The qPlus sensor, a powerful core for the atomic force microscope.

Atomic force microscopy (AFM) was introduced in 1986 and has since made its way into surface science, nanoscience, chemistry, biology, and material science as an imaging and manipulating tool with a rising number of applications. AFM can be employed in ambient and liquid environments as well as in vacuum and at low and ultralow temperatures. The technique is an offspring of scanning tunneling microscopy (STM), where the tunneling tip of the STM is replaced by using a force sensor with an attached tip. Measuring the tiny chemical forces that act between the tip and the sample is more difficult than measuring the tunneling current in STM. Therefore, even 30 years after the introduction of AFM, progress in instrumentation is substantial. Here, we focus on the core of the AFM, the force sensor with its tip and detection mechanism. Initially, force sensors were mainly micro-machined silicon cantilevers, mainly using optical methods to detect their deflection. The qPlus sensor, originally based on a quartz tuning fork and now custom built from quartz, is self-sensing by utilizing the piezoelectricity of quartz. The qPlus sensor allows us to perform STM and AFM in parallel, and the spatial resolution of its AFM channel has reached the subatomic level, exceeding the resolution of STM. Frequency modulation AFM (FM-AFM), where the frequency of an oscillating cantilever is altered by the gradient of the force that acts between the tip and the sample, has emerged over the years as the method that provides atomic and subatomic spatial resolution as well as force spectroscopy with sub-piconewton sensitivity. FM-AFM is precise; because of all physical observables, time and frequency can be measured by far with the greatest accuracy. By design, FM-AFM clearly separates conservative and dissipative interactions where conservative forces induce a frequency shift and dissipative interactions alter the power needed to maintain a constant oscillation amplitude of the cantilever. As it operates in a noncontact mode, it enables simultaneous AFM and STM measurements. The frequency stability of quartz and the small oscillation amplitudes that are possible with stiff quartz sensors optimize the signal to noise ratio. Here, we discuss the operating principles, the assembly of qPlus sensors, amplifiers, limiting factors, and applications. Applications encompass unprecedented subatomic spatial resolution, the measurement of forces that act in atomic manipulation, imaging and spectroscopy of spin-dependent forces, and atomic resolution of organic molecules, graphite, graphene, and oxides.


I. INTRODUCTION
A. Atomic force microscopy (AFM) The first instrument that allowed to image a surface at atomic resolution was the scanning tunneling microscope (STM). STM was invented by Gerd Binnig and Heinrich Rohrer in 1981, and the author still remembers the day in the fall of 1985 when he first heard about this incredible instrument in a hallway of ETH Zurich as an undergraduate student. In those days, it was taught early on in school that while the existence of atoms is unquestioned, it is impossible to "see" them except for the atomic structure of sharp tips in a field ion microscope. Today, high school students perform STM experiments, and it is hard to recollect the strong skepticism against the true resolution of atoms that prevailed in the scientific community before the invention of STM. It was strongly doubted that the challenges posed by mechanical and thermal vibrations that oppose stable tunneling across a vacuum gap between a sharp tip and a flat sample could ever be mastered. Any remaining doubts about the capability to resolve atoms

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scitation.org/journal/rsi evaporated when Binnig, Rohrer, Gerber, and Weibel showed a real space image of the silicon (111)-(7 × 7) surface in 1983. 22 Binnig and Rohrer received the Nobel Prize in Physics already in 1986 for the invention of STM, shared with Ernst Ruska, the inventor of the electron microscope. The challenges of implementing STM in the early years are recollected in the Nobel lecture of Binnig and Rohrer. 23 Although the technical challenges to create a scanning tunneling microscope that operates in an ultrahigh vacuum (UHV) to provide clean and well defined surfaces and tips were huge, the physics of the quantum mechanical tunneling effect is highly supportive to the concept of a microscope based on electron tunneling. Heinrich Rohrer named the achievement of vacuum tunneling 21 as the birth of STM in a noteworthy talk at the 1991 International Conference on STM in Interlaken, Switzerland. STM relies on the quantum mechanical tunneling current between two biased electrodes, the tip and the sample. This tunneling current increases by a factor of ten when decreasing the tip-sample distance by one Å (100 pm). Even for a relatively blunt metal tip close to a flat surface, chances are high that one atom in the tip apex sticks out by, say, a third of an atomic diameter, thereby carrying the major part of the tunneling current and thus enabling the spectacular atomic resolution.
The monotonic exponential increase of the tunneling current enables a simple implementation of a distance control loop, where the logarithm of the ratio between the actual tunneling current and its setpoint yields a linear and monotonic input for a distance regulator.
Careful observations during operation of the STM revealed to Binnig that atomic forces are present (see also Refs. 46 and 209), and in 1985, Binnig suggested to utilize these forces and invented a new type of microscope, the "Atomic Force Microscope" (AFM), 24 a microscope that should extend the atomic resolution capability of STM to insulating samples as reported in the opening paragraph of Ref. 203. A first working version of AFM was introduced by Binnig, Gerber, and Quate in 1986, 25 in a publication that has been cited close to ten thousand times since. Conceptually, the atomic force microscope might be viewed as a highly refined stylus profilometer, an instrument that maps a surface using a cantilever with a sharp tip that scans across a surface and records its topography line by line. However, the spatial resolution provided by stylus profilometry is much lower than needed for reaching the atomic scale, and documented evidence that profilometry might ever reach atomic resolution levels is not known to us. Therefore, AFM has its origin more in the scanning tunneling microscope (STM) which has opened direct access to matter on the single atom scale. Although the AFM did not achieve atomic resolution from the start, it became much more widespread in use than STM because it works in any environment and can image conductors and insulators. An estimated ten thousand atomic force microscopes are in use worldwide today, and in 2016, 30 years after their seminal paper, 25 Binnig, Quate, and Gerber were rewarded with the Kavli Prize for Nanoscience. Many review papers 64,86,118 and books [167][168][169] are available that discuss the key elements of the rise of AFM.
In the development of STM, profound experimental challenges have been mastered such as establishing mechanically stable vacuum tunneling junctions on a picometer lengthscale, sub-Angstrom scanning precision, as well as the preparation of atomically sharp tips and clean flat samples. Nevertheless, reaching atomic resolution by AFM required overcoming additional profound challenges that are rooted in the added complexity of the control signal, the force. While the tunneling current in vacuum is 1. monotonic with distance, 2. extremely short range and thus originating mainly from the tip's front atom, and 3. ranging from pA to hundreds of nA and easily measured experimentally, the tip-sample force shares none of these three characteristics of the tunneling current. First, the tip-sample force is not monotonic, and it is usually initially attractive, turning repulsive when atoms get in contact. Second, the force is composed of strong long-range components due to van der Waals and other interactions, and the chemical forces that allow for atomic resolution are often much smaller in magnitude. Third, the experimental difficulties to measure forces in the pico-to nanonewton regime are much harder than those encountered when measuring currents in the pico-to nanoampere regime. For operation in ambient conditions, contamination layers add to the complexity of the tip-sample interaction. For these reasons, the spatial resolution of AFM did not reach the spatial resolution of STM for a long time, and initially, common experience and expectations held that the AFM will stay behind the resolution of STM forever. Today, the AFM usually exceeds the spatial resolution of STM, and the key element of the AFM is the force detector with its tip and deflection sensor, 56 which is the subject of the present article.
B. Principle and operating modes of atomic force microscopy

Quasistatic mode
The quasistatic mode is the simplest mode of AFM, where the cantilever scans the surface similar to the stylus in the profilometer or the needle of a record player. In the static (or dc-) mode, the tip-sample force F ts leads to a deflection q of a cantilever beam with force constant k given by Early after the introduction of AFM, reports about the achievement of atomic resolution appeared. While atomic lattices were shown in these early reports, steps or defects were not observed, and the appearance of atomic lattices was explained with an egg-carton effect where the tip assumes a negative of the sample after repeated scanning (see, e.g., Fig. 1 in Ref. 69).
One key challenge of obtaining true atomic resolution in the quasistatic mode is the presence of strong long-range forces that lead to an uncontrolled "jump-to-contact" of the cantilever when it is brought close to the surface. It is possible to obtain true atomic resolution in quasistatic mode for certain

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REVIEW scitation.org/journal/rsi conditions. In one experiment, 68 KBr(001) was resolved atomically in a low-temperature UHV microscope using a special trick: when the cantilever had landed on the KBr surface after jump-to-contact, the load on the cantilever due to the strong van der Waals forces was reduced by pulling the sample back to a point just before the contact between the cantilever and the sample was lost again. Then, the sample was moved laterally from the damaged cantilever landing zone to an area that was pristine and scanning was resumed at a repulsive interaction of an estimated 1 nN yet a net attractive interaction. Despite the small repulsive interaction, the 5 nm scanframe was slightly depressed and a small superstructure was induced in the scanned area (see Fig. 20 in Ref. 86). In 1993, Ohnesorge and Binnig obtained true atomic resolution in ambient conditions on calcite with quasistatic AFM in contact-and noncontact modes, even resolving an atomic step. 179 Here, the trick was to immerse the surface and the cantilever in water, thus preventing jump-to-contact by strongly reducing the van der Waals interaction. 131

Dynamic modes
The quasistatic mode is conceptually simple. However, it has a few drawbacks. First, it is hard to measure static deflections of small springs with a good signal-to-noise ratio (SNR). Second, the contact between the tip and the sample can lead to wear and tear. Third, the chemical bonding forces that enable atomic contrast are typically overwhelmed by large long-range van der Waals forces. These three challenges can be met by operating in a dynamic mode, in particular, with a dynamic mode that operates at small amplitudes (see Fig. 1).
The original publication about the AFM already discussed dynamic modes of operation of AFM. 25 Currently, most AFMs operating in ambient conditions use amplitude modulation (AM) mode. In this mode, the cantilever is driven to oscillate at a constant frequency close to the eigenfrequency f 0 of the cantilever. The tip-sample interaction changes the oscillation amplitude, and this change is used as a feedback signal. Both dissipative and non-dissipative parts of the interaction force change the resulting amplitude, and a separation of those two force parts is not straightforward. Often, the phase image that shows the difference in phase between the driving signal and the oscillation provides greater spatial resolution. The reason for the widespread use of the AM mode rests on two characteristics. First, it is relatively simple to measure the amplitude at the operational frequency of the cantilever-a lock-in technique allows for excellent precision. Second, a monotonic feedback signal is available in AM-AFM: The setpoint of the amplitude is smaller than the amplitude of the free oscillation, and the tip-sample interaction upon reducing the distance reduces the amplitude in a more or less monotonic fashion (except for a subtle artifact 10 ).
The technically more advanced frequency modulation mode (FM-AFM) 2,47 strictly separates dissipative and nondissipative interactions-dissipative interactions lead to an increase in the driving amplitude that excites the oscillation of the cantilever, and non-dissipative interactions lead to a frequency shift.

FIG. 1.
Long-and short-range contributions to the frequency shift ∆f as a function of amplitude for a force that consists of a short range force (plotted in magenta) F sr (z) = F 0 sr · exp(−z/λ sr ) and a long-range force (blue) F lr (z) = F 0 lr · exp(−z/λ lr ). Here, we have adapted a typical example of F 0 lr = 3 · F 0 sr and λ lr = 30 · λ sr . For amplitudes A that are small compared to the range of the interaction λ, the frequency shift is proportional to the force gradient F/λ; for large amplitudes, it is expressed by the normalized frequency shift γ = ∆f/f 0 kA 3/2 ∝ F √ λ. The influence of short-and long-range forces can thus be adjusted by the choice of amplitude A: small amplitudes result in a good sensitivity for short-range forces, and large amplitudes emphasize long-range forces. Guide to read this plot: for a specific type of cantilever or sensor, the frequency shift for infinitely small amplitudes amounts to 100 Hz due to short range forces and 10 Hz due to long range forces. Upon increasing the amplitude to λ sr , the short range contribution starts to drop, while the frequency shift due to long range forces remains constant. Once the amplitude is significantly larger than λ lr , the frequency shift contribution due to long range forces is 16 times larger than the short range contribution, i.e., the short range versus long range contributions change from 10 to 1/16 as the amplitude is increased. Inset: Schematic presentation of the tip and the sample, reprinted with permission from Binnig et al., Phys. Rev. Lett. 56, 930 (1986). Copyright 1986 The American Physical Society.

Stability criterion
Attractive forces between the tip and the sample can lead to an uncontrolled jump-to-contact of the cantilever when approaching it to a sample. 30 In the quasistatic mode, jumpto-contact is prevented if the stiffness k of the cantilever fulfills the following condition: where V ts is the potential energy between the tip and the sample. In the dynamic modes, jump-to-contact can be prevented for any cantilever, provided the amplitude A is large enough, 78 k · A > max ( ∂V ts ∂z ). (3)

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REVIEW scitation.org/journal/rsi a large amplitude of 34 nm to stabilize the relatively soft cantilever (k = 17 N/m) at the close tip-sample distances required to obtain atomic resolution. 71

II. PHYSICS OF FREQUENCY MODULATION ATOMIC FORCE MICROSCOPY
Of all physical observables, time and its inverse, frequency, can be measured by far at the greatest precision. 224 Today, frequency modulation AFM (FM-AFM) 2,47 has emerged as the AFM technique that enables high spatial and force resolution with a straightforward separation of conservative and dissipative interactions. In FM-AFM, the cantilever is driven to oscillate by applying positive feedback to obtain a constant amplitude A and the conservative force between the tip and the sample alters the frequency from the natural eigenfrequency f 0 of the cantilever to f = f 0 + ∆f, while dissipative forces induce the amplitude feedback to adjust the drive signal A dri e to keep A constant.
A. Signal-Frequency shift and its dependence on tip-sample forces

Frequency shift as a function of tip-sample forces
In frequency modulation atomic force microscopy, the oscillation of the force sensor is driven by feeding back its deflection, phase shifted by +π/2, to an actuator that shakes the base of the cantilever with a drive amplitude A dri e . An automatic gain circuit adjusts A dri e to establish a constant oscillation amplitude A. The oscillation frequency corresponds to f 0 , the eigenfrequency of the cantilever, when the tip is far from the surface and the force gradient field between the tip and the sample is close to zero. For smaller distances, the tipsample force gradient increases and leads to a frequency shift ∆f = f − f 0 . With f = f 0 + ∆f and f 0 = 1 2π k/m * , the frequency shift is given by ∆f = f 0 2k k ts (4) with 84 k ts (z, A) = 2 π 1 −1 k ts (z + ζA) 1 − ζ 2 dζ.
This equation shows that the frequency shift is a function of the amplitude. Two extreme cases allow a simplification. When the amplitude is small compared to the range of the tip-sample interaction λ, the gradient approximation holds, i.e., k ts (z, A λ) ≈ k ts (z).
When the amplitude is large compared to the range of the tip-sample interaction λ, it is useful to introduce a largeamplitude approximation where the frequency shift is given by with the "normalized frequency shift" γ ts ≈ 0.4F ts √ λ. 78 In FM-AFM, the signal is a frequency shift ∆f. This frequency shift depends on the tip-sample interaction and the stiffness k, eigenfrequency f 0 , and amplitude A of the cantilever. For a force that follows an exponential distance dependence F(z) = F 0 exp(−z/λ), we find ∆f = f 0 kA F 0 e −(z+A)/λ I 1 (A/λ), (8) where I 1 (A/λ) is the Bessel function of the first kind, a special version of the Kummer function. 83 We can rewrite Eq. (8) such that its resemblance to the gradient approximation is directly evident, The first factor in this equation is the gradient approximation, while the fraction 2I 1 (x)e −x /x with x = A/λ can be expanded as 2I 1 (x)e −x /x = 1 − x + 5/8x 2 + O(x 3 ). For a minimum distance between the tip and the sample of z, the tip oscillates within the interval [z, z + 2A], and at the optimal oscillation amplitude A opt ≈ 1.545λ, 92 we obtain an average tip-sample force gradient that is approximately one third of the peak force gradient at distance z because 2I 1 (1.5451)e −1.5451 /1.5451 ≈ 0.33.

