulation frequencies f_{m} exceeding the cantilever corner frequency f_{0}/(2Q_{0}) as seen in Figure 3b. This approximation covers most of the practically relevant spectral range as the corner frequency is smaller than 1 Hz for highQ cantilevers. Combining Equation 1 and Equation 3 yields a simple yet accurate expression for the power spectral density of the total displacement noise in an FMAFM system operated under highQ conditions [4]:
To obtain the noise power spectral density of the frequencyshift signal present at the demodulator output, the demodulator amplitude response for noise is applied, and we find [4]
As apparent from Equation 5, the contribution of the thermal noise to the total noise is independent of the modulation frequency f_{m}, whereas the detectionsystemnoise power contribution is amplified by the square of the modulation frequency. We further note that the total noise power in Δf depends on the reciprocal of the squared cantilever oscillation amplitude.
The frequencyshift noise spectral density and its components as described in Equation 5 are shown as a function of the modulation frequency f_{m} in Figure 4 for typical experimental conditions neglecting bandwidth limitations. This result clearly points to the experimental parameters determining the frequencyshift noise: the thermal limit is defined by the temperature T and cantilever properties, namely the ratio f_{0}/(k_{0}Q_{0}). For a cantilever with given f_{0} and k_{0}, it is most important to yield a high effective Qfactor that may considerably differ from the intrinsic Qfactor [9] if one is interested in reducing the thermalnoise limit to the lowest possible value. The noise contribution from the detection system depends on the required bandwidth B (range of f_{m}) and the quality of the detection system represented by . Overall, scales with the inverse of the cantilever oscillation amplitude A. In Figure 4, the thermal noise limit is shown for typical cantilever properties and for T = 300 K as dashdotted lines representing different Qfactors. From Figure 4 we can deduce the displacement noise floor of the detection system that must not be exceeded for a thermalnoiselimited measurement. We define the bandwidth for a thermalnoiselimited measurement by the frequency where the contributions of and to the total frequencyshift noise spectral density are equal. This frequency corresponds to the crossing point between the dashed and dashdotted lines in Figure 4.
It follows that operation at the thermal noise limit can only be obtained if the bandwidth B of the demodulator is set close to
where the noise spectral density is treated here as a constant. This is fully justified by its whitenoise character around the cantilever resonance. Note that this bandwidth limitation is solely based on noise considerations and does not reflect other bandwidth requirements, such as the stable operation of the PLL. However, there is a bandwidth limitation in any real system and has to be considered as a hypothetical quantity that is rarely accessible. In any PLL system of practical use, the detection bandwidth is defined by internal filters, loopgain settings and time constants that are normally accessible to the user for an optimisation of the signal processing. Thus, a complete PLL is modelled by using G_{PLL} = G_{filter} × G_{demod}, with G_{filter} being the amplitude response for the aforementioned filters (see Figure 1). Taking the amplitude response of the full PLL system into account, we obtain for the accessible noise power spectral density
at the output of the bandwidthlimited PLL system. The experimental determination of an unknown amplitude response G_{filter} is described in Section 3 of Supporting Information File 1. To characterise the demodulator output noise with a single number, we define δf_{tot} as the root mean square (RMS) of the overall frequencyshift noise:
where the integration can practically be limited to an upper frequency limit related to the detection bandwidth B. This is fully justified as filtering in the demodulator always yields a lowpass characteristic. A discussion of the RMS noise figure and its calculation by using approximations for the demodulator bandwidth is presented in Section 4 of Supporting Information File 1.
Experimental
Noise measurements are performed with three NCAFM systems, named systems A, B and C in the following. All systems are well decoupled from mechanical vibrations by spring suspension and eddycurrent damping systems. As an additional precaution, connections between the electronics and piezos are removed during noise measurements to ensure that measurements are not affected by any spurious electrical signals exciting the cantilever. All systems investigated here are based on the optical beamdeflection scheme for measuring the cantilever displacement. Therefore, the laserlight power P_{pd} reaching the photodetector is a parameter characterising the system. P_{pd} is calculated from the sum signal of the PSD, which in turn depends on the spectral sensitivity of the photodiode, the used laser light wavelength and the DC transimpedance of the preamplifier. The calibration of the detection system is described in Section 1 of Supporting Information File 1.
System A is a roomtemperature UHV AFM/STM (Omicron NanoTechnology GmbH, Taunusstein, Germany) equipped with an easyPLL (Nanosurf AG, Liestal, Switzerland) for frequency demodulation. The AFM/STM setup has been modified by replacing the light source (lightemitting diode exchanged with a laser diode) and using optimised preamplifiers. Preamplifiers have been optimised for lownoise operation at frequencies around 100 kHz and 300 kHz, respectively, and are exchanged depending on the eigenfrequency of the cantilever. Details on this modification and the frequency response of the preamplifiers can be found in [7]. The light source is a 48TESOT (Schäfter+Kirchhoff GmbH, Hamburg, Germany) and emits light at a wavelength of 685 nm, while the PSD has a spectral sensitivity of 0.45 A/W at this wavelength. Noise spectra are recorded with an SR770 spectrum analyser (Stanford Research Systems, Inc., Sunnyvale, CA, USA).
System B is a UHV VT AFM/STM (Omicron NanoTechnology GmbH, Taunusstein, Germany) equipped with an easyPLL plus (Nanosurf AG, Liestal, Switzerland) as the demodulator. This system uses a light source having a wavelength of 830 nm, while the spectral sensitivity of the PSD is 0.57 A/W at this wavelength. Noise spectra are measured using the zoom FFT module of a HF2LI lockin detector (Zurich Instruments AG, Zurich, Switzerland) for spectral analysis.
System C is a UHV 750 variable temperature STM/AFM with a PLLPro2 (software version 0.20.0) as the demodulator (RHK Technology, Inc., Troy, MI, USA). The light source is a laser source type 51nanoFCM (Schäfter+Kirchhoff GmbH, Hamburg, Germany) operated in the constantpower mode with radiofrequency modulation to reduce the coherence length to about 300 μm. The laserlight wavelength is 639 nm, and a maximum output power of 5 mW is available at the fibre end while the PSD has a spectral sensitivity of 0.4 A/W at this wavelength. A homebuilt preamplifier (lowbandwidth preamplifier) or the preamplifier supplied by the manufacturer (highbandwidth preamplifier) is used depending on the bandwidth requirements. The frequency response of both preamplifiers is shown in Figure 5. To measure noise spectra, the SR770 spectrum analyzer (Stanford Research Systems, Inc., Sunnyvale, CA, USA) is used.
Force sensors are commercial silicon cantilevers (Nanoworld AG, Neuchâtel, Switzerland). For our investigations, we use a set of cantilevers covering a large range of resonance frequencies, Qfactors and stiffness, to explore the impact of these parameters on the noise figures. Cantilever specifications are compiled in Table 1 and Table 2 (cantilevers D and AO are type FM, cantilevers AQ, AR and V are type NCH, cantilever AP is type NCVH and cantilever AL is type Arrow™ according to the commercial classification scheme).
Table 1: Fundamental properties of the cantilevers used for noise analysis. Length l, mean width and thickness t are provided by the manufacturer. The stiffness k_{dim} is calculated from the cantilever dimensions [10]. Typical properties of a qPlus sensor are taken from [11] for comparison.
cantilever  l (μm)  (μm)  t (μm)  k_{dim} (N/m) 

AO 3  224  30  3.0  3.0 ± 0.9 
D 5  229  30  2.9  2.5 ± 0.8 
AR 17  127  27  3.6  26 ± 5 
V 15  125  26  3.7  29 ± 6 
AQ 10  123  29  4.5  60 ± 10 
AP 5  40  24  2.0  130 ± 50 
AL 3  35  42  0.7  9 ± 3^{a} 
qPlus  2400  126  214  1800 
^{a}Value provided by the manufacturer.
The cantilever eigenfrequencies of the fundamental and the first and second harmonic mode are determined by measuring resonance curves and fitting the amplitude response function to the data as described in [9]. This procedure also yields quality factors Q_{n}, while the stiffness k_{dim} is calculated from cantilever dimensions and material properties [10] and used as a good approximation to the modal stiffness k_{0} [12].