Amplitude dependence of short-range force contributions to the frequency shift ∆f
For a total force given by a sum of long-range and shortrange forces F(z) = F 0 lr exp(−z/λ lr ) + F 0 sr exp(−z/λ sr ), we obtain the total frequency shift with Eq. (10). The frequency shift is given by ∆f = f 0 kA F 0 lr e −(z+A)/λ lr I 1 (A/λ lr ) + F 0 sr e −(z+A)/λsr I 1 (A/λ sr ) . (10) The amplitude dependence of the frequency shift is displayed in Fig. 1. For amplitudes A that are smaller than the range of the short-range forces λ, the frequency shift is constant. However, once A reaches λ, the frequency shift starts to drop at a high rate proportional to 1/A 3/2 . Figure 2 highlights the relevant observables that govern STM and AFM in a logarithmic display. The black curve shows

FIG. 2.
Logarithmic plot of the tunneling current I, van der Waals force F dW , and chemical bonding force F chem composed of attractive (F att ) and repulsive (F rep ) components. The long-range van der Waals force typically dominates the interaction force by a large margin, so it masks the chemical bonding forces that enable true atomic resolution of AFM.

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scitation.org/journal/rsi the tunneling current with its typical factor ten increase for each distance reduction of 100 pm. A strong distance dependence is important for high spatial resolution. The distance control loop features a logarithmic amplifier that computes an error signal from the logarithm of the ratio between the tunneling current and its setpoint s error = log(I/I set ). This signal s error is smaller than zero if the current is below its setpoint and greater than zero otherwise. The error signal s error enters a PI (proportional-integral) controller, enabling a stable and simple feedback loop. FM-AFM measures forces that are typically composed of long-and short-range contributions, where the long-range contributions arise from van der Waals and electrostatic forces. These long-range forces do not contribute to atomic resolution and should be filtered out. The long-range force has a very weak distance dependence but a large magnitude compared to the short-range attractive (green) and repulsive (red) forces. This graph outlines an important consequence of the choice of amplitude in an FM-AFM experiment: choosing a small amplitude increases the contribution of short-range forces to the frequency shift, and the use of large amplitudes results in a large contribution of long-range forces to the frequency shift. When using soft cantilevers, a relatively large amplitude might have to be chosen in order to obtain a stable oscillation; see Eq. (3). Figure 3 is a display of currents and force gradients. We notice a striking difference between the role of short-range and long-range components: the force gradient of the shortrange forces is larger than the gradient of the long-range force for small distances. Also, the slope of the repulsive Pauli force is typically larger than even the slope of the tunneling current.

FIG. 3.
Logarithmic display of tunneling current I, gradients of van der Waals force k dW , and chemical bonding force k chem composed of attractive (k att ) and repulsive (k rep ) components. The gradient of the forces, in particular, the repulsive component shown in red, dominates the interaction for small distances. In frequency modulation AFM, the frequency shift is proportional to the force gradient if the amplitude is small compared to the interaction length; therefore, the use of small amplitudes results in a strong contribution of the repulsive Pauli exclusion forces that occur when atoms are touching each other. Therefore, a direct coupling of the force gradient to the frequency shift provided by small amplitude FM-AFM is an ideal way to maximize the sensitivity to short-range forces.

Deconvolution of forces from frequency shifts
The origin of the experimentally observed frequency shift is force gradients in the tip-sample force. For infinitely small amplitudes A, the frequency shift is proportional to the force gradient, but for larger amplitudes, a convolution as explained in Eq. (5) takes place. The reverse process, a deconvolution, can be obtained by a matrix inversion as introduced in 2001 in Ref. 84 or by using the more popular Sader-Jarvis algorithm 206 that was introduced three years later. In principle, the matrix inversion is accurate, but it is quite sensitive to the accuracy of the oscillation amplitude. 256 For example, if a frequency shift spectrum has been recorded at an amplitude of A = 50 pm, but the deconvolution is processed for A = 55 pm, spikes and errors can result. The Sader-Jarvis algorithm 206 is more stable against inaccuracies of the oscillation amplitude but may result in deconvolution errors up to 5%. Recently, Sader et al. found that some inversion problems of frequency shift into force are even ill-posed, and well-posed inversions may only be possible by choosing amplitudes A that are either larger or smaller than the specific thresholds, as outlined in Refs. 207 and 208, depending on the characteristics of the force law.

B. Noise
If the frequency of the force sensor could be measured with infinite accuracy, infinitely small force gradients could be measured. In practice, there is noise, and four relevant noise contributions can be distinguished. Thermal noise has already been discussed in the first article about FM-AFM by Albrecht et al. 2 A different source of noise (detector noise) has been described by Dürig in 1997 [Eq. (11) in Ref. 48]. For large bandwidths B, i.e., for large scanning speeds, deflection detector noise is dominant as it increases with B 3/2 . Two other noise sources, thermal noise and oscillator noise, increase with the square root of bandwidth B. The fourth noise source is due to sensor frequency drifts that can be caused by temperature changes. As we measure an average force gradient in FM-AFM, the noise in this figure is given with Eq. (4),

Thermal noise
The thermal noise of a force sensor at a bandwidth B is already discussed in Eq. (1) of the first article about frequency modulation AFM, 2

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REVIEW scitation.org/journal/rsi is given by kA 2 /2 when it oscillates at amplitude A. The relative thermal frequency noise is therefore related to the square root of the quotient between thermal energy and total energy multiplied by the ratio between bandwidth and eigenfrequency divided by the quality factor. Ultimately, the relevant quantity in an AFM measurement is the force gradient between the tip and the sample, and the thermal noise in force gradient measurement is given by The absolute thermal frequency noise after Eq. (12) is given by an integral of a thermal noise density n ∆f thermal from modulation frequency f mod from 0 up to bandwidth B with With this integral, the noise density of thermal noise is constant with respect to frequency (white thermal noise),

Deflection detector noise
The precision of all length measurements, including the measurement of the deflection of a cantilever, is compromised by noise. The precision of the deflection detection can be described by a deflection detector noise density n q . For example, with n q = 100 fm/ √ Hz, the error in deflection measurement is δq = n q √ B with δq = 100 fm at a bandwidth of B = 1 Hz and δq = 1 pm at B = 100 Hz. Typically, the oscillation frequency of the cantilever varies very little around the eigenfrequency f 0 , and we can therefore assume a constant deflection detector noise density n q for frequencies around f 0 . If n q was zero, a single oscillation cycle would allow an infinitely precise measurement of the oscillation period Ξ and its inverse, the oscillation frequency f. Figure 4 shows two oscillation cycles of a cantilever as it can be observed on an oscilloscope. The oscillation period Ξ is given by the time difference between two consecutive zero crossings, and the uncertainty of each zero crossing is given by the error in deflection measurement δq = 2n q √ B divided by the slope of the deflection curve that is given by ω · A with ω = 2π · f and oscillation amplitude A. If only one oscillation cycle was available for frequency measurement, the relative precision of the frequency measurement would be However, if the bandwidth is smaller than f 0 , a larger number of oscillation cycles given by f 0 /B is available for frequency measurement, and the precision is greatly enhanced, yielding a frequency uncertainty due to detector noise given by   FIG. 4. Typical cantilever deflection signal as it appears on an oscilloscope. The deflection of the cantilever q (t) is subject to an error δq = 2 · n q B 1/2 . 43 Therefore, the oscillation period of a single oscillation cycle is subject to an error δΞ given by δq divided by the slope of the deflection curve at the zero crossing (see text). The oscillation frequency is given by the inverse time lag between two consecutive zero-crossings with positive velocity. If N oscillation cycles are used for frequency measurement (corresponding to a bandwidth B = f /N), the accuracy increases by a factor N and the frequency error due to detector noise is proportional to B 3/2 ( A more precise calculation 50,105,146 yields a slightly larger prefactor (0.816 vs. 0.637), With Eq. (11), we find The noise density of detector noise is increasing linearly with frequency (blue noise),

Oscillator noise
In 2009, Kobayashi et al. 146 described an additional contribution to frequency noise in FM-AFM that is inversely proportional to Q and therefore is significant, in particular, in low Q environments. This contribution is not explicitly temperature dependent and thus can become significant at low temperatures where thermal noise becomes small. The origin of this noise is a consequence of driving the cantilever with its own phase shifted (π/2) and noisy (due to a finite n q ) oscillation signal. The lower the Q value, the more of this noise pushes the cantilever at the correct phase. Therefore, this noise contribution is proportional to n q and inversely proportional to Q, With Eq. (11), we find

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Similar to thermal noise, oscillator noise is proportional to the square root of the detection bandwidth B and inversely proportional to amplitude. While thermal noise also decreases with increasing Q proportional to 1/ Q, oscillator noise is proportional to 1/Q; thus of all noise sources, it is most sensitive to the quality factor. The noise density of oscillator noise is constant with respect to frequency (white noise),

Frequency drift noise
All of the three noise sources discussed so far increase with bandwidth. In principle, those noise sources can be made infinitely small by reducing the bandwidth and thus the scanning speed. Thermal drift prevents the acquisition of images or spectra at infinitely small speeds. First, the scanning probe microscope itself will drift with changes in temperature or for other reasons such as piezo creep. Second, the cantilever will not maintain a constant oscillation frequency. Changes in temperature, aging, or other factors cause the oscillation frequency to drift with time. Thermal drift is a challenge for room temperature measurements and, in particular, for hightemperature measurements. The oscillation frequency of a sensor will change slightly even in liquid helium environments, e.g., induced by slight changes in the boiling temperature induced by variations in ambient pressure.
We can compute the power spectral density of the frequency drift noise contribution by taking a Fourier transform of the square of the frequency drift. If the sensor frequency drifts linearly with time at a drift rate r, we find δf(t) = r · t within a time interval [−τ/2. . .τ/2]. With Ω = 2π/τ, we can express the time dependence of the frequency as δf 2 (t) = ∞ n=0 a n cos (nΩt) (24) with Fourier coefficients a n = Ω π τ /2 t=−τ /2 r 2 t 2 cos (nΩt)dt (25) and a n = (−1) n r 2 τ 2 π 2 n 2 .
We can now interpret |a n | as the equivalent power component at a frequency f mod = n/τ in a frequency interval of 1/τ. Therefore, the power spectral density (power per frequency) becomes (27) and Thus, a linear frequency drift leads to 1/f noise in the frequency spectrum of the phase-locked-loop (PLL) output. The magnitude of this noise component depends on the drift rate of the frequency r and the measurement period τ. The time period τ is at least the time it takes to complete one image. Thus, for fast measurements, frequency drift noise can be reduced provided that the frequency detector (PLL) is reset before an image is taken. To obtain the effect of this noise on the force gradient measurement, we need to multiply n ∆f ( f mod ) by 2k/f 0 [see Eq. (4)] to obtain The noise density of drift noise is inverse with frequency (1/f or pink noise), The effect of thermal drift noise for qPlus sensors and needle sensors will be discussed in detail in Subsection IV B 4.

Summary of noise calculations
To assess the various noise contributions regarding their impact on the precision of measuring the tip-sample force gradient, we express the four noise sources in terms of k ts . The errors in measuring k ts are evaluated by squaring the density expressions, integrating them in a modulation frequency interval from f mod from 1/τ to B where τ is the time needed to acquire the image, δk ts sensor = 8 3 For the first three noise sources, we can set the lower modulation frequency to zero, but drift noise would diverge for infinitely long measurement times, requiring a finite lower threshold B min = 1/τ, As B 1/τ, the noise in k ts due to drift is proportional to the drift rate r times the time it takes to acquire the image τ. If we assume to acquire an image with N pixels, we find τ ≈ N/B and These four noise sources are statistically independent, and the net effect of statistically independent variables is computed by taking the square root of the sum of squares.
The summary of noise calculations allows us to identify important conclusions about ideal properties of the force sensor. The first three noise sources are all inversely proportional to A, i.e., they appear to suggest the use of infinitely large amplitudes. However, we will see in Sec. II C that for

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scitation.org/journal/rsi maximizing the signal-to-noise ratio, finite amplitudes on the order of the decay length of the forces are optimal. The stiffness k affects all noise sources: thermal noise increases as √ k, and oscillator noise, frequency drift noise, and drift noise are proportional to k. Therefore, we recommend that k should be large enough to enable stable sensor oscillations as discussed in Eqs. (2) and (3) at the optimal amplitude but should otherwise be as small as possible. High Q-values are desirable to minimize thermal and oscillator noise. The frequency drift noise can be minimized by using a sensor material with little frequency change as a function of temperature and by operating the sensors in a thermally stable environment, preferentially at temperatures at or close to the turnover temperature T p [see Eq. (77)]. The noise performance of the electrical amplifier is critical to minimize deflection detector noise.

C. Signal-to-noise ratio
Using the calculations for the frequency shift and for noise, we can now calculate the signal-to-noise ratio. When increasing the amplitude from zero, we found in Eq. (8) that the frequency shift remains almost constant until the amplitude reaches the range of the interaction and then decreases proportionally to 1/A 3/2 . The noise in the frequency measurement of the sensor is inversely proportional to A; therefore, the signal-to-noise ratio (SNR) can be expressed as This function is plotted in Fig. 5; it has its maximum at κA = 1.5451. . .; thus, the optimal signal-to-noise ratio is reached FIG. 5. Signal-to-noise-ratio (SNR) as a function of the product between decay constant κ and amplitude A or ratio A/λ with decay length λ = 1/κ. Optimal SNR is obtained for A opt = 1.545λ, equivalent to κA opt = 1.545. The decrease in SNR for non-optimal amplitudes is asymmetric with log(A), choosing A = 0.1A opt or A = 100A opt both yield approximately twenty percent of the ideal SNR. for amplitudes that correspond to the decay length λ = 1/κ of the tip-sample force, 80 or more precisely, A opt ≈ 1.545λ. In theory, this ideal amplitude applies to all sensors in FM-AFM that probe interactions of range λ, provided the sensor stiffness is sufficient to enable stable oscillation close to the surface. 78 While the ideal amplitude for an optimal signal-to-noiseratio is A/λ ≈ 1.545, we still have to consider the influence of the choice of amplitude to the signal strength of the desired observable. For example, if the interaction is composed by a repulsive interaction with a decay length of 20 pm and an attractive interaction with a decay length of 50 pm, the ideal amplitude for maximizing SNR for the repulsive branch would be about 30 pm, while the optimal SNR in the attractive branch would result for A ≈ 75 pm. Even when probing the repulsive regime, using an amplitude even smaller than 30 pm can be beneficial to suppress more of the attractive contribution (see Fig. 1).

D. Dissipative forces
When the tip-sample forces vary between forward and backward traces during tip oscillation, the force field is no longer conservative. Generally, dissipative processes lead to an energy loss, while in some cases with nonzero tip-sample bias voltages, discharging effects can also cause an energy transfer from the tip-sample system to the cantilever. In this case, the amplitude feedback circuit needs to alter its drive signal. When the cantilever oscillates in a conservative force field, the drive amplitude for an optimally adjusted phase is given by where Q is the quality factor of the sensor. For a proper adjusted phase in the oscillation controller, the phase of the cantilever lags by φ = −π/2 with respect to the drive; i.e., for a drive signal given by the cantilever oscillates according to The purpose of the drive signal is to make up for energy that is lost by internal friction in the force sensor. This loss per oscillation cycle is given by If the tip-sample force is not conservative, the integral over one oscillation cycle (running from q = −A to q = A and back to q = −A) is nonzero, and the drive amplitude needs to adjust from A dri e to A drive given by

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The energy dissipation can then be extracted from A drive by [similar to Eq. (4) in Ref. 35 for the special case φ = π/2 as noted above] Monitoring A dri e thus allows us to record the dissipation as a function of position in addition to conservative forces.