Results and Discussion
The noise analysis is performed in two steps. First, we measure the displacement noise spectral density and, second, we investigate how it is propagated to the frequencyshift noise spectral density . The displacement noise is measured by a spectrum analyser connected directly to the output of the preamplifier (see Figure 1). The measurement range of the spectrum analyser is set to a few kilohertz around the cantilever resonance frequency to obtain high spectral resolution. The spectral density of the noise in the signal V_{z} is measured and converted to the displacementnoise spectral density in units of fm/ by the calibration procedure outlined in Section 1 of Supporting Information File 1. Figure 6 shows a representative result obtained with system C. The measured displacement noise spectral density is shown in Figure 6a (solid lines), together with the thermal noise contribution (dashdotted line) calculated from the given cantilever properties by using Equation 2. The noise floor of the detection system (dotted lines) is measured beside the resonance peak where thermal noise becomes negligible (solid and dashed lines are identical). For a study on how the noise of the detection system propagates through the demodulation system, different noise levels are artificially created by using white noise from a waveform generator DS345 (Stanford Research Systems, Inc., Sunnyvale, CA, USA) added to the displacement signal V_{z}. The curve with = 108 fm/ represents the noise floor of the setup while the other curves show artificially increased noise levels.
To measure the frequencyshift noise , the cantilever is excited to an oscillation with typically 10 nm amplitude and the spectrum analyser is connected to the output of the demodulator (see Figure 1) to measure the voltage noise in the Δf signal. The demodulator is adjusted to zero mean frequency shift and the measurement range of the spectrum analyser is set to the frequency region between 0 and 3 kHz. The measured voltage noise is multiplied by the known conversion factor of the demodulator (e.g., 30 Hz/V) to obtain the frequencyshiftnoise spectral density in units Hz/. In Figure 6b, this quantity is shown for the same three levels of artificial detectionsystem displacement noise at the input of the demodulator as supplied for the measurement in Figure 6a. Measurements (solid lines) are compared to calculated curves (dashed lines) based on the values obtained from the measurements shown in Figure 6a. The curve (f_{m}) is determined from the cantilever properties and the filter settings of the PLL demodulator (thermal contribution in Equation 7) and represents the ideal case of the thermal noise of the cantilever without any detectionsystem noise. The trailing edge on the right side is caused by the attenuation through the lowpass filter with amplitude response G_{filter} (see Section 3 of Supporting Information File 1 for details).
The dashed lines are model curves calculated using Equation 7 with the measured noise contribution and the calculated thermal noise contribution . The measured noise curves (solid lines) are in good agreement with the model (dashed lines). A disturbing side peak, which can be observed on top of in Figure 6a for a low detectionsystem noise floor, similarly appears in the corresponding curve in Figure 6b. Such peaks are due to electromagnetic emission from switching power supplies and other devices present in the laboratory environment. As the propagation of displacement noise to frequencyshift noise is well reproduced by experimental data, the latter can be used to obtain the level of the noise floor from a noise measurement in the lowfrequency region of the Δf signal. By inversion of Equation 7, the displacementnoise spectral density = can be obtained from if the system frequency response and are known:
At the frequency f_{m} = , where f_{m} × G_{filter}(f_{m}) has its maximum, the total noise is dominated by the noise from the detection system. At this point, Equation 9 can be simplified when assuming << yielding
This approximation defines an upper limit = for the detectionsystem noise spectral density.
In this manner, we investigate the noise characteristics of the three NCAFM systems using different preamplifiers and various cantilevers at different eigenmodes; the corresponding results are listed in Table 2. The detection noise measured directly in the displacement signal V_{z} as shown in Figure 6a is denoted as while the same quantity obtained from the frequencyshift noise by using Equation 9 is denoted as . The upper limit derived from Equation 10 is denoted as . The latter is a useful approximation that can easily be calculated without knowledge of the cantilever properties.
Table 2: Cantilever properties and noise figures for systems A, B and C. f_{n} and Q_{n} are the eigenfrequencies and Qfactors for the nth eigenmode of the cantilever. Noisefloor values are directly determined from the displacement signal V_{z}, while and are extracted from the Δf noise at the demodulator output as described in the main text. P_{pd} is the total light power on the PSD. For system C, measurements are performed with two different preamplifiers. Missing values are due to frequencyrange limitations of the spectrum analyser. In the case of higher harmonics, we cannot easily calculate the modal cantilever stiffness, as it strongly depends on the tip mass, which is generally not known [13]. Therefore, determining requires the knowledge of the stiffness and is, thus, only calculated for the measurements at the fundamental resonance frequency. Typical properties of a system operated with a qPlus sensor are taken from [11] for comparison.
cantilever  f_{n}  Q_{n}  P_{pd}  

(fm/)  (fm/)  (fm/)  (μW)  
System A  
AQ 10  f_{0} = 361,599 Hz  Q_{0} = 21,200  275  278  97  
System B  
V 15  f_{0} = 279,451 Hz  Q_{0} = 47,200  125  119  124  105 
System C, lowbandwidth preamplifier  
D 5  f_{0} = 68,353 Hz  Q_{0} = 118,000  115  122  130  120 
AO 3  f_{0} = 68,183 Hz  Q_{0} = 173,700  237  223  226  106 
AR 17  f_{0} = 276,360 Hz  Q_{0} = 39,200  97  98  120  
System C, highbandwidth preamplifier  
AO 3  f_{0} = 68,183 Hz  Q_{0} = 173,700  416  417  105  
AO 3  f_{1} = 437,086 Hz  Q_{1} = 48,500  93  105  
AO 3  f_{2} = 1,235,138 Hz  Q_{2} = 15,200  51  105  
AR 17  f_{0} = 276,360 Hz  Q_{0} = 39,200  258  259  120  
AR 17  f_{1} = 1,730,811 Hz  Q_{1} = 6,300  99  120  
AP 5  f_{0} = 1,996,199 Hz  Q_{0} = 32,400  302  309  77  
AL 3  f_{0} = 1,316,757 Hz  Q_{0} = 16,600  845  892  18  
qPlus system  
qPlus  f_{0} = 32,768 Hz  Q_{0} = 5,000  62 
Table 2 allows a comparison of the noise floor for different NCAFM systems and demonstrates the influence of cantilever properties on the noise figures. The best values for the noise floor achieved here are around 100 fm/ as measured for cantilever V 15 in system B and cantilevers D 5 and AR 17 in system C. These cantilevers have a length in the range of 100 μm to 250 μm. Exchanging the preamplifier may cause a large difference in the noise floor. This can be observed for cantilevers AO 3 and AR 17 in system C where the noise floor is doubled by changing from the lowbandwidth to the highbandwidth preamplifier. The benefit of the highbandwidth amplifier is the possibility to operate cantilevers at their higher resonance frequencies, where the displacementnoise floor significantly decreases even for a similar voltage noise caused by the laserpowerdependent photodiode shot noise due to different amplitude calibration factors for the corresponding modes. However, due to the length of only 40 μm and 35 μm, for the highfrequency cantilevers AP 5 and AL 3, the laser adjustment becomes difficult, yielding only 77 μW and 18 μW laser power on the PSD compared to about 100 μW for other cantilevers. Therefore, the detectionnoise floor for these cantilevers is much higher than for larger cantilevers. With an improved laserspot adjustment, however, a noise floor close to 100 fm/ should be possible.
In Figure 7, we illustrate the choice of optimum filter settings for a thermalnoiselimited detection. For that purpose, the frequency noise originating from the same detection system but passed through different PLL filters is shown. As these measurements are performed in system C, the filters are modelled as a closed loop where the settings of the PI controller have a significant effect on the frequency response and need to be individually adjusted for each setting of the loop filter order o and cutoff frequency f_{c} (see Section 3 of Supporting Information File 1 for details). The optimum settings for each loop filter used in the following are listed in Table S4 in Section 3 of Supporting Information File 1. In Figure 7a, we display the noise spectral characteristics of the Δf signal, while Figure 7b shows a plot of the total noise represented by the RMS value of the Δf signal as a function of the cantilever oscillation amplitude. Using a bandwidth of B_{−3dB} = 385 Hz (f_{c} = 1 kHz, o = 5), the total noise exceeds the thermal noise level by half an order of magnitude. Choosing a much lower bandwidth of B_{−3dB} = 48 Hz (f_{c} = 125 Hz, o = 5) decreases the frequency range where the signal is not attenuated below the one defined by the thermal noise limit. The optimum filter setting for the noise floor present in this measurement is a filter setting with a bandwidth of B_{−3dB} = 103 Hz (f_{c} = 125 Hz, o = 1), where the total noise does not significantly exceed the thermal noise and the signal is not unnecessarily attenuated. For all filter settings investigated here, experiment (solid lines) and model (dashed lines) agree well with each other.