E. Higher harmonics
The oscillation frequency is the main observable in FM-AFM. If the cantilever vibrates in a nonharmonic potential, a shift in frequency is not the only change in cantilever oscillation. In addition, we observe the emergence of higher harmonics a 2 , a 3 . . . in the cantilever oscillation with its main amplitude A = a 1 . Dürig 49 has found a highly elegant method to calculate the higher harmonics in the cantilever motion that links the higher harmonics to an expansion of F ts in terms of Chebyshev polynomials.
The cantilever motion is assumed to be periodic; therefore, it is expressed as a Fourier series with fundamental frequency f, The static deflection a 0 is the average tip-sample force divided by the stiffness of the cantilever, while a 1 is just the unperturbed oscillation amplitude A. For n > 1, the amplitudes a n are given by a n = 2 πk Because f ≈ f 0 and a 1 = A, a n = 2 πk In 2004, we 112 realized that integration by parts allows us to express the higher harmonics in a more intuitive way, a n = 2 πk The nth harmonic is a convolution of the nth force gradient with a weight function with a sharpness that increases with n. Figure 6 shows these weight functions. For exponential force laws, explicit solutions are available. With F(z) = F 0 exp(−κz), we find where I n (z) is a modified Bessel function of the first kind. 1 The results of these calculations are also applicable for amplitude modulation AFM (Ref. 20). Because of the link of higher harmonics to higher force gradients, the method is well FIG. 6. Weight functions n (q /A) to derive frequency shift ∆f (semicircle) and higher harmonics a n (more and more bell-shaped with larger n). The frequency shift of an oscillating cantilever is given by a convolution of the force gradient k ts = −dF ts /dz with the semicircular weight function w 1 (u) = (1 − u 2 ) 1/2 [see Eq. (5)], while the nth harmonic a n is given by a convolution of the nth derivative of the force d n F ts /dz n with the corresponding weight function w n (u) = (1 − u 2 ) n−1/2 [see Eq. (47)]. The horizontal axis displays the ratio u = q /A between cantilever deflection q and amplitude A.
suited for very high resolution. However, even for small distances with strong interactions, higher harmonics are often very small on the order of 100 fm, although piezoelectric detection as present in qPlus sensors increases the magnitude of higher harmonics. 91

F. Simultaneous work function measurements
The tunneling current follows an exponential distance law where the decay constant κ is linked to the work function Φ, with electron mass m e . When performing simultaneous STM and AFM, the oscillating tip of course modulates the tunneling current and the extent of the modulation allows information about the work function. 115,124 III. THE qPLUS SENSOR AND ITS OBJECTIVE TO EMBODY AN IDEAL SENSOR FOR FM-AFM As FM-AFM utilizes frequency to measure force gradients, the frequency stability of the sensor is critical. It is worthwhile to study the frequency standards that have been developed in the art and science of watchmaking. Quartz oscillators come in various geometries, mainly determined by the oscillation frequencies. Quartz watches typically use tuning forks that oscillate at a frequency of 2 15 Hz [see Figs. 7(b) and 8(a)], and tuning fork geometries are available in eigenfrequencies up to several 100 kHz, although some use a second flexural mode to reach high Q factors for frequencies above 100 kHz. For higher frequencies, length extensional resonators [LER, see Fig. 8 higher frequencies in the range up to 100 MHz and higher are using plates or disks that vibrate in a shear mode (not shown here). The quartz watch is a promising candidate to check if its oscillator core can be transformed to a useful sensor in AFM. A quartz tuning fork not only keeps time at an excellent precision but also provides an electrical deflection signal, uses little power, and therefore does not produce large heat loads for low temperature microscopes.
A key design element of a quartz tuning fork is its high symmetry, where two prongs oscillate opposite to each other. These two prongs form two coupled oscillators. In an AFM, we only have one tip, so the attachment of a tip breaks the symmetry. While the mass of the tip can be compensated, the tip-sample force gradient would only act on one of the two coupled oscillators, leading to a drastic reduction in the Q factor, beating modes and other problems. In the first designs of the qPlus sensor, one of the prongs was attached to a very heavy substrate such that only a quartz cantilever and thus a single cantilever remained. Thus it is more  Table I Fig. 11) are connected, as are the side electrodes (blue, named B in Fig. 11). (c) shows the distribution of the electric field when the beam is deflected. (d) shows the strain distribution of the deflected beam. The needle sensor (e) is built by attaching a light tip to one prong of the length extensional resonator and has a longitudinal vibrational eigenmode. Because of the longitudinal mode, the cross section of the beam (f) develops an almost uniform electric field distribution (g) with uniform strain distribution (h). [(i)-(l)] show a combination of the standard qPlus sensor and a LER sensor. The top and bottom electrodes shown in ( j) are split into two with electrode A (red), electrode A (green), and electrodes B (blue). This design allows us to excite two different modes, a bending mode in the z-direction that is read out by measuring the difference between the charges generated by electrodes A + A and B, and a longitudinal mode in the x-direction that is measured by the difference between the charges generated by electrodes A and A .

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scitation.org/journal/rsi accurate to find this new name "qPlus sensor" rather than calling it "tuning fork sensor" (see also the discussion in Ref. 92 regarding coupled and single oscillators). The detrimental impact of mass imbalance on the quality factor of tuning forks has been studied in Ref. 204. Due to their simple use, quartz tuning forks were adopted early in scanning probe microscopy. Dransfeld et al. 45,99 used quartz tuning forks as a sensor in near field acoustic microscopy in 1989, and in 1993, Bartzke et al. transformed a 1 MHz quartz length extensional resonator into the so-called needle sensor by attaching a tip to one of the extensional beams. 13,14 Tuning forks were also successfully used in scanning near field optical microscopy by Karrai et al. 136,137 Quartz tuning forks have been used successfully as sensors in AFM using normal forces in the group of Siria. 31 The group of Jhe studied the influence of a liquid layer in ambient conditions manifested in shear force interactions, 8,143,153 as well as the wetting mechanism of titania 152 using quartz tuning fork sensors. Figure 8 shows the geometry, the electrode placement, and the electric field in the oscillating quartz sensor for the qPlus geometry, the length extensional resonator, and a novel biaxial qPlus sensor that is currently in development. A standard qPlus sensor [see Fig. 8(a)] is created by attaching one of the prongs of the tuning fork to a substrate and attaching a tip to the other prong. The prong without displayed electrodes is fixed to a massive substrate (not shown here, see Fig. 10). The custom built qPlus sensors only have electrodes on their single beam (see Fig. 11). The arrangement of the electrodes in Fig. 8(b) becomes clear when looking at the stress distribution in the beam when it is bent downwards (C), where tensile stress (D) occurs in the upper half of the cross section and compressive stress in the lower half. Therefore, the electric field vector is pointing to opposite directions in the upper and lower halves of the cross section and the surface charges on the red electrodes on the top and bottom have an equal sign. Lateral electrodes are needed to collect the opposite charges.
Quartz tuning forks and quartz length extensional resonators consist of two coupled oscillators. The high Q value they reach is due to the perfect antisymmetric oscillation. The invention of the qPlus sensor transformed the two coupled oscillators that build up a tuning fork to one single high Q oscillator. The analysis of the first successful noncontact AFM experiment of 1995 71 published in 1997 78 made it clear that small amplitudes and stiff cantilevers were needed. It turned out that the spring constants of the quartz tuning forks used in Swatch watches (see Fig. 7) were very close to the ideal stiffness that was wanted for an atom-resolving cantilever. 80 The first qPlus sensors were built using tuning forks and immobilizing one of the beams by attaching it to a heavy substrate; later custom designed versions have only a single oscillating beam from the start.
A. Calculation of stiffness, eigenfrequency, and sensitivity

qPlus sensor
For a rectangular beam with width , thickness th, and length L, the spring constant k is given by 33 where E Y is Young's modulus. For a harmonic oscillator, the eigenfrequency is given by f 0 = k/m * , where m * is the effective mass. For a beam with length L, width , height th (see Fig. 8), a constant cross section, and constant mass density ρ, the effective mass is given by m * = 0.243 m = 0.243 · ρ · · th · L; 33 thus, the fundamental eigenfrequency f 0 is given by where s is the speed of sound in quartz given by v s = E Y /ρ. The calculation of the sensitivity has been performed for the qPlus sensor in Ref. 81 and for the needle sensor (LER) in Ref. 92, and we reproduce the calculation that leads to the sensitivities from Refs. 81 and 92 in the remainder of this subsection. When a force F = kz is acting on the upper prong, the strain at the upper side is given by where E Y is Young's modulus and J is the moment of inertia J = · th 3 /12. This strain causes a stress σ mech = E which leads to a surface charge density σ charge = σ mech d 21 , where d 21 is the piezoelectric coupling constant (d 21 = 2.31 · 10 −12 C/N for quartz). 33 The lower side of the bent prong also has a charge density σ charge (both and the z-component of the surface normal vector have opposite signs) and contributes an equal amount to the total charge q. Integrating σ charge from x = 0 to x = L e (=length of the electrodes) and y = − /2 to /2 yields q/z = 12 · d 21 · k · L e (L − L e /2)/th 2 .
With z = A cos(2πfτ) and L e ≈ 1.6 mm, the expected output voltage per deflection is S ≈ 2πf · R · 2.8 µC/m, and with f = 25 800 Hz and R = 100 MΩ, we find S theory V ≈ 45 µV/pm, Standard qPlus sensors with the dimensions listed in Table I yield S theory q qPlus = 2.8 µC/m. It is important to note that the calculated sensitivity is based on an idealized homogeneous field distribution, as shown in Fig. 3(e) of Ref. 92, while the actual field looks more like Fig. 8(c). In practice, the actual sensitivity typically only reaches about half of this theoretical value. 255

Needle sensor (length extensional resonator)
The needle sensor consists of two coupled beams that oscillate opposite to each other [see Fig. 8(e)]. The longitudinal stiffness of k of each of the two bars that constitute the needle sensor is given by with Young's modulus E, length L, width , and thickness th of each quartz beam. The fundamental eigenmode is a

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REVIEW scitation.org/journal/rsi 2πf 0 /k of the quartz oscillators used. The qPlus sensor has a single beam with length L, width , and thickness (height) th, while the needle sensor features two coupled beams that oscillate in a longitudinal direction opposite to each other (see Fig. 8). Note that the experimental sensitivity reaches typically only 50% of the theoretical value listed above for qPlus sensors, while it matches the theoretical value for the needle sensor. longitudinal standing wave with a node at the root of each beam and its end at a maximal deflection; thus, the length of one beam L corresponds to a quarter wavelength λ/4. Because the velocity of sound is v s = E/ρ with mass density ρ, the eigenfrequency is given by To calculate the sensitivity of the needle sensor, we note that the deflection of a cross section at a distance z from the mount is given by when the ends of the device oscillate at amplitude A. The strain as a function of z-position is then given by The strain leads to a mechanical stress σ mech given by The piezoelectric effect causes the emergence of a surface charge density σ el given by where d 21 = 2.31 pC/N is the transverse piezoelectric coupling coefficient of quartz, 246 which is equal to the longitudinal piezoelectric coupling coefficient d 11 . It is important to note that d 21 is essentially constant over the temperature range from 1.5 K to room temperature. 246 When the charge density is integrated over the surface of the sensor, the total charge q el at a given deflection A is given by Thus, the sensitivity is given by With Eq. (56), we can express Eq. (63), The electrodes extend almost to the end of the beams (L e = 1.1 mm, L = 1.34 mm); therefore, the sine in the equation above is almost one (exact value 0.960 685 188), and with L/th = 1340/70, we find S theory q LER ≈ 19 · d 21 · k . With the stiffness k = 540 kN/m, we find a theoretical sensitivity of S theory q LER = 45 µC/m.

Biaxial qPlus sensor
Figure 8(i) shows the geometry of a biaxial qPlus sensor, where the conventional axis is mapping the z-direction and the longitudinal axis covers the x-direction of the tip-sample interaction. The stiffnesses, eigenfrequencies, and sensitivities can be calculated using the formulas for the normal qPlus and LER sensors. The only differences are that (a) needle sensors use two coupled beams with stiffness k (540 kN/m for the dimensions shown in Table I) where the effective stiffness is k = 2k and for the biaxial qPlus sensor, the stiffness in the xdirection is k x = k and (b) the sensitivity of the biaxial qPlus sensor in the x-direction is only half of that of the needle sensor because it only uses one beam. Currently, this sensor is in development, and results are in preparation. The design of this new sensor is described here to motivate the discussion of the properties of the needle sensor here as these are needed for the x-axis of the biaxial qPlus sensor.

B. Sensor and its fabrication
To build a qPlus sensor from a quartz tuning fork, we need a heavy substrate to fix one of the two tines and a tip needs to be mounted on the free prong. The first generation of qPlus sensors was built by attaching one leg of an unspecified

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scitation.org/journal/rsi tuning fork as used in quartz wristwatches to a small piece of Pyrex glass (Fig. 9). The second generation already featured the tuning forks of the Swatch brand as well as a custom made ceramic substrate 32 with the electrical connections shown in Fig. 10. The fabrication of sensors starts with the assembly of the quartz tuning fork with a rigid insulating substrate, as shown in Fig. 10. Both conductive and nonconductive glues are used in assembly. In 2011, newer versions of the qPlus sensor were invented. [74][75][76] These sensors only have a single quartz beam and provide separate electrodes for STM current, excitation, and other purposes. The third generation of qPlus sensors (Fig. 11) has an integrated separate electrode to bias the tip electrically without having to route an extra wire. We also switched to a ceramic substrate with a rectangular shape (as used in the lateral force sensor of Fig. 13) and smoother edges than those used in generation two. 11 The fourth generation of qPlus sensors (Fig. 12) has a total of four electrodes: two for differential deflection detection, one to bias the tip, and one for deflection excitation.
To be able to measure lateral forces, 85,260-262 the sensor can be rotated by 90 • , as shown in Fig. 13. In this case, the tip is aligned with the beam of the qPlus sensor. Table I shows the geometrical parameters of a variety of qPlus sensors along with stiffness, eigenfrequency, and sensitivities. The stiffness k is a very important parameter for extracting force data from experimental frequency shift. The connection between the geometrical parameters presented in Eq. (51) is linear with sensor width , cubic with thickness th, and inverse cubic with length L. Therefore, a 3% error in measuring the linear dimensions of the sensor would result FIG. 9. First generation qPlus sensor. 79 A tuning fork extracted from an encapsulated tuning fork with a length of one prong of L = 3.0 mm, thickness t = 330 µm, and width = 120 µm was used, yielding a stiffness of k = 3143 N/m. One tine of the tuning fork was glued to a small piece of Pyrex glass, the Pyrex glass was glued to a washer made of stainless steel and mounted to the end of a piezoelectric tube scanner. 26 Reproduced with permission from F. J. Giessibl, Appl. Phys. Lett. 73, 3956 (1998). Copyright 1998 AIP Publishing LLC.

FIG. 10.
Second generation of qPlus sensors. 87 Here, type E158 tuning forks from the Swiss company Micro Crystal 163 were separated from a wafer (a) and glued onto a specially designed piece of aluminum oxide (CeramTec 32 ) with two electrical connections. (b) The tuning fork is glued to a substrate made of alumina (Al 2 O 3 ) that contains gold plated nickel electrodes to provide for electrical contacts. The tuning fork is glued to the substrate with a nonconductive glue EPO-TEK T7110 or EPO-TEK H70E. 54 (c) Third step of assembly: a tip is added to the free prong of the tuning fork, and the contact leads of the substrate are connected to the electrodes of the tuning fork using a conductive glue EPO-TEK E4110 or EPO-TEK EC101. 54 The electrical contact between the tip and the electrode is also done using EPO-TEK E4110. Here, we show three different tips: an etched tungsten tip on the left, a splinter of silicon in the center, and a piece of iridium that was chipped off a 250 µm wire using a wire cutter. Reproduced with permission from Giessibl  in a 21% error of the stiffness k. Sader et al. 205 proposed to express the stiffness in terms of mass density, volume, and eigenfrequency, FIG. 11. Third generation of custom built qPlus sensors (Type M5B in Table I).
This sensor has top and bottom electrodes A and side electrodes B for differential deflection measurement as well as a center electrode C to bias the tip for STM operation and (optional) to excite the sensor oscillation.

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REVIEW scitation.org/journal/rsi FIG. 12. Forth generation qPlus sensor with a total of four electrodes (Type S1.0B in Table I where ρ is the mass density of the cantilever material. The eigenfrequency of the sensor can be measured with outstanding accuracy, and only three geometrical parameters in linear power remain (length L, width , and thickness th). If again each geometrical parameter is measured with an accuracy of 3%, the accuracy of the stiffness of the sensor is now precise

FIG. 13.
Sensor for lateral force microscopy based on a third generation qPlus sensor (Type S1.0 in Table I) and a rectangular ceramic substrate. 11 The tip is aligned with the quartz beam, and the sensor is rotated by 90 • to measure lateral forces. 263 to 9%. Of course, the eigenfrequency needs to be measured before mounting the tip. Also, one could take the mass of the gold coating into account. Typically, the electrodes have a 500 nm thick gold coating on a much thinner adhesive layer of, e.g., chromium-as the density of gold is about 8 times larger than the density of quartz, the mass of the sensor beam might increase another percent or so. Experimental measurements of k for the E158 type tuning forks listed in Table I show a stiffness of k = 1900 N/m with respect to the end of the beam, 162 and finite element analysis came to a similar result. 37 However, given that we usually mount a tip with a wire diameter of d ≈ 100 µm to the end of the quartz beam with length L, this stiffness has to be corrected by a factor L 3 /(L + d/2) 3 and a value of k = 1800 N/m is a very good estimate.