In Figure 7b, measured values δf_{tot} (circles, squares, triangles) are compared to calculated values δf_{tot} derived from Equation 8 (solid lines) and δf_{th} defining the thermal limit of the RMS frequencyshift noise (dashdotted lines). While there is a large difference between thermal noise and total noise for the largebandwidth filter setting (f_{c} = 1 kHz, o = 5), this discrepancy becomes smaller and finally negligible on further reduction of the bandwidth. Note, however, that the settings yielding the smallest RMS noise are not the optimum as the corresponding filter does not only reduce the noise but attenuates the NCAFM signal more than necessary for thermalnoiselimited operation.
The RMS value of the total noise is an important figure of merit of the NCAFM detection system, as it defines the minimum detectable frequency shift. Figure 7b is an excellent demonstration of the potential of small amplitudes for atomic resolution measurements as it is known that the atomic contrast increases with reduced cantilever oscillation amplitude [11,14]. For a measurement with B_{−3dB} = 385 Hz (red line, f_{c} = 1 kHz, o = 5), one would choose an amplitude of 5 nm or above to reduce the noise; however, this would also reduce the atomic contrast compared to a lower amplitude measurement. For a measurement with B_{−3dB} = 103 Hz (blue line, f_{c} = 125 Hz, o = 1), one can take full advantage of the increased atomic corrugation for the smaller amplitude, as the total noise is even below the thermal noise for the larger amplitude.
In Figure 8a, different cantilevers are compared regarding their total RMS frequency shift noise δf_{tot} (solid lines) as well as the thermal frequency noise δf_{th} (dashdotted lines). Here, the same bandwidth of B_{−3dB} = 258 Hz is chosen for all simulations to facilitate the comparison of the cantilevers. Regarding thermal noise, all cantilevers except AL 3 exhibit an RMS noise below 0.5 Hz for amplitudes larger than 1 nm. The total noise values are ordered by the level of the corresponding noise floor , dominating the total noise for a bandwidth larger than the thermallimit bandwidth. Note that the thermalnoise contribution of AL 3 is even larger than the total noise of the other cantilevers. These results are compared to typical values for a qPlus sensor with parameters taken from [11]. The thermal noise δf_{th} of the qPlus sensor is an order of magnitude below the values for the cantilevers. Including the noise of the detection system, δf_{tot} of the qPlus sensor is nearly identical to the thermal noise δf_{th} obtained for cantilever D 5 (curve not shown) and, therefore, only half of the noise level of the best cantilevers.
For a valid comparison of measurements obtained under different experimental conditions, however, it is important to compare limits in the normalised frequency shift γ rather than the plain frequency shift Δf. Based on the concept of the normalised frequency shift [15], we define a normalisedfrequencyshift RMS noise as
to compare the noise characteristics of cantilevers independently of their stiffness and resonance frequency and display the corresponding data as a function of the cantilever oscillation amplitude in Figure 8b. Regarding the thermal contribution δγ_{th} to the normalised frequencyshift noise, cantilevers D 5 and AO 3 exhibit the best performance but are closely followed by cantilever AL 3. The δγ_{th} value of AR 17 is even larger than the total noise δγ_{tot} of cantilever D 5. This is presumably due to the large ratio k/f_{0}. Although cantilever AL 3 has the largest detectionsystem noise floor, its δγ_{tot} is quite close to that of cantilever D 5. On the other hand, the qPlus sensor has a noise level δγ more than two orders above the results for the cantilevers due to its exceptional k/f_{0} ratio. Therefore, the advantageous noise figures of the qPlus sensor documented in Figure 8a can only be exploited if the sensor is operated at very low amplitudes.
As cantilevers D 5 and AL 3 have thermal bandwidth limits of = 95 Hz and = 85 Hz according to Equation 6, they are best suited for thermalnoiselimited operation. Operating them with a filter B_{−3dB} = 103 Hz (f_{c} = 125 Hz, o = 1) yields noise limits of δγ_{tot} = 0.69 aN and δγ_{tot} = 0.84 aN, respectively, for an oscillation amplitude of A = 5 nm. Assuming, the detection noise floor of AL 3 could be decreased to 130 fm/ as for cantilever D 5, thermalnoiselimited operation with a bandwidth of = 586 Hz and δγ_{tot} = 1.62 aN would be possible for an oscillation amplitude of 5 nm and a filter setting of B_{−3dB} = 646 Hz (f_{c} = 1000 Hz, o = 3). This means that by switching from cantilever D 5 to AL 3, the usable bandwidth could be increased by a factor of six at the cost of increasing δγ_{tot} by a factor of two. In comparing such numbers, one should, however, consider that the assumed oscillation amplitude of 5 nm may be at the limit of stable operation [15], specifically for the soft cantilever D 5. In conclusion, the highfrequency and relatively stiff cantilever AL 3 represents an excellent choice for highspeed measurements with small amplitudes and good noise performance, while the larger and softer cantilever D 5 is the better choice for slower measurements with best possible noise performance.
Conclusion
We investigated the relation between the displacement noise in NCAFM measurements and the corresponding frequencyshift noise at the output of the demodulator and demonstrated that predictions based on the demodulator transfer function and filtering are well reproduced by experiments. For a quantitative analysis of the noise, a precise amplitude calibration of the detection system relating electrical signals to the mechanical oscillation of the cantilever is inevitable. The displacement noise of an NCAFM system can be measured directly with a spectrum analyser at the output of the detection system, and the thermal component of the displacement noise extracted from such spectra agrees well with spectra derived from a model of thermal cantilever excitation. The noise contribution of the detection system can be obtained from the whitenoise floor of the measured spectra. The knowledge of the detectionsystem transfer functions allows one to predict the frequencyshift noise from the measured displacement noise, and by inversion, a measurement of the detectionsystem noise from the frequencyshift noise is possible. While the former analysis requires a spectrum analyser with very high resolution and an operating range that includes the eigenfrequency of the cantilever, the latter procedure requires only a measurement of the frequencyshift noise with a device covering the frequency range between a few hertz and about 10 kHz at moderate frequency resolution. Therefore, a rather complete noise characterisation with a simple spectrum analyser as integrated in many NCAFM systems is possible for a calibrated system.
The framework of modelling noise in the NCAFM system in combination with the experimental practice described here provides a clear guideline for system design and the choice of experimental parameters for thermalnoiselimited operation. The analysis shows that for a noiseoptimised NCAFM measurement, the right choice of the cantilever is most important, and obtaining a high effective Qfactor should be given great attention to keep the level of thermal noise at a minimum. The bandwidth of thermalnoiselimited operation is determined by the noise generated in the detection system. By an appropriate choice of PLL filter settings, one can make full use of this bandwidth without attenuating the NCAFM signal while very efficiently eliminating most of the detection system noise. We find that with a technically optimised system and an appropriate choice of experimental parameters, roomtemperature thermalnoiselimited NCAFM measurements are possible over a bandwidth of 100 Hz and a detection limit smaller than 0.7 aN for the normalised frequency shift operating at an amplitude of 5 nm.
Supporting Information
Supporting Information File 1: Experimental details and theory.  
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Acknowledgements
The authors gratefully acknowledge experimental support from Stefan Kuhn and HansHermann Pieper and most helpful discussions with Steffen Porthun, Sadik Hafizovic, Kei Kobayashi and Hirofumi Yamada. Substantial help from Alfred Ziegler with the mathematics of Section 2 in Supporting Information File 1 is gratefully acknowledged. The project has been generously supported by Nanoworld Services GmbH.