C. The probe tip of the sensor
Traditional silicon cantilevers are only about 0.2 mm long, and it is not feasible to attach a tip by hand. The large size of the qPlus sensor allows us to attach a variety of tips to them. Figure 10(c) already shows three examples where tungsten, silicon, and iridium tips are attached to the end of a qPlus sensor with conductive glue such that they can be used for STM and AFM simultaneously. The rigid properties of the qPlus sensor also allow us to mount single crystal tips from a cleavable material such as NiO and cleave the tips in situ 268 or clean by applying large voltage pulses. 119 Metal tips can be characterized at the atomic level with a technique called COFI (Carbon Monoxide Front Atom Identification). In this method, the front end of the tip is probed by a CO molecule that is bonded, e.g., to a Cu(111) surface, where it stands upright and exposes the O atom to the tip. The CO molecule probes the tip structure and allows us to count the number of tip atoms (Ref. 257 and correction in Ref. 53) and even to determine, within limits, the chemical species of the tip's front atom. 121 Gross et al. 95 found that the adsorption of a CO molecule on a metal tip increases the spatial resolution of AFM for organic molecules dramatically. The CO molecule adsorbs on the metal tip with the C atom bonded to the metal, exposing the O atom as the front atom of the tip. These tips provide atomic resolution of organic molecules, and they even image metallic clusters at unprecedented resolution. 53 Mohn et al. 166 discovered that other inert tip terminations such as Br, Cl, and Xe also lead to enhanced contrast. Subsequent studies 16,164 found that CuO tips offer a similar spatial resolution without the aberrations caused by lateral deflections of the CO molecule. Temirov et al. 235 found that STM with a trapped H 2 molecule in the junction provides a spatial resolution similar to AFM with CO terminated tips.
Regarding the length of tips, we recommend to keep them as short as possible as the longer they are, the more lateral motion is added to the normal oscillation. 162,243

D. Amplifier and wiring
For achieving low detector noise, the sensor needs to produce a large ac-current when oscillating at a given amplitude.

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The amplifier needs to transform this ac-current into an acvoltage with as little noise as possible. Thus, the combination of sensor and amplifier determines the detector noise. The sensor produces a certain charge per deflection, called calculated charge sensitivity S theory q that is measured in Coulombs per meter. When the sensor oscillates, this charge turns into an ac current, yielding a current sensitivity S The amplifier transforms this current into a voltage, yielding a voltage sensitivity of S = R · S theory I = R · 2πf · S theory q , where R is the gain of the amplifier.
In the first publication of the qPlus sensor in 1998, a highgain instrumentation amplifier (Analog Devices 9 AD624 wired to a gain of 1000) was used. 79 Atomic resolution with the qPlus sensor was achieved by choosing a current-to-voltage converter 81 based on a 100 MΩ feedback SMD (surface mounted device) resistor and an AD744 operational amplifier chip. 9 Performance was further enhanced by using the AD823 operational amplifier chip 9 with a lower quiescent current. Today, we use a differential amplifier setup based on a schematic described in Refs. 133 and 144 with 10 GΩ SMD feedback resistors and an AD8616 operational amplifier chip, 9 and the complete setup and assembly are explained in Ref. 125 and Fig. 14. FIG. 14. Low noise and low power qPlus amplifier providing a differential deflection measurement. The amplifier is mounted directly onto a 4 K stage that minimizes the Johnson noise of the resistors, and the AD8616 operational amplifier is held by its stainless steel electrical wires to minimize thermal conduction. The length and width of the wires are chosen such that the internal heat dissipation of the amplifier leads to a temperature of the chip of about 80 K. 125  It is important to note that the distance between the sensor and the amplifier should be as small as practical. To be more precise, the capacity of the cable that connects the sensor and the amplifier should have a capacity as low as possible. The reason for this is that amplifiers have a certain input voltage noise and, when the input connects to a capacitive load, that voltage noise turns into a current noise, see Ref. 92. The capacity of a coaxial wire with length l, radius of inner wire r i , and radius of outer shield r a is given by where 0 = 8.85 pF/m is the permittivity of vacuum and r is the relative dielectric constant of the insulation material used in the wire. In room temperature and 4 K instruments, the distance between the sensor and the amplifier is often only a few centimeters and the connection between the sensor and the amplifier can be done by two wires that are as thin as practical (possibly only 50 µm) and spaced by a few centimeters, resulting in a capacity of only 1 pF or so. At ultralow temperatures, the distance between the amplifier and the sensor needs to be larger as outlined in the last chapter.

E. Sensor excitation
The simplest excitation scheme uses the xyz-scan piezo such as a tube scanner with a modulated excitation signal added to the z-high voltage signal. Some tube scanners do not bring the four lateral electrodes all the way to the end but FIG. 15. Schematic for driving a qPlus sensor of generation 3 or 4. The tip is connected through a separate wire to the bias voltage. For generation 3 sensors, a transducer (typical gain 100/1-1000/1) adds a small excitation voltage to the bias. For a Q of 100 000, a drive voltage of 10 µV (10 mV-100 mV before the transducer) is sufficient to result in an oscillation amplitude of 100 pm. For generation 4 sensors, the drive voltage connects directly to terminal D, and the sample bias connects to terminal C of the sensor. The deflection measurement is performed with a differential amplifier that is connected to deflection electrodes A and B (see Fig. 14). The drive voltage V dri e usually couples to the deflection electrodes through parasitic capacitive coupling, but as the deflection is measured differentially, an equal coupling of the drive voltage to electrodes A and B is easily achieved by appropriate routing of the wires. The positive and negative power supply voltages for the deflection amplifier are batteries that are referenced to the bias voltage V bias . Thus, all three (or four) sensor electrodes are at a potential very close to V bias , and the dc offset of the deflection amplifier of V bias is subtracted by a differential amplifier at the output to result in a pure ac deflection output signal V deflection . The tunneling current I is measured at the sample, in our case mainly with a 100 MΩ gain such that V I = −100 MΩ · I.

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scitation.org/journal/rsi feature a small ring electrode at their end that is used for excitation. Other schemes use a separate dither piezo mounted onto the end of the xyz-scanner. The newest design of qPlus sensors integrates a separate excitation electrode on the quartz beam with a total of four electrodes, two for a differential excitation measurement, one for the STM current signal, and one for the excitation; see Fig. 12. A schematic that explains the wiring of an electrically excited qPlus sensor is displayed in Fig. 15.

IV. EXPERIMENTAL NOISE MEASUREMENTS OF QPLUS SENSORS
So far, we have only considered theoretical calculations to compare the noise characteristics of qPlus sensors and needle sensors (length extensional resonators or LER) studied here. Now, we supplement the calculations by measurements. Some of the following explanations and text are adopted from an earlier work that explicitly compares qPlus and needle sensors. 92

A. Thermal noise peak
The measurement of the thermal noise peak is a comparatively simple and highly informative measurement. It shows how well sensor and amplifier perform and yields very important data such as the eigenfrequency f 0 of the sensor, its quality factor Q, the experimental voltage sensitivity S exp v , and the deflection noise density n q . First, we measure the thermal noise peak of the qPlus sensor with sensors of standard dimensions as shown in Fig. 16. The equipartition theorem states that each degree of freedom of an oscillator carries a thermal energy k B T/2, where k B is Boltzmann's constant and T is the temperature in Kelvin. For a one dimensional oscillator like the standard qPlus sensor, we find the thermal amplitude by equating the average potential energy to the thermal energy, yielding At room temperature, we find a thermal rms-amplitude of A rms = 1.52 pm or peak amplitude of A 0 p = 2.14 pm for k = 1800 N/m. In contrast to a qPlus sensor, a tuning fork has two coupled beams that oscillate in a common antiparallel mode such that 2 · kA 2 rms /2 = k B T/2, yielding a thermal rms-amplitude of A rms = 1.07 pm or peak amplitude of A 0 p = 1.52 pm. The needle sensor also is a coupled oscillator; therefore, 2 · k A 2 rms /2 = k B T/2, yielding a thermal rms-amplitude of A rms = 62 fm or peak amplitude of A 0 p = 88 fm.
The power spectral density in Fig. 16 was recorded by connecting the output of the amplifier to the input of an FFT (Fast Fourier Transform) analyzer (Agilent 35670A Dynamical Analyzer). The input of the amplifier was connected to a standard qPlus sensor of type S1.0 (no tip attached). From Fig. 16, we can calculate the sensitivity as well as the deflection detector noise density by following the procedure published in Ref. 81.
The noise of the deflection measurement depends on the sensor, the cable, and the amplifier. A standard qPlus sensor oscillates at 30 kHz and generates a charge of 1.44 µC/m. For the commercial Femto HQA-15M-10T charge amplifier, 60,148 we measured a noise density of n amp = 122 zC/ √ Hz when the input was connected to a 1 m coaxial cable with a capacitance of 100 pF and n amp = 86 zC/ √ Hz without cable. Thus, a standard qPlus sensor would yield n q = 122 zC/ √ Hz/1.44 µC/m = 85 fm/ √ Hz when connected with a cable with a capacity of 100 pF and n q = 62 fm/ √ Hz when connected directly to the amplifier. For our homebuilt amplifier based on the operational amplifier AD823, 9 the noise at 4 K typically drops to 50%, 110 yielding n q = 31 fm/ √ Hz at 4 K. Our newest design 125 has reached optimal levels down to n q = 16 fm/ √ Hz and standard levels of n q = 22 fm/ √ Hz at 4 K. These homebuilt amplifiers are vacuum and UHV compatible and therefore can FIG. 16. Thermal spectrum of a qPlus sensor without a tip and standard dimensions at room temperature and ambient pressure. A homebuilt preamplifier was used. The eigenfrequency is f 0 = 32 698 Hz, the Q-factor is 2900, the charge sensitivity of the sensor is S be connected closely to the sensor, thereby greatly reducing C cable . The power spectral density in Fig. 16 was recorded by connecting the output of a home-built UHV compatible amplifier to the input of the Nanonis OC4 PLL and recording its FFT (Fast Fourier Transform) output. The input of the amplifier was connected to a qPlus sensor without a tip with standard dimensions using a short cable with a length of approximately 0.1 m (capacity approximately 10 pF). The experimental result is S exp qPlus = 1.44 µC/m-about 51% of the theoretical value. The deviation between the theoretical and experimental values is probably due to edge effects-the calculation of the sensitivity is based on a homogeneous field distribution and an electrode configuration in the quartz crystal, while the actual field distribution is perturbed by edge effects [see Fig. 8 The validity of the equipartition theorem is verified by measuring the thermal spectrum as a function of temperature, as displayed in Fig. 17. 255 While the measurement of the thermal noise peak is simple, it needs to be used with care at low temperatures. As the thermal excitation becomes smaller with temperature, other excitation sources such as mechanical vibrations can rise above the thermal excitation, and it is advisable to use a different method for amplitude calibration at low temperatures.
B. Comparison of the four theoretical noise sources to experimental qPlus results

Thermal noise
Thermal noise depends on three sensor properties: eigenfrequency f 0 , stiffness k, and quality factor Q. In summary, thermal noise is small for cantilevers that are as soft as possible while stiff enough to prevent jump-to-contact at small amplitudes and that have a high eigenfrequency and a high quality factor, For the qPlus sensor, Q ≈ 3000 at room temperature, reaching up to 200 000 at 4 K 120 with type qPlus S1.0 or even more than one million 190 with type qPlus M4 (see Table I

Detector noise
In summary, we find that using a very large spring constant for sensors with piezoelectric detection is not a significant disadvantage regarding deflection detector noise because although the frequency shift that a sensor is subject to is proportional to 1/k, the sensitivity is proportional to k, and the two effects cancel.
For quartz sensors, the deflection noise depends on the charge that is generated per deflection and the gain and noise of the preamplifier. Current-to-voltage converters convert the current provided by the quartz sensor to a voltage. However, the frequency response of the current-to-voltage converter is not independent of frequency but is given by where R is the resistance of the feedback resistor and C is its parasitic capacitance. The lower line in Fig. 19 shows the theoretical frequency response of an ideal operational amplifier with R = 100 MΩ and a parasitic capacitance of C = 0.2 pF. The gain is flat for frequencies smaller than the corner frequency f c1 = 1/(2πRC) = 7.96 kHz. For f f c1 , the gain is given by V out ≈ −I/(i2πfC)-inversely proportional to f. A sinusoidally varying charge Q ch = Q 0 exp(i2πft) corresponds to a current I =Q ch = Q 0 i2πf exp (i2πft); thus, the gain can be expressed as V out = −Q ch /C. Therefore, this amplifier is called a "charge amplifier" for frequencies significantly larger than f c1 . Simple amplifiers as the one shown in Fig. 18 often display a second corner frequency f c2 not very much higher than f c1 and for frequencies beyond f c2 the gain decays proportional to 1/f 2 . A popular commercial amplifier 60 that is used here for the needle sensor has an f c2 at around 15 MHz and is therefore suited well for high-frequency sensors.
The question is now when is it advisable to use a currentto-voltage converter and when is it favorable to use a charge amplifier. Figure 19 shows that the current-to-voltage converter operates similar to a charge amplifier for sufficiently large frequencies, i.e., it does not provide a linear relationship between current and output voltage independent of frequency, but its gain rolls off as 1/f with increasing frequency f. While one can increase f c1 by reducing the value of the feedback resistor R, a reduction in R increases the current noise. The tradeoff between noise and bandwidth leads to an optimal amplifier type for a given operating frequency. Indeed, we found out that our home-built current-to-voltage converter has a better signal-to-noise ratio for frequencies between

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scitation.org/journal/rsi FIG. 18. Schematic of a quartz sensor, cable, and current-to-voltage converter that is often used for amplifying deflection data from quartz sensors. 92 The gain of the amplifier is given by where V out is given by a complex number to indicate the phase shift. Typically, amplifiers of qPlus sensors are operated far beyond f c1 , i.e., f /f c1 1, so they essentially act as charge amplifiers. The capacity of the cable should be as low as possible-cable capacity increases noise in the amplifier output. If the amplifier is vacuum compatible, it can be placed close to the sensor, thus reducing cable capacity and noise. The sensor can be excited electrically as shown in this figure or mechanically-the drive signal is grounded in this case. 60 kHz and 100 kHz, while the FEMTO amplifier 60 works better for higher frequencies. 38 For charge amplifiers, the deflection detector noise density can be expressed by where n amp is the noise density of the preamplifier and S q is the charge sensitivity (charge per deflection) as calculated for the needle sensor in Eq. (64) and for the qPlus sensor in Eq. (55). With the calculations in the previous chapter, we find This equation shows that the deflection detector noise is small for small spring constants, small amplifier noise, large sensitivity, and large eigenfrequency. Thus, the figure of merit for the sensor is not how much charge per deflection it generates (expressed by S q ) but the charge per force given by S q /k when a charge amplifier is used. If a current amplifier with sufficient bandwidth is available, the figure of merit is the current that the sensor generates per force, expressed as S q f 0 /k. We can also express this in noise density, For both needle and qPlus sensors, the sensitivity is proportional to k. We find for the needle sensor that δk ts needle sensor = 8 3 for the ideal case of L e = L. For the qPlus sensor, we find δk ts qPlus sensor = 8 3 again assuming the ideal case of L e = L. Thus, deflection detector noise depends on the properties of the sensor and the amplifier. If we assume a charge noise density of n amp = 90 zC/ √ Hz (such as achieved by the commercial FEMTO amplifier 60 when loaded with a 1 m coaxial cable corresponding to a 100 pF cable capacitance), we can now calculate an explicit number for the deflection detector noise contribution to the force gradient noise with A = 100 pm and the geometrical values after Table I and for the qPlus sensor, we find a theoretical deflection detector noise contribution of δk ts qPlus sensor = 25.7 µN/m B 3/2 For a bandwidth of 100 Hz, the theoretical deflection detector noise contribution is thus 33.2 mN/m for the needle sensor and 25.7 mN/m for the qPlus sensor. However, we have based this calculation on the theoretical sensitivity of the sensors, and we found out experimentally that while the sensitivity of the needle sensor matches theory, the qPlus sensor develops only about 50% of the theoretical sensitivity. As deflection detector noise depends dramatically on bandwidth, it can be reduced substantially by bandwidth reduction. At low temperatures, where slow scanning is possible, the bandwidth can be reduced to 1 Hz or less and tiny force gradients can be detected in this case. For a bandwidth of 1 Hz, the deflection detector noise contribution is thus 33.2 µN/m for the needle sensor and 25.7 µN/m for the qPlus sensor. However, at very low bandwidth, the remaining three noise sources are typically much larger than the deflection noise.

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Oscillator noise for quartz sensors
As oscillator noise is white, it is hard to distinguish it from thermal noise. For the Q values from above, we find room temperature values of δk ts osc = 4.6 (mN/m)/ √ Hz for the needle sensor and δk ts thermal = 0.6 (mN/m)/ √ Hz for the qPlus sensor. At T = 4 K, the contribution of oscillator noise to the minimal detectable force gradient is δk ts osc = 1.4 (mN/m)/ √ Hz for the needle sensor and δk ts thermal = 9.5 (µN/m)/ √ Hz for the qPlus sensor. Again, these calculations refer to A = 100 pm.

Frequency drift noise for quartz sensors
Temperature variations cause a drift in eigenfrequency. For silicon cantilevers, the relative frequency variation is linear with temperature with a value of −35 ppm/K at room temperature. 100 Thus, a hypothetical Si cantilever with k = 1 kN/m (this large stiffness would be required to enable stable oscillation at small amplitudes) would be subject to a k ts drift of Quartz sensors show a quadratic frequency shift with temperature, and the eigenfrequency varies with temperature as an inverted parabola centered around the turnover temperature T p , 163 The turnover frequency depends on the crystal cut (see Fig. 9 in Ref. 170). Tuning fork crystals are often cut to yield a turnover temperature T p = 298 K, as this is close to the temperature of a watch in thermal equilibrium with a wrist. Length-extensional-resonators, by contrast, are often oriented such that their turnover temperature is around 313 K, 163 probably because 1 MHz crystals are typically not worn on the wrist but built into printed circuit boards that have higher operating temperatures than the human body. This thermal frequency drift causes a thermal drift in force gradient measurement given by Although the temperature stability of quartz is excellent with very small values of χ = 35 · 10 −9 K −2 , 163 the net effect on the precision on the measurement of k ts is proportional to the effective stiffness of the sensor k, and drift noise can be a challenge for very stiff sensors such as the needle sensor. The quadratic dependence of the frequency variation with temperature is only valid for temperatures around T p . For the temperature range from 300 K to 4 K, the frequency variation has been measured by Hembacher et al. 110 and fits well to a semi-empirical relation given by with a total relative frequency change of −1620 ppm over the temperature range from 300 K to 4 K (see Fig. 20)  Although the relative frequency shift is much smaller for quartz sensors than for silicon cantilevers, the effect on the measured force gradient scales with stiffness k.
between room temperature and absolute zero. However, while this approximate formula models the data measured by Hembacher et al. 110 quite precisely down to liquid helium temperatures, a detailed measurement of the eigenfrequency of several types of quartz sensors has shown that the frequency shows a minimum around 23 K and an increase as the temperature decreases to 4 K at a rate of approximately −7.5 ppm/K at 4 K. Figure 21 shows the dependence of the  We now analyze the effect of temperature drift on the measured tip-sample force gradient. First, we look at the frequency drift of the sensor for a given rate of temperature change. Figure 22(a) shows temperature versus time for a constant drift rate of dT/dt = 125 µK/s at T − T p = 10 K over a time interval of 10 minutes. The frequencies of quartz sensors vary according to Eq. (77) by a rate r ns = 100 µHz/s for the needle sensor and r qP = 3.3 µHz/s for the qPlus sensor.

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Because the frequency drift rate is proportional to f 0 , the force gradient noise due to thermal drift is proportional to the stiffness of the sensor k, and thus this noise source is 600 times larger for the needle sensor than for the qPlus sensor.
To obtain the variation in the measured tip-sample force gradient k ts , the relative frequency shift has to be multiplied by 2k. For the qPlus sensor, we obtain a k ts drift of −27 (µN/m)/mK, and for the needle sensor, we get −16 (mN/m)/mK. Because variations in ambient pressure cause a slight change in boiling temperature of the helium bath, we expect that temperature drift can become a significant issue for the needle sensor even at low temperatures.
We can estimate the influence of temperature changes at 4.2 K by noting that a change in ambient pressure of 1 hPa results in a change in boiling temperature of helium of about 1 mK. The change in sensor frequency is then about 7.5 ppb, i.e., a frequency change of −0.25 mHz. This is still small compared to the expected frequency stability of the PLL detector.  Even oven-controlled quartz oscillator as used in Nanonis control systems 173 have typical frequency errors on the order of 20 ppb. Generally, frequency drift noise is low, even more so at low temperatures. Frequency drift noise can be minimized by using a PLL with an oven controlled quartz time base and by allowing it enough time to settle between powering on of the PLL and starting the measurement.
C. Comparison between experimental and theoretical noise data Figure 23 shows the calculated (smooth lines) and experimental ( jagged lines) power spectral density of the force gradient noise n kts as a function of modulation frequency f mod . This graph is produced by inserting the output of the phaselocked-loop detector to a FFT analyzer (Agilent) and multiplying the frequency shift by the corresponding scaling factor (k ts = 2k/f 0 · ∆f, thus n kts = 2k/f 0 · n ∆f ). The total noise of the force gradient measurement is given by The calculated graphs include deflection detector noise (linear with f mod ), thermal noise (constant with f mod ), and oscillator noise (also constant with f mod ). Frequency drift noise, which is large for long measuring times (i.e., small f mod ), is not included in the calculation but clearly apparent in the measurement by the increase of the experimental needle deflection detector noise density for small f mod . The increase in noise for small f mod in the needle sensor data is related to the strong influence of thermal frequency drift. A random walk in temperature would lead to a random walk in frequency, and the Fourier transform of a random walk function leads to a 1/f power distribution. 245 Table II lists the individual noise contributions and typical parameters for qPlus sensors and needle sensors at ambient conditions, UHV, and low temperatures. Figure 24 displays the four noise sources and their contributions to the frequency spectrum of the PLL output for typical qPlus sensors at liquid helium temperature. Thermal noise and oscillator noise are constant with respect to frequency. Due to the high Q value, oscillator noise is negligible here-even smaller than thermal noise. However, although   (20) connects the slope of the experimental n ∆f curve above to n q , yielding n q ≈ 50 fm/ √ Hz. Note that the linear increase in noise with frequency rolls off at the bandwidth setting of the PLL detector.

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thermal noise is proportional to (T/Q) 0.5 , thermal noise (red) is noticeable at a density of about 1 mHz/ √ Hz. Drift noise is apparent at very low frequencies-the noise density reaches a minimum at about 0.75 Hz, showing that noise could be decreased until the time per pixel reaches 1/0.75 s corresponding to an acquisition time of more than 1 h for a 64 × 64 pixel image. For even lower scanning speeds, noise would go up again. Detector noise is clearly dominant and, in theory, increases linearly forever with higher modulation frequencies.
In practice, the finite bandwidth of the PLL detector sets an end to that increase. Figure 25 shows an experimental noise spectrum (see the figure caption for sensor parameters). The linear increase in noise starts to roll off at about a frequency of 50 Hz, corresponding to the bandwidth setting of the PLL detector that provides a low pass filter of at least second order that turns the linear increase of noise with f mod to a decaying noise for frequencies above the bandwidth setpoint.

D. Identification of sensor noise in an experimental FM-AFM image
The four noise sources that have been identified above can be seen directly when performing a Fourier transformation of the frequency shift signal. Thus, one either needs a FFT Analyzer or a control software that allows us to display the frequency shift signal in frequency space. Interestingly, one can also partially distinguish the noise sources in postprocessing of the images. When looking at a constant-height image of a flat and periodic surface such as shown in Fig. 26(a), we use imaging processing software such as WSXM 122 to create a FFT (Fast Fourier Transform) image in (b). Figure 26(b) essentially shows six peaks that correspond to the reciprocal unit cell vectors of the CaF 2 lattice imaged in (a). Figure (c) shows a Gaussian filtered image, using a 10 × 10 Gauss filter. The original image had an average frequency shift of −6.5 Hz with a corrugation of ±2.1 Hz, and the corrugation is reduced to ±1.25 Hz due to Gauss filtering. Figure 26

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scitation.org/journal/rsi we see only noise in Fig. 26(d) because the expected 20 mHz corrugation is masked by noise. The FFT image in Fig. 26(e) shows that this noise is mainly due to detector noise, as the detector noise density increases linearly with frequency [see Eq. (72)]. The V-shaped FFT spectrum shows that detector noise strongly dominates overall noise. As detector noise rises sharply with frequency, it can be curtailed by bandwidth reduction. The region around the center of the FFT spectrum in Fig. 26(b) still shows the six data peaks that also appear in Fig. 26(b), although they are more than a factor of 100 higher in (b). In principle, the V-shaped detector noise floor is also present in B. Low-pass filtering in the time domain can be performed after the data have been taken by low pass filtering in the spatial frequency domain as when using a Gaussian filter. Figure 26(f) is a double low pass filtered version of Fig. 26(d)it clearly shows the atomic structure similar to Fig. 26(c), yet at only about 1% of contrast as it is recorded 250 pm further away (actually, the contrast is even smaller due to applying a Gauss filter).

V. APPLICATIONS
The qPlus sensor has many applications, and today, hundreds of microscopes are in use that utilize this sensor. Therefore, this section provides some of the typical applications that have emerged so far in our group as well as other groups from various parts of the world. As this article is mainly concerned with instrumentation, the applications are structured by environment: ambient, ultrahigh vacuum (UHV), low temperature UHV, and ultralow temperature UHV. Incidentally, this is also more or less in line with the historic development.
A. Ambient and liquid environments

First results in ambient conditions
The first images using the qPlus sensor were obtained in ambient conditions on a test grating and a CD in intermittent contact mode (see Fig. 27). The forces that were used in these early images were relatively large-at least 30 nN repulsion or more, but the scanning speed was relatively fast, up to 100 µm/s. The oscillation amplitudes in these early experiments were very large-250 nm and needed for stability due to several reasons that will be discussed below (poor amplifier, analog amplitude control with little dynamic range). After these experiments, we moved to vacuum (Subsection V B 1) and obtained atomic resolution within about a year. We revisited the challenge of obtaining atomic resolution in ambient conditions using the qPlus sensor after a detailed understanding about the imaging contrast was obtained in controlled vacuum conditions. 267

Atomic resolution of KBr, calcite, and graphene
In ambient conditions, surfaces are continuously changing because of adsorption and desorption and they are usually covered with a film of adsorbates. This was the main reason, why it took a few years after great success in vacuum to obtain atomic resolution. Nevertheless, true atomic resolution in ambient conditions, where samples are usually covered by a contamination layer, as shown in Fig. 28, or immersed in liquid, is possible as shown by Iichii et al. 129 and Wastl et al. 252 These contamination layers are a severe challenge because they cause a strong damping of the cantilever, as shown in Fig. 29. For the wet sample, the drive signal had to increase from less than 1 mV when the oscillating tip is outside of the adsorption layer at z = 9 nm to about 40 mV at z = 0 nm for the immersed tip [see Fig. 29(d)], while for a dry sample, the drive only had to increase from about 1 mV at z = 2 nm to about 2.5 mV at z = 0 nm [see Fig. 29(e)]. The analog amplitude controllers we used in the early days of qPlus did not provide enough dynamic range to bridge these large changes in damping. That is why it took 13 years for us after having obtained atomic resolution in late 1999 in vacuum 81 to achieve the same feat in ambient environments. The bottom part of Fig. 29 shows the damping and frequency shift for a dried sample, where the drive signal only had to increase by a factor of three. Figure 30 shows atomically resolved data of KBr. The corresponding publication 252 also describes experiments where craters were made into KBr and the healing process was observed later. Other samples, where atomic resolution in ambient conditions was demonstrated, include graphene 253 and calcite. 254

Atomic resolution of lipid bilayers in thin liquid films
High-resolution imaging of soft biological samples with atomic force microscopy (AFM) is challenging because they must be imaged with small forces to prevent deformation. Typically, AFM of those samples is performed with soft silicon cantilevers (k ≈ 0.1-10 N/m) and optical detection in a liquid environment. Several complex biologically relevant solutions

Atomic resolution in bimodal AFM
As outlined in the second chapter, silicon cantilevers are not stiff enough to allow stable operation at small amplitudes. Garcia et al. 65 found a workaround: multifrequency AFM, and in particular bimodal AFM, where the cantilever is oscillated with a large amplitude at the fundamental mode and with a small amplitude at the simultaneously excited higher flexural mode. The advantage is that the higher flexural mode is more sensitive to the short range interaction.
We operated a qPlus sensor simultaneously in the fundamental mode 32(a) as well as in the second flexural mode with one node [see Fig. 32(b)]. 178 Although the stiffness of the sensor is about 40 times higher in the second flexural mode, we could obtain atomic resolution both in the fundamental mode [see Fig. 32(c)] and in the second flexural mode [see Fig. 32(d)]. However, we found that when both modes are excited, simultaneous atomic resolution in the fundamental and the higher flexural modes is only possible if the sum of the two amplitudes is approximately below 100 pm. As a consequence, the excitation of the fundamental mode reduces the signal-to-noise ratio in the higher flexural modes (and vice versa), and although bimodal AFM is a workaround to prevent jump-to-contact with soft cantilevers, a more straightforward approach is to use cantilevers with sufficient stiffness to prevent instabilities at small amplitudes right away.
However, there is a possible highly interesting application of bimodal AFM. If the mass of the tip is chosen appropriately, the node of the second flexural mode in Fig. 32(b) can move close to the tip position, and the motion of the tip is mainly lateral. This would allow a simultaneous detection of normal forces with the fundamental mode and lateral forces with the second flexural mode. Currently, we are investigating this mode and the future will tell if this approach is feasible.

Magnetic force microscopy (MFM)
Force gradients due to the magnetic interaction between the tip and the sample are very small compared to chemical bonding forces. To provide a large signal, i.e., a large frequency shift, the sensor should be as soft as possible to allow operation at the ideal amplitude without suffering from jump-tocontact. As the decay length of magnetic dipole forces is on the order of the domain size, and the ideal amplitude is on the order of the decay length, large amplitudes on the order of 50 nm are ideal in conventional magnetic force microscopy (MFM) and soft Si cantilevers are ideally suited for this. The qPlus sensor is not ideal for these types of measurements as the sensor should be as soft as possible while maintaining stability at the optimal oscillation amplitude. Nevertheless, Schneiderbauer et al. 210 could demonstrate magnetic resolution on a hard disk with milli-Hertz frequency resolution in Fig. 33. The advantage of using the qPlus sensor for MFM is that the same setup that performs the MFM measurement can be used to perform atomically resolved AFM-, STM-, and possibly even exchange force-measurements, once the domain structure has been measured by MFM. The transition from performing MFM to atomically resolved AFM would then merely require amplitude reduction from the ideal MFM value of about 50 nm to the ideal atomic value of about 50 pm.

Subatomic spatial resolution on silicon and CoSm
In 2000, only two years after the introduction of the qPlus sensor, we reported about the observation of subatomic features by force microscopy. 82 In this experiment (see Fig. 34), a Si covered tungsten tip mounted on a qPlus sensor imaged the Si(111)-(7 × 7) surface. Each adatom appeared as two crescents, interpreted as two dangling bonds. The tip and sample switched roles here, as the dangling bonds on the Si adatoms imaged the front atom of the tip just as the tip  imaged the surface. The data were explained using the angular dependent Stillinger-Weber potential. The claim of having achieved subatomic resolution raised lively discussions. 127 While the explanation of the data was confirmed by density functional theory, 123,270 the science of tip preparation in 2000 was not as sophisticated as today, where subatomic resolution with atomically characterized tips has been obtained, as discussed in Subsection V C 7. On a side note, the eminent visual artist Gerhard Richter used the orbital image of Si as an inspiration for his offset print "Erster Blick (First View)." 176,199 Figure 35 is another example where the adatoms of the Si(111)-(7 × 7) produce repeated subatomic images of the front atom of a Co 6 Fe 3 Sm tip. 114 Figure 36 shows the atomically resolved CaF 2 (111) surface, highlighting a total contrast inversion due to a tip changea common occurrence when imaging at close distance at room temperature. Figure 37 displays NiO(100) at room temperature-as the Neel temperature of NiO is significantly above room temperature, we tried to measure spin contrast due to exchange interaction-this would have produced a 2 × 1 superstructure. Both samples have been revisited using low temperature AFM: CaF 2 (111) was imaged over days with the same tip, and at very high precision 157 and spin contrast, measurements including force spectroscopy on Ni(100) were achieved as well. 190

Friction studies by lateral AFM on Si(111) and Si(100)
In 2002, we performed first frictional studies by lateral force microscopy using a laterally oriented qPlus sensor at room temperature and measuring the energy it takes to excite a Si adatom. 85 The dissipation data were explained using the Tomlinson-Prandtl model 194,240 of friction as a plucking action on single molecules. Figure 38 shows a refined experiment on a sample that shows a clear directional dependence of lateral stiffness: Si(100). 260 In Si (100), dimers form at the surface that changes their direction by 90 • when going over an atomic step.