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3.  Fukuma, T.; Kimura, M.; Kobayashi, K.; Matsushige, K.; Yamada, H. Rev. Sci. Instrum. 2005, 76, 053704. doi:10.1063/1.1896938 
4.  Kobayashi, K.; Yamada, H.; Matsushige, K. Rev. Sci. Instrum. 2009, 80, 043708. doi:10.1063/1.3120913 
4.  Kobayashi, K.; Yamada, H.; Matsushige, K. Rev. Sci. Instrum. 2009, 80, 043708. doi:10.1063/1.3120913 
2.  Kobayashi, K.; Yamada, H.; Itoh, H.; Horiuchi, T.; Matsushige, K. Rev. Sci. Instrum. 2001, 72, 4383–4387. doi:10.1063/1.1416104 
9.  Lübbe, J.; Tröger, L.; Torbrügge, S.; Bechstein, R.; Richter, C.; Kühnle, A.; Reichling, M. Meas. Sci. Technol. 2010, 21, 125501. doi:10.1088/09570233/21/12/125501 
4.  Kobayashi, K.; Yamada, H.; Matsushige, K. Rev. Sci. Instrum. 2009, 80, 043708. doi:10.1063/1.3120913 
1.  Albrecht, T. R.; Grütter, P.; Horne, D.; Rugar, D. J. Appl. Phys. 1991, 69, 668–673. doi:10.1063/1.347347 
3.  Fukuma, T.; Kimura, M.; Kobayashi, K.; Matsushige, K.; Yamada, H. Rev. Sci. Instrum. 2005, 76, 053704. doi:10.1063/1.1896938 
4.  Kobayashi, K.; Yamada, H.; Matsushige, K. Rev. Sci. Instrum. 2009, 80, 043708. doi:10.1063/1.3120913 
3.  Fukuma, T.; Kimura, M.; Kobayashi, K.; Matsushige, K.; Yamada, H. Rev. Sci. Instrum. 2005, 76, 053704. doi:10.1063/1.1896938 
3.  Fukuma, T.; Kimura, M.; Kobayashi, K.; Matsushige, K.; Yamada, H. Rev. Sci. Instrum. 2005, 76, 053704. doi:10.1063/1.1896938 
3.  Fukuma, T.; Kimura, M.; Kobayashi, K.; Matsushige, K.; Yamada, H. Rev. Sci. Instrum. 2005, 76, 053704. doi:10.1063/1.1896938 
7.  Torbrügge, S.; Lübbe, J.; Tröger, L.; Cranney, M.; Eguchi, T.; Hasegawa, Y.; Reichling, M. Rev. Sci. Instrum. 2008, 79, 083701. doi:10.1063/1.2964119 
8.  Rode, S.; Stark, R.; Lübbe, J.; Tröger, L.; Schütte, J.; Umeda, K.; Kobayashi, K.; Yamada, H.; Kühnle, A. Rev. Sci. Instrum. 2011, 82, 073703. doi:10.1063/1.3606399 
10.  Lübbe, J.; Doering, L.; Reichling, M. Meas. Sci. Technol. 2012, 23, 045401. doi:10.1088/09570233/23/4/045401 
11.  Giessibl, F. J.; Pielmeier, F.; Eguchi, T.; An, T.; Hasegawa, Y. Phys. Rev. B 2011, 84, 125409. doi:10.1103/PhysRevB.84.125409 
9.  Lübbe, J.; Tröger, L.; Torbrügge, S.; Bechstein, R.; Richter, C.; Kühnle, A.; Reichling, M. Meas. Sci. Technol. 2010, 21, 125501. doi:10.1088/09570233/21/12/125501 
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15.  Giessibl, F. J. Phys. Rev. B 1997, 56, 16010–16015. doi:10.1103/PhysRevB.56.16010 
11.  Giessibl, F. J.; Pielmeier, F.; Eguchi, T.; An, T.; Hasegawa, Y. Phys. Rev. B 2011, 84, 125409. doi:10.1103/PhysRevB.84.125409 
11.  Giessibl, F. J.; Pielmeier, F.; Eguchi, T.; An, T.; Hasegawa, Y. Phys. Rev. B 2011, 84, 125409. doi:10.1103/PhysRevB.84.125409 
14.  Giessibl, F. J.; Bielefeldt, H.; Hembacher, S.; Mannhart, J. Appl. Surf. Sci. 1999, 140, 352–357. doi:10.1016/S01694332(98)005534 
12.  Lübbe, J.; Temmen, M.; Rahe, P.; Kühnle, A.; Reichling, M. Beilstein J. Nanotechnol. 2013, submitted. 
13.  Melcher, J.; Hu, S.; Raman, A. Appl. Phys. Lett. 2007, 91, 053101. doi:10.1063/1.2767173 
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