Submolecular resolution of structure and work function on organic molecules
Submolecular resolution of organic molecules is now a standard practice at low temperatures, where CO terminated tips can be formed. At room temperature, those tips are not stable. Nevertheless, we did obtain high resolution STM, AFM, and decay constant images of semiconducting organic dye molecules (perylenetetracarboxylic dianhydride, or short PTCDA) in Ref. 124, see Fig. 39. While we cannot be sure about the structure and chemical identity of the tip, it seems likely that we picked up a PTCDA molecule with the tip and that this molecule was oriented such as to expose a corner that terminates with a CO molecule. Similar effects may have played a role in the work of Jarvis et al., 134 who provided atomically resolved images of NTCDI at T = 77 K with a qPlus sensor and a nominal metal tip that might also have been contaminated with the molecule it was imaging-NTCDI also is rectangular and exposes CO terminations at the corners.

C. Ultrahigh vacuum at liquid helium temperature (4 K)
Liquid helium temperatures are required to induce some phase transitions, e.g., to obtain a transition from the normal conducting to the superconducting state for classic superconductors. However, even when low temperatures are not required to cause a specific thermally induced state of the sample, low temperatures have two key advantages for high precision imaging: (a) tip stability and (b) the possibility of slow scanning to reduce noise by recording at low bandwidth B.
Tip stability is a function of temperature; tips usually change frequently at room temperature, but they can stay unchanged at 4 K for weeks of imaging. The CO tip termination that has been proven so successful in imaging organic matter is only stable for temperatures of liquid helium or slightly above.
When imaging at room temperatures, a relative drift between the tip and the sample induced by temperature gradients and thermal expansion normally requires a certain minimal scanning speed to limit the distorting effects of thermal drift. At low temperatures, distortions are close to zero, and the scanning speed and detection bandwidth of the PLL detector can be reduced to very low values. The all-electric deflection detection and the low power requirement of the amplifier have led to a wide utilization of the qPlus sensor at low temperatures. This subsection is organized in the following way: we start by describing the first steps of qPlus operation at liquid helium temperatures that were conducted at the University of Augsburg from 2002. Then, a description of collaborations with the low-temperature STM laboratories at IBM Almaden in San Jose, USA, and Rüschlikon, Switzerland, follows. In parallel, several companies adopted the qPlus sensor for their commercial low temperature scanning probe microscopes, leading to an increasing number of external results. The early Augsburg results will be followed by new experiments that were conducted after my transition to Regensburg in 2006. From 2010 on, several groups installed commercial 4 K qPlus based STM/AFMs in their laboratories and many interesting results were obtained in parallel around the globe, and it is difficult to sort these results in a timeline. However, many of these groups focused on specific sample systems and the ordering is following a more or less arbitrary list of results obtained by various research groups.

Simultaneous STM and AFM on graphite
Graphite has been studied from the early days of scanning probe microscopy. When our first 4 K qPlus microscope became operational in Augsburg around 2002, graphite was one of the first samples we investigated. 111,113 Due to its stacking symmetry in the highest available grade of HOPG (highly oriented pyrolytic graphite), only one of the two basis atoms in one unit cell appears in an STM image, while both appear in AFM at close distance with repulsive interaction. The explanation is given by the electronic structure of graphite. In an isolated carbon atom, the six electrons display a 1s 2 2s 2 p 2 distribution, following Hund's rules. In graphite, three of the four electrons in the second shell hybridize to sp 2 orbitals that bond covalently with their three nearest neighbor atoms. The fourth electron of the second shell is in the 2p z state. The two atoms in the unit cell of graphite are different-the so-called α atoms (or A atoms) have a direct neighbor in the atomic plane underneath and the 2p z states of these α atoms overlap, leading to a slightly lower energy than the one of the β atoms (or B atoms) that do not have direct neighbors underneath and therefore expose a slightly higher energy. Therefore, the electronic states at the Fermi energy display a local maximum on top of the β atoms. As the STM image is a map of the charge density at the Fermi level, STM shows only the β atoms. By contrast, AFM (in the repulsive regime) is sensitive to the total charge density and therefore shows both α and β atoms. Figure 40 shows the experimental data and simulation below. The repulsive forces that are imaged in the experimental AFM image (b) are increasing with the charge density; thus, a charge density plot is a good approximation for a

Higher harmonic AFM
As outlined in Subsection II E, the tip-sample interaction not only changes the oscillation frequency of the force sensor but also introduces higher harmonics if the tip-sample force is nonlinear. Although the magnitude of the higher harmonics is small, they are even more sensitive to short range interactions than the frequency shift for small amplitudes. It is interesting to note that only every second atom of the graphite surface images a tungsten tip atom, and it is therefore probable that the bonds between the surface atoms under the tip have been rehybridized to a diamond-like bonding under the pressure of the tip. Wright and Solares 266 calculated the contrast for such a tip terminated by a tungsten atom oriented in a (001) direction and confirmed the fourfold symmetry, but some open questions remain.

Current-induced electrostatic forces
In scanning tunneling microscopy, atomic resolution is usually possible over a quite large distance range. At a metallic single atom point contact, the conductance of the junction is approximately G 0 = 1/12.9 kΩ, leading to a current of I = 1 µA for a bias of V bias = 12.9 mV. As the tunneling current typically decays by a factor of 10 for every distance increase of 100 pm, a distance of 600 pm is easily possible as a current of 1 pA can still be measured conveniently. In AFM, the situation is different. Initially, atomic resolution was only possible at very small distances where the forces are maximal and sufficiently large to be detectable. Thus, when approaching the probe of a combined STM/AFM from far to the sample, one usually first detects a tunneling current long before forces become noticeable. We were quite surprised when we performed a combined STM/AFM experiment on Si(111)-(7 × 7) shown in Fig. 42(a). At a tunneling current of about 2 nA and the bias of V bias = 1.5 V in Fig. 42(a), we estimate an average distance on the order of 500 pm. Nevertheless, the frequency shift channel on the right shows a pronounced repulsion on top of the adatoms. As we could not explain the origin of this strong repulsive force immediately, we named it "phantom force." This force was apparently related to the tunneling current, demonstrated by Fig. 42(b) where again STM and AFM channel were recorded at zero bias. Approaching the tip 340 pm closer to the surface in Fig. 42(c), again at zero bias, of course also did not show any current but the attractive forces on top of the adatoms, i.e., the adatoms appeared dark, indicating attraction. Based on the correlation between tunneling current and frequency shift shown in the bottom section of Fig. 43, we speculated that the phantom force may have its origin in the limited conductivity of our semiconducting sample. When a weakly conductive sample is subject to a local current injection, the surface potential can change over a larger area, reducing the voltage differential between the tip and the sample and thus reducing the electrostatic attraction, feigning a repulsive force that is proportional to the tunneling current. We tested this hypothesis with a setup outlined in the top of Fig. 43. In this experiment, a switch allows us to either directly connect the sample to the currentto-voltage converters (a) or to put a resistor R 1 = 10 MΩ (b) or R 2 = 30 MΩ in series with the sample. If our hypothesis was correct, a resistor in series to the resistance of the sample should increase the phantom force effect, verified by the increasing slope of the ∆f(I) curves shown in the bottom of

Measurement of forces that act during atomic manipulation
In 1990, Donald Eigler and Erhard Schweizer used the tip of a 4 K STM to move individual Xe atoms on a Ni(110) surface and spelled out "IBM" at the IBM Research Laboratory in Almaden, 51 fulfilling Richard Feynman's dream in his December 1959 lecture There's Plenty of Room at the Bottom: An Invitation to Enter a New Field of Physics. 61 Toward the end of his talk, Feynman made the following statement: "But I am not afraid to consider the final question as to whether, ultimately -in the great future -we can arrange the atoms the way we want; the very atoms, all the way down! What would happen if we could arrange the atoms one by one the way we want them (within reason, of course; you can't put them so that they are chemically unstable, for example)." This last comment "within reason, of course" is very important. Eigler and Schweizer used Xe atoms that bond mainly by the weak van der Waals interaction to the surface, and they picked the Ni (110) surface that has grooves in it in contrast to densely packed (111) surfaces to prevent thermally activated motion of the Xe atoms on the surface. The driving forces behind atomic manipulation were not accessible by STM. Therefore Andreas Heinrich, Eigler's successor at IBM, was very eager to add AFM capability to STM to measure the forces that drive atomic manipulation. This upgrade was relatively simple by replacing the tunneling tip of Eigler's STM with a qPlus sensor such that combined STM and AFM became possible. 239 In this study, we could finally measure the forces that act in atomic manipulation (Fig. 44). In atomic manipulation by STM, the tip remains steadily close to the atom that is manipulated. In combined STM/AFM, we need to oscillate the tip such that the distance between the tip and the atom varies by 2A ≈ 100 pm. Nevertheless, atomic manipulation with a vibrating tip was possible with a similar behavior as with a steady tip. We label selected line scans with the closest approach z during the oscillation [see note (15)  the potential energy. The lateral derivative of the inverted potential energy yields the lateral forces. 117 It turned out that the required forces depend both on the substrate and on the adsorbed species-it is easier to move a Co atom across Cu(111) than a CO molecule, and Pt(111) is stickier than Cu (111). 239 More recently, our Regensburg group found evidence for a lowering of the manipulation threshold due to the presence of the probe tip, 52 and very recently, we demonstrated atomic manipulation with CO terminated tips. 18 Atomic inlays created by controlled atom manipulation have been obtained using conventional Si cantilever AFM. 228

Carbon monoxide front atom identification (COFI)
The carbon monoxide front atom identification (COFI) method was introduced in 2012 257 and uses a CO molecule that bonds upright to a closed packed metallic surface, e.g., Cu (111), as shown in Fig. 45. A constant-height STM image shows a dip in the current, and a simultaneously recorded frequency shift image shows much more structure. The initial interpretation concluded that all three tips shown in Fig. 45 were single tips, based on the similarity of their STM data. A later study 53 has proved that tip 2 was a dimer tip, and tip 3 was a trimer tip. This 2015 study points out the erroneous interpretation from 2012 in the abstract and provides a detailed comparison of the profound similarities between single-, dimer-, and trimer tips interacting with CO/Cu (111) and a CO tip that images single adatoms, dimers, and trimers as shown in Fig. S11 of Ref. 53.
In addition to resolving the structure of the front section of the tip, COFI also allows us to draw information about the chemical identity of the tip, as found in Ref. 121. Figure 2 in this publication shows that tips that have a Cu atom at the apex display a maximal attractive force of about 130 pN to CO/Cu (111), while Fe and W terminated tips show a maximum of about 250 pN. This chemical identification by the maximal attractive force follows the work of Sugimoto et al. who distinguished three atomic species silicon, tin, and lead by their force profile. 229

Metallic surfaces and metal clusters
In preparation to the studies that later determined the forces that act in atomic manipulation at the IBM Almaden Research Laboratory, 239 we accidentally picked up CO on the tip and obtained high resolution images of the Pt(111) surface in an unpublished experiment of June 2007, shown in Fig. 46. The dark spots in the STM channel coincide with the dark spots in the frequency shift channel, and they are located at the positions of the Pt surface atoms. The reason of contrast inversion in the STM channel is due to the p x , p y states that lead to reverse contrast, 33 and the dark spots above Pt in the AFM  (111) acts as a probe to inspect the tip. For low bias voltages, CO appears dark. Because the width of the minima over CO was similar for all three tips, the initial assumption was that all three tips expose a single atom at the front that only changes by its bonding orientation indicated by the orientation of the bcc Wigner-Seitz cell in the right top inset of the left column images. A later study 53 has confirmed that panels (a)-(c) had a single atom tip but also has clearly shown that tip 2 in panels (d)-(f) had two atoms at the front and tip 3 in panels (g)-(i) had three front atoms. Center column: constant-height data of tunneling current in nA. channel denote increased attraction. The increased attraction of CO tips over metallic surface atoms was later also observed on Cu(111) 53,213 and Cu (100). 211 The measurement of the distance dependence of current and forces of single adatoms on a metallic surface allowed to precisely determine the force versus distance dependence of metallic bonding forces-it turned out that they have a similar exponential distance dependence as the tunneling current that decreases to 1/10 for every distance reduction of 100 pm. 238 In scanning tunneling microscopy, metal clusters consisting only of a few atoms usually cannot be resolved atomically. Instead, the number of atoms involved can be inferred by their height (see the top row in Fig. 47). A qPlus sensor with a CO terminated tip resolves the atoms one by one, as shown in the second and third rows of Fig. 47. 53 The bottom row shows the adsorption sites, calculated by DFT.

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Wedge-shaped Pb islands grown on silicon were studied by Mao et al. 158

Subatomic spatial resolution on copper and iron adatoms
Three examples of silicon or graphite samples that produce repeated subatomically resolved images of the tip's front atom had been collected in the first five years of using the qPlus sensor in AFM. Non-spherical images of the tip's front atom had been found in a possible Si front atom in Ref. 82, in a possible Sm front atom, 114 and in a possible W front atom. 112 While these results were promising steps forward, they revealed the structure of the tip atom being imaged by the sample. The real purpose of microscopy is to image a sample, not to inspect the probe tip. Therefore it was an important step to create very well defined tips and image a precisely defined adatom as a sample at subatomic resolution. Starting in 2012, we imaged single adatoms on flat surfaces with CO terminated tips, discussed in Ref. 53. Figure 48(a) shows a Cu adatom on Cu(111) at very close imaging distance. We see a repulsive center and a repulsive ring. Figure 48

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Kelvin probe measurements
In parallel to the introduction of the qPlus sensor to the IBM Almaden laboratory, the technology was also transferred to Gerhard Meyer's group at IBM Rüschlikon. In 2004, Repp et al. had found a way to charge gold atoms on a NaCl layer. 197 Meyer et al. repeated these experiments with the added AFM capabilities offered by the qPlus sensor. A Kelvin probe measurement would allow to confirm that charge transfer, predicted by density functional theory before, 197 actually took place. The combined STM/AFM experiment in Ref. 94 as outlined in Fig. 49 confirmed the charging of the single gold adatom by a shift of Kelvin parabolas for charged versus uncharged gold adatoms. In a combined STM and AFM constant-height experiment, the charged Au atom showed less current in the STM image than the neutral Au atom because the charged Au atom sinks into the surface as depicted in the inset of Fig. 49(a). However, the charged Au atom shows a more negative frequency shift than the neutral Au atom due to the larger electrostatic interaction, also indicated by the more negative frequency shift of the Au − Kelvin parabola in Fig. 49(a).
Kelvin probe measurements using the qPlus sensor have also been performed on Si surfaces 19 and on Pb islands. 156 Equalizing the local contact potential by Kelvin probe microscopy is a standard procedure to minimize electrostatic interaction in imaging insulators, e.g., in imaging NiO(001). 190

Atomic resolution of organic molecules, graphene, and graphene nanoribbons
In 2009, Leo Gross et al. discovered that picking up a CO molecule on the metal tip of a qPlus sensor resulted in a dramatic increase in resolution in the AFM channel 95 of an organic (pentacene) molecule displayed in Fig. 50(a). The group used a qPlus sensor with a PtIr tip and noticed that a functionalization of the AFM's metal tip by picking up CO as previously described for STM by Bartels et al. 12 improved the resolution profoundly. The inertness of the oxygen termination of a CO molecule is important for accessing the repulsive regime and for achieving high resolution. Mohn et al. found that functionalization of metal tips with inert gases also provides inert tips that allow us to probe the repulsive regime. 166 This beautiful image of pentacene was followed by the detection of the bond order in a C 60 molecule 97 shown in Figs. 50(b) and 50(c). Bonds that are part of two hexagons, labeled h, are of greater bond order than bonds that are part of a pentagon and a hexagon, labeled p. At medium tip height in Fig. 50(c), bonds appear with different brightness, with bonds of greater bond order appearing brighter. At small tip heights in Fig. 50(d), bonds appear with different apparent lengths, with bonds with greater bond order appearing shorter.
Imaging of molecules with a qPlus sensor terminated by a CO tip opened a new field that is covered in recent overviews by Pavlicek and Gross 184 and Gross et al. 98 Figure 50(g) shows molecular compounds of heavy oil. For each of the eight different samples investigated on the order of 100, molecules were imaged. A typical molecule is displayed for each mixture. Analysis of the structures found by AFM provided insight into the molecular geometry, aromaticity, types and locations of heterocycles, occurrence, length and connectivity of alkyl chains, and content of archipelago-type architectures. Figure 50(h) shows molecules that undergo a reversible Bergman cyclization. The Br atoms from dibromoanthracene (DBA) are dissociated to form first a radical and then a diradical. The latter can be reversibly switched into a diyne and back by tunneling electrons at V > 1.6 V. With an applied voltage below 1.6 V, the molecule remains stable and can be imaged by AFM (bottom panel, AFM at V = 0 V). A potentially unknown molecule from the deep sea was identified, 96 reaction products of molecules were identified, 42 and more studies of oil compounds that were analyzed at the submolecular level can be found in Refs. 58 and 59.
Imaging works well for flat molecules, but Jascha Repp's group even succeeded in imaging of "butterfly"-shaped dibenzo[a,h]thianthrene (DBTH) molecules that extend in the third dimension, 183 a much harder task. Pavlicek et al. showed in this experiment (see Fig. 51) that AFM data reveal the handedness of the molecule, while STM does not. Albrecht et al. from the same group identified the conformational response of individual nonplanar molecules. 4 The group of Qiu in Beijing was one of the first users of a commercial qPlus based AFM and reported a strange appearance of lines between adjacent 8-hq molecules on surfaces 269 that were interpreted as evidence for hydrogen bonds, shown in Fig. 52. The interpretation of these lines as a possible evidence for hydrogen bonds is a subject of an ongoing fruitful debate; see Refs. 57, 101, 103, and 165.
The possibility to resolve the structure of organic molecules as demonstrated by Gross et al. is also very helpful when molecules undergo transformations due to chemical reactions. Crommie and Fisher from the University of Berkeley first imaged the precursors of a chemical reaction, then applied heat to induce a chemical reaction, cooled down for imaging again, and imaged the products in Fig. 53(a). The same group imaged oligomers by combined AFM and scanning tunneling spectroscopy in Fig. 53(b) and even studied the anchoring of molecules to graphene nanoribbons in Fig. 53(c). Molecular reactions were also studied by Albrecht et al. using qPlus based AFM. 3 Figure 54 shows images of graphene nanoribbons imaged with a CO terminated metal tip (a) and a plain metal tip (b). 231 As already apparent in imaging organic molecules, the CO tip termination has the advantage that CO is very inert, allowing to probe the repulsive regime. The metal tip shows an inverted contrast; i.e., attraction between C atoms and the metallic tip atom prevails. The edge of the nanoribbon exerts strong attractive forces to the metal tip-this is much less pronounced for CO terminated tips. More data on using the qPlus sensor to study graphene and graphene nanoribbons from this group 28,132,244 and other researchers are available in the literature. 138,142,147,202,222 Schwarz et al. studied the structure of hexagonal BN on Cu(111) using a combination of qPlus AFM and X-ray standing waves. 216 The group of Jelinek in Prague is also an early adaptor of the qPlus AFM technique. Figure 55(a) from this group shows simultaneously recorded STM, AFM, and inelastic tunneling spectroscopy (IETS) data of iron-phthalocyanine (FePc) on Au(111). 39 Figure 55 The group of Tautz at the Forschungszentrum Jülich was also quite an early user of the qPlus sensor. Figure 56 shows three central fields of their study consisting of controlled pick up of large molecules from surfaces (a), the utilization of molecules attached to the tip as sensitive probes (b), and the controlled deposition of vertical molecules that can act as controlled emitters of electrons. Their study 236 shows in Fig. 56(a) how a PTCDA molecule (see figure caption) is peeled off a metal surface. The forces needed to peel off the molecule were quite constant as demonstrated by 226 traces in Fig. 56(a) middle and a 2D histogram of 226 ∆f(z) traces. 247,248 Fitting ∆f(z) with a molecular mechanics model reveals the evolution of the junction geometry and helps quantify long-range van der Waals 249 as well as the short range interactions acting between the molecule and the surface. 247 Figure 56(b) illustrates scanning quantum dot microscopy (SQDM)-a tool for the nanoscale imaging of electrostatic potential Φ where a Kelvin parabola is recorded at a specific spot on the sample, and characteristic dips occur at a voltage V − corresponding to electron removal and at voltage V + when adding an electron. The molecule peeled off the surface remains hanging on the tip, acting as a quantum dot (QD) that can be charged with single electron precision if a sufficient bias is applied to the tip-surface junction. 67,237,250 The electrostatic potential is then given by Φ ∝ −V − /(V + − V − ) [see the center part of Fig. 56(b) and caption]. As an application, the quadrupole potential of a flat-lying PTCDA molecule probed at a distance of 17 Å from the Ag(111) surface is depicted in the right part of Fig. 56  In Fig. 56(c), it is demonstrated how moving the tip along a special 3D trajectory 66,154,155 enables a single PTCDA molecule to be erected onto a pedestal of two silver adatoms into a stable, vertically standing configuration. 55 In the middle of Fig. 56(c), it is demonstrated that a standing PTCDA also acts as a quantum dot (QD): An electron can be added to the QD if a large negative bias is applied to the surface. Due to field-emission, this electron leaves the QD quickly and moves toward the tip. The right part of Fig. 56(c) shows the intensity map of the field-emission current imaged 7 nm away from the surface, caused by the quantum-mechanical interference of single election trajectories.

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A vast array of applications of qPlus based AFM is found in the structural arrangement and reactions of molecules on surfaces; see, e.g., Refs. 27, 107, 182, and 251.

Van der Waals forces
The van der Waals interaction is an important force in nature that acts between all matter. For noble gases, it is the  Fig. 57(a). 139 They also studied the sliding forces of a graphene nanoribbon on Au (111) in Fig. 57(b) 140 as well as boron-and nitrogen-doped graphene nanoribbons in Fig. 57(c). 138,142 Figure 57

Silicon
The imaging of the Si(111)-(7 × 7) reconstruction has been an important benchmark test of the AFM utility as a tool for surface science. 88 The first result [see Fig. 58(a)] was obtained with a self-sensing silicon cantilever 71 in 1995, originating from covalent bonding between tip and sample. 187 Much nicer results that even showed evidence for the presence of the rest atoms were obtained later at low temperatures in 2000 150,151 and at room temperature with very sharp Si cantilevers. 50 The first successful imaging of Si with the qPlus sensor was obtained in 2000 at room temperature, 82 showing even strong indications for subatomic resolution where the Si adatoms repeatedly image two tip orbitals that originate in one Si tip atom [see Fig. 58 Low temperature qPlus AFM with CO terminated tips that proved so successful in imaging organic molecules 95 has also helped obtain clearer images of silicon. Figure 58(c) shows an image of silicon, obtained with a CO terminated tip, featuring very small local maxima for the twelve adatoms and even clear local maxima for the six rest atoms in each unit cell. 53 The group of Moriarty at the University of Nottingham has studied the switching of the dimer buckling structure on Si(100) shown in Fig. 59(a) via direct mechano-chemical interaction with the apex of a scanning probe tip. The insets in Fig. 59(b) show the position of the atoms in the tip-sample junction at each stage of the manipulation. In Fig. 59(c), reverse imaging of a C 60 molecule attached to the scanning probe tip occurs similar to the imaging process in Fig. 58 (b) or in the comparison of COFI and Si adatom imaging in Ref. 258. Figure 59(d) shows the measurement and calculation of the chemical force responsible for sub-molecular atomic contrast. Figures 59(e) and 59(f) describe the interaction of C 60 terminated tips with C 60 molecules on the surface.
The group of Wolkow 126,196 worked on silicon based atomic logic circuits on the atomic scale and used a low temperature qPlus based microscope. Figure 60 shows the construction of an OR gate on a silicon surface by dangling

Topological insulators
Topological insulators are materials that conduct electrical currents on the surface but not inside. Spin and momentum are locked in the topologically protected conductive surface states. 104 The surface of the topological insulator TlBiSe 2 has been resolved atomically with the qPlus sensor in 2015. 192 The surface was imaged before by STM, displaying only wormlike structures that did not reveal the true structure of the cleaved surface. The AFM data shown in Fig. 61 indicate that the layered material cleaves along Tl planes, leaving approximately half of the Tl atoms on either cleavage plane.

Atomically resolved damping
As outlined in Subsection II D, FM-AFM is very sensitive to local dissipation, i.e., hysteresis in the force versus distance curve. Such a hysteresis occurs in particular over loosely bonded atoms. The topological insulator TlBiSe 2 consists of stacked layers. Cleaving occurs preferentially on Tl layers, leaving two new surfaces with about a 50% occupation of the Tl sites, as shown in Fig. 61. At room temperature, these Tl atoms are quite mobile such that the surface appears to be flat (Fig. 5 in Ref. 192). While these Tl atoms freeze out at 4 K to form wormlike structures, the atoms can still be deflected laterally as the vibrating tip approaches it. These deflections equilibrate when the tip oscillates away from the Tl adatoms, causing moderate dissipation of up to about 20 meV per oscillation cycle. In the first proposal on highly spatially resolved dissipation measurements, 41 induced electric currents were discussed as the origin of damping. Apparently, this is not the case in Fig. 62, as voltage and current are 20 times larger in the lower row compared to the top row, although the measured dissipation plotted in the right column is quite similar.

Imaging involving superconductors
Jorge Hirsch proposed an experiment regarding screening of charges by a superconductor, 116 where the Thomas-Fermi screening length that shields charged impurities in a metal with its typical range of less than 100 pm would increase to the London penetration depth in a superconducting transition. We have attempted to check this effect and so far could not confirm it 188 because the difference in frequency shift versus distance curves in Fig. 63 between normal and superconductive tips is not larger than the experimental errors. Possibly, the density of normal conductive electrons was still high to observe a clear effect at the minimal temperature of 2.4 K we could reach so far. However, we could demonstrate that the tip of a qPlus sensor is colder than the transition temperature of niobium (9.5 K for bulk T c ) and was indeed superconductive. An analysis of the dI/dVspectrum (see the inset in Fig. 63) shows that the tip is at the same temperature as the sample, within fractions of 1 K. To our knowledge, this is the first demonstration of a superconductive tip for an operating atomic force microscope.

Water
Water is one of the smallest molecules and its chevronlike structure and permanent dipole moment leads to fascinating structural properties. Resolving its structure and adsorption geometries is an interesting challenge for AFM. The group of Sugimoto in Japan has imaged (Fig. 64) a pentagonal chain of water molecules on a Cu (110) surface, where the AFM channel on the right side shows much more features than the STM channel in the center. 223 Ying Jiang's group from Beijing, China, has imaged water dimers [ Fig. 65(a)], tetramers [ Fig. 65(b)], 185 and the ion hydrate Na + · 4D 2 O [ Fig. 65(c)] adsorbed on the NaCl(001) surface 186 using a CO terminated metal tip. The submolecular features originate from the highorder electrostatic force acting between the quadrupole-like CO-tip and the strongly polar water molecules. The crooked depressions (see dashed curves) and the bright protrusions (see white arrows) in the AFM image arise from the positively charged hydrogen and the negatively charged oxygen, respectively.

Lateral force microscopy and friction measurements
While lateral forces can be derived when the tip-sample potential energy landscape is available, as possible in lowtemperature force spectroscopy used in measuring force needed to move an atom, 239 a direct measurement is often very weak interaction between one CO molecule attached to a tip and the other to a sample has been measured by lateral force microscopy; 261 see Fig. 66. The interaction of two CO molecules is on the order of a few milli-electron volts-a few orders of magnitude weaker than the interactions between atoms that form covalent bonds. A precise measurement of such small interactions is challenging, in particular, the separation from long-range van der Waals background forces. 230 The great advantage of lateral force microscopy is that the van der Waals interaction between a tip and a flat sample acts only in the z-direction. Therefore, even weak lateral forces are not concealed by a van der Waals background. The measurement shown in Fig. 66 enabled to determine the lateral stiffness of a CO attached to a tip at 0.24 N/m in agreement with the findings in an algorithm that allows us to correct for the CO-bending distortions that occur when imaging organic molecules. 174 A recent review of LFM can be found in Ref. 262.

Oxides
Atomic imaging of insulators is one of the key possibilities that were opened up by AFM. Thin oxide films such as titania have been studied by STM for a long time by the group of Diebold 44 and others. Diebold and Setvin et al. have started to apply AFM to understand the structural properties of oxides, 220 as shown in Fig. 67. Rutile TiO 2 is one of the most studied model systems in oxide surface science; well-defined surface oxygen vacancies (V O ) are ideal for investigating defect related surface chemistry; see the structural model in Fig. 67(a). Figure 67(a) shows a structural model of rutile TiO 2 , a model system in the surface science of oxides. Figure 67 coordinated Ti5c atoms. AFM reveals the two-fold = O 2c atoms shown in Fig. 67(c). The anatase phase of TiO 2 is shown in Fig. 67(d).
Adsorbed O 2 appears neutral and singly charged, showing vastly different force spectra in Fig. 67 a polar perovskite oxide, KTaO 3 (001). Upon cleaving, islands are visible, with steps that have a height of half unit cell in Fig. 67(g). With nc-AFM, these are identified as KO terraces with an undisturbed (1 × 1) structure. Upon gentle heating, V O s form at the center of the KO islands in Fig. 67(h), and further heating induces a well-defined labyrinth structure in Fig. 67(i). These and other features are explained as mechanisms for compensating the polarity in the system. 221 The group of Schwarz in Yale looked at surface oxidized Cu(100) samples in Fig. 68(a). The total force contrast is 22 pN, and atomic corrugation is clearly observable on the plane of closest approach. The average force for each horizontal plane has been subtracted from the data to improve contrast. force based sensor to study the atomic structure of glasses and silica. 29 The IBM Almaden group investigated the structure and thickness of insulating films on metals that are used to provide isolated carriers for atomic spin structures. 15

Exchange force microscopy
Nickel oxide (NiO) is a very special metal oxide because of its antiferromagnetic structure. Shortly before atomic resolution by AFM was obtained, Mukasa et al. proposed to probe magnetic exchange forces by force microscopy. 172 More than a decade passed, until Kaiser, Schwarz, and Wiesendanger succeeded to image NiO and show the 2 × 1 magnetic superstructure in 2007. 135 However, unit cell averaging was necessary to clearly show the 2 × 1 magnetic superstructure and a 5 T magnetic field had to be applied to stabilize the magnetization of the iron covered Si tip. In 2013, the initial attempt to resolve the antiferromagnetic order on the NiO surface In contrast to forces, the maxima in the tunneling current coincide with the copper atoms on the surface. The presence of linear defects on the surface is highlighted by the dashed white ellipses. The total current contrast is 7 pA. (d) 2D map of interaction forces (6.55 × 6.55 nm 2 ) extracted from a 3D data set recorded on rutile TiO 2 (110). The dashed yellow circle highlights an oxygen vacancy, whereas no perturbation is detected in the force data around the orange dashed circle. The total force contrast is 100 pN. (e) 2D map of tunneling currents recorded simultaneously with the force map of (d). While the oxygen vacancy is observed again, a subsurface, interstitial hydrogen atom causes an additional perturbation of the tunneling current at the location highlighted by the dashed orange circle. The total current contrast is 368 pA. The data in (a)-(c) have been acquired at T = 5 K using a qPlus sensor with A ≈ 1.00 nm, k ≈ 2000 N/m, and f 0 = 29 177 Hz. 16 The data in (d) and (e) have been acquired at T = 77 K using a qPlus sensor with A ≈ 1.50 nm, k ≈ 2000 N/m, and f 0 = 25 328 Hz. 17 Reprinted with permission from U. Schwarz. Copyright Udo Schwarz, Yale University, New Haven, USA.

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at room temperature with a qPlus sensor 212 was repeated at low temperatures. 190 First experiments using a bulk iron tip resolved the antiferromagnetic order at close distance, but spin contrast was lost again at very close distance (supplemental to Ref. 190). The use of a hard magnetic tip material (CoSm) provided much better data, as shown in Fig. 69. Spin contrast was clearly visible in the raw data of Ref. 190, and force spectroscopy was performed to measure the magnitude of the exchange interaction. In addition, evidence for super exchange was found-the height of the O atoms in Fig. 69 also shows a 2 × 1 structure, in agreement with calculations by Granovskij et al. 93 Exchange force microscopy combined with spin-polarized scanning tunneling microscopy to independently characterize the geometric as well as the electronic and magnetic structures of nonflat surfaces has recently been demonstrated in the group of Khajetoorians. 106 D. New frontiers

Atomic force microscopy combined with inelastic electron tunneling spectroscopy
Inelastic electron tunneling spectroscopy (IETS), pioneered by the group of Ho, 227 extends the capabilities of STM to measuring excitations. A further combination of STM with IETS and AFM provides new possibilities. The intensity of IETS signals usually varies from tip to tip. COFI analysis of tips allows us to repeatedly poke and check tips to confirm a single atom tip. Figure 70 shows five different experiments with five different tips that have all been checked by COFI to have a single atom tip, and those tips all produce highly similar IETS spectra with similar intensities.
A second example where the additional input provided by AFM is helpful is to assess the force that the probe tip exerts on vibrating molecules. 181 Figure 71 shows the five contributions that the force exerted from the probe tip to a vibrating CO molecule entails. First, the tip-sample force directly adds to the restoring forces provided by the molecular bonds themselves. Second, the force that acts on the molecule influences the strengths of bonds within the molecule and to the substrate. This is exactly what was found in the experiment-the direct impact of the forces did not suffice to explain the experimental changes in vibrational energies, and bond weakening due to the forces exerted by the tip was evident; see

Ultrahigh vacuum at ultralow (mK) temperature and high magnetic fields
Currently, many exciting studies of qPlus based STM/AFM are performed in vacuum at liquid helium temperatures. Extending its operation into the mK regime is promising for a variety of reasons. When studying devices at ultralow temperatures, 145,226 it would often be desirable to approach and navigate on sections of the device that are nonconductive. Therefore, AFM would be mandatory. Another example is the study of the possible different screening of normal versus superconductive states that we attempted at temperatures down to 2.4 K, 188 where lower temperatures are needed to find out if Jorge Hirsch's theory 116 on screening in the superconductive state is accurate. When getting from the room temperature of 300 K to a liquid helium temperature of 4 K, it may seem as a minor step to conquer the last percent toward absolute zero. Figure 73(a) shows a linear plot of the temperature scale, where it appears like there is not much difference between 4 K and ultralow temperatures at 10 mK. However, many processes in nature are governed by the Boltzmann factor exp(−E/k B T). Therefore, a comparison of temperatures needs to take the ratio into account. The step from 300 K to 4 K reduces temperature by a factor of 4/300 = 1/75, while the step from 4 K to 10 mK corresponds to a factor of 1/400. Therefore, the instrumental challenges to go from 4 K to the mK regime are much harder than to go from room temperature to 4 K. Furthermore, liquid helium with its boiling temperature of 4.2 K is amply available in many laboratories with commercially available helium liquefiers, while complicated refrigerators are needed to cool down to the mK regime. This larger challenge is visually evident on the logarithmic scale in Fig. 73(b).

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Combinations of high magnetic fields and ultralow temperatures are, e.g., relevant when studying magnetic excitations with inelastic electron tunneling spectroscopy. Magnetic energy levels are given by multiples of µ B B, and the thermal energy is given by the Boltzmann constant times temperature k B T. The thermal broadening in inelastic electron tunneling spectroscopy is about 5.4k B T. 149 As the Bohr magneton µ B = 9.27 · 10 −24 J/T is a very small number, it takes high magnetic fields and ultralow temperatures for the two energies to match. The green line in Fig. 73(b) indicates the magnetic field strength B as a function of temperature T where 5.4k B T = µ B B. One application of ultralow temperature IETS is to study Landau levels in graphene. 226 Ultralow temperatures are a special challenge for scanning probe microscopy because the cooling power of mK cryostats is limited and the requirements of vibration isolation and ultralow temperatures often encounter conflicting objectives. The 10 mK STM of the Stroscio group at NIST Gaithersburg 225 has recently been upgraded to allow simultaneous STM/AFM operation. 217 The special challenge of a 10 mK AFM is the low cooling power of the cryostat and the conflicting needs of the amplifier to require an operating temperature of at least 50 K and a close distance to the sensor. A close distance is helpful to obtain low noise because the capacity of the wire that connects the amplifier and the sensor adds apparent current noise via the input voltage noise of the amplifier. In the case of the NIST microscope, a distance of about 1.2 m between the sensor and the amplifier is required. In most 4 K AFMs, the distance between the sensor and the amplifier is less than 0.1 m-less than one tenth of the distance of the NIST microscope. The wiring between the sensor and the amplifier has two components that contribute to its capacity: length and capacity per length. The capacity of a coaxial wire is given in Eq. (66), showing that a small inner radius r i , a large outer radius r a = 550, and a low value for the relative dielectric constant of the insulation material r of the wire are crucial. To minimize cable capacity at NIST given the long distance, we designed a cable with a very low capacitance per length of about 25 pF/m and had this cable professionally manufactured. 175 This bicoaxial cable featured two coaxial leads within one external shield. Each of the two coax cables had an inner wire with radius r i = 50 µm and a shield with radius r a = 550 µm. The insulating material between the leads is PTFE (polytetrafluorethylene-Teflon) with a dielectric constant of r = 2 at the operating frequency of the sensor. According to Eq. (66), this cable with r a /r i = 11 yields a theoretical capacitance of 23 pF/m close to the experimental value. Typical capacities per length of coaxial cables are 100 pF/m, so the effort to design a custom cable has proven to be worth the cost and effort. In many 4 K systems, where the amplifier can be close to the sensor, the connection between the sensor and the amplifier is just done by a pair of thin wires that float in vacuum. In that case, the ratio r a /r i is on the order of 10 to 100 with the dielectric constant of vacuum r = 1, yielding even lower capacities per length and much lower total capacities due to possible lengths on the order of 0.1 m.
The NIST team performed preliminary measurements applying a magnetic field up to 15 T with sensor type qPlus S1.0B (see Table I). The sensor still operates, but the Q value drops to about 20 000 at | B | = 8 T. An even stronger damping effect had been reported previously by James Hedberg 108 from Peter Grütter's group at McGill University using qPlus sensors made of tuning forks. In these experiments, Hedberg found a drop in Q to only 1700 at a field of 8.5 T. A plot of with three examples of microscopes: a room temperature AFM without a magnetic field (green squares), a 4 K AFM with a 5 T field (blue triangles), and a 10 mK AFM with a 15 T field (red circles). (a) On a linear temperature scale, the difference between a 4 K AFM and a 10 mK AFM seems little. (b) A logarithmic temperature scale is more appropriate to reflect the instrumental challenges of a 4 K microscope versus a 10 mK microscope-the step from 4 K to 10 mK is much larger on a logarithmic scale than from 300 K to 4 K. Magnetic fields allow us to study magnetic excitations on energy scales of µ B B, with the Bohr magneton µ B = 9.27 · 10 −24 J/T. The energy resolution of tunneling spectroscopy is on the order of 5.4k B T 226 with the Boltzmann constant k B . The green line in (b) is given by µ B B = 5.4k B T; i.e., to be able to resolve energies on the order of µ Bohr B, a microscope needs to provide a sufficiently large ratio between magnetic field and operating temperature, indicated by the region above and to the right of the green line in (b).

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damping that is proportional to 1/Q showed a proportionality to the square of the magnetic field B. While the lowering in Q leads to increased oscillator noise (scaling as 1/Q) and to a lesser extent increased thermal noise (scaling as T/Q), it does not prevent one to obtain atomic resolution. The origin of this magnetic field induced damping is not clear yet, Hedberg discussed (p. 89 in Ref. 108) four potential origins: (1) (anti)ferromagnetic metals in the tuning fork electrodes, (2) eddy currents in the tuning fork electrodes, (3) eddy currents in the tungsten tip, and (4) intrinsic dissipation of the quartz crystal.
The following analysis suggests that eddy currents in the electrodes of the sensor that are strained with the oscillation of the beam are a relevant channel for damping. Faraday's law states that a voltage is induced in a closed conductor given by the time derivative of the magnetic flux (see Chap. 17 in Vol. II of Ref. 62). The induced voltage leads to eddy currents in a current loop, and these currents will cause dissipation. Eddy currents are therefore induced in conductors that move through a magnetic field gradient. Figure 74(a) shows a qPlus sensor with a magnetic field B applied in the z-direction, i.e., parallel to the tip and its motion. The oscillating magnetic flux in the sensor electrode in Fig. 74(b) induces a voltage (see the caption of Fig. 74) given by The strain at deflection A of the sensor is not constant; according to Eq. (53), it is maximal at the root of the beam at x = 0 and falls to zero proportional to x − L. With Eq. (53), F = k · A, and Eq. (51), we find For an amplitude of A = 50 pm and a sensor S1.0B with L = 2.36 mm and th = 214 µm, we obtain a maximal strain at x = 0 Due to the oscillation of the tip, the area covered by the electrode and thus the magnetic flux Φ is not constant but changes slightly according to Φ(t) = | B | · w · L e · (1 + cos (2πft)) where is the strain at the top and bottom parts of the sensor beam when it is deflected at amplitude A.

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REVIEW scitation.org/journal/rsi amounting to = 7.2 · 10 −10 , resulting in a peak voltage of V induced 0 = 0.42 nV for | B| = 10 T. The current loop indicated in Fig. 74(b) has an approximate resistance of R ≈ ρ el · 2 · (L + w) w/2 · th Au , (83) where th Au is the thickness of the Au metallization in the electrodes and ρ el is the specific resistivity of the electrode material (typically 300 nm Au on top of 25 nm Cr). At room temperature, the specific resistivity of gold is ρ el ≈ 20 nΩm (similar to Ag and Cu). However, at low temperatures, resistance goes down by about two orders of magnitude (according to Ref. 128, ρ el Au ≈ 0.2 nΩm between 1 and 10 K). However, defects are a main source of resistance and the actual value depends a lot on the pureness of the sample. Assuming a specific resistivity of ρ el Au = 0.2 nΩm, we find a resistance of R = 51 mΩ for this current loop. Due to the sinusoidal time dependence of the induced voltage, the average power dissipation of these eddy currents is P = V 2 induced /(2R) = 1.7 aW and the energy loss per oscillation cycle is ∆E loss = P/f = 57 yJ = 0.71 meV when considering both top and bottom electrodes. Equation (40) can be reversed to yield an expression for the Q factor as a function of ∆E loss , where E qPlus = kA 2 /2. For an oscillation amplitude of A = 50 pm and k = 1800 N/m, we find E qPlus ≈ 14 eV; thus, if the Q factor was infinite without the magnetic field, Q would go down to about 125 000 due to this eddy current damping according to Eq. (84). This is still large compared to the experimental drop in Q to 1700 in Hedbergs case or 20 000 at NIST. However, a change in magnetic flux and thus an induced voltage in the sensor electrodes also occurs if the magnetic field in the cryostat is even only slightly inhomogeneous. If the magnetic field changes by a similar magnitude as the strain ( = 7.2 · 10 −10 for the example above) on the lengthscale of the sensor's oscillation amplitude (typically 50 to 200 pm), there will be a similar damping effect. If the field would change by a relative amount of 7.2 · 10 −9 over a distance of 2 · A = 100 pm, corresponding to a field gradient of 7.2 · 10 −9 · 10 T/100 pm = 7.2 T/cm, Q would drop to 12 500 due to this effect. Moreover, in an inhomogeneous magnetic field, all conductive and moving parts of the sensor (the tip in particular) would be subject to eddy current damping. Both suggested eddy current damping mechanisms, the strained electrode and the inhomogenic field contributions, yield damping that is proportional to the square of the magnetic field as experimentally observed.
The homogeneity of the magnetic field in a high-field superconductive magnet directly determines the field dependent damping by eddy currents. American Magnetics 5 quoted homogeneities of ±0.1% in a one-centimeter diameter spherical volume as routine. For a 10 T magnet, this homogeneity corresponds to a field gradient of 0.002 · 10/0.5 T/cm = 40 mT/cm, still small compared to the gradient quoted above. However, the sensor is mounted on a microscope, and the magnetic permeability of the materials used in the microscope will change the local magnetic field and cause larger field gradients.
In summary, it appears reasonable that eddy current damping is the main reason for reduced Q values at high magnetic fields. However, more studies are needed to verify this mechanism. If this mechanism is correct, the use of the electrode and tip material with a higher resistivity, smaller tips and possibly changing the geometry of electrodes by applying cuts to prevent the occurrence of large current loops should greatly reduce the drop of Q in high fields.

VI. SUMMARY AND OUTLOOK
The hurdles that had to be overcome in the introduction of the STM were deep and profound. Perhaps the largest hurdle was to imagine that a tip and a sample could be positioned and scanned at the required accuracy and to actually do the experiment. Many scientists and engineers considered this to be plainly impossible at the time the STM was introduced. However, the physics of the tunneling current as the STM's control signal, an exponential decay versus distance at a rate of a decade per Angstrom, helped immensely. The rapid decay meant that even a relatively blunt tip would enable atomic resolution because even then the tunneling current was focused on the front atom of the tip. The monotonic exponential decrease in the tunneling distance with distance allows us to create a very simple distance control loop. Neither of these two beneficial properties of the control signal holds for the AFM. Instead, four basic challenges that relate to the physics of the tip-sample forces emerged: (1) a soft spring used as a force sensor suffers from a jump-to-contact phenomenon, (2) tip-sample forces are not monotonic with distance, initially attractive, and then repulsive, making it difficult to establish a feedback loop, (3) long range forces of potentially large magnitude are overlayed on the short range forces that are needed for obtaining atomic images, and (4) the measurement of forces in the nN and pN regime is more difficult than measuring currents of magnitudes ranging from nA to pA.
Today, many companies provide commercial low temperature STM/AFMs that utilize the qPlus sensor. This helps the scientific community to study matter on the atomic scale with much greater precision and versatility, and breakthrough results are published monthly around the globe using this new technology.

ACKNOWLEDGMENTS
Tiny machinery has fascinated me as long as I can think. I thank Gerd Binnig for having offered me the opportunity to explore atomic force microscopy as a graduate student at the IBM Physics Group Munich thirty years ago. Gerd had received the Nobel Prize in Physics together with Heinrich Rohrer for the invention of STM in 1986. He resisted the many temptations to constantly travel around the world giving talks that go along with this highest honor and moved on to set very ambitious new goals, offering a choice of three potential PhD projects to me: (a) sequencing DNA by STM, (b) designing a tunneling detector to measure gravitational waves, or (c)

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scitation.org/journal/rsi explore and realize the potential of AFM to obtain true atomic resolution. I chose the latter, and that kept me busy for the rest of my life so far. Although Gerd started a highly successful company for computerized image analysis from 1995, he kept an open ear and door to discuss the further evolution of force microscopy even today. Christoph Gerber helped to set up the IBM Physics Group in Munich and set an example for boundless energy in cheerfully pursuing big scientific goals. He also helped me to return to science after almost 5 years in industry by establishing contact with Jochen Mannhart. I first met Calvin F. Quate in the late 1980s when he visited our group in Munich. He supplied us with cantilevers and served on the board of Park Scientific Instruments (PSI) where I worked as a senior scientist after receiving a PhD in Munich.
In his visits to PSI, Cal would often drift into the UHV research area, patting my back and stating: "You can do it, Franz," "it" meaning to atomically resolve the iconic silicon 7 × 7 surface by AFM. He also worked with us in the late 1990s on a few projects involving the imaging of graphite. Theodor Hänsch had hosted the IBM physics group in his laboratory space at Ludwig Maximilians University Munich from 1988 until 1995. Ted also loaned me some equipment in 1996 when I took the first steps of the qPlus development in my home laboratory in Munich. McKinsey&Company, Inc., supported the first steps to the invention of the qPlus sensor by offering paid and unpaid vacation in the summer of 1996. At a time, where the potential of the qPlus sensor were mainly pipe dreams, Jochen Mannhart offered me a permanent position in Augsburg. He cheerfully encouraged exciting new AFM projects and was fully engaged with brains, hands, and heart. Diploma and PhD students Stefan Hembacher, Markus Herz, Martin Breitschaft, and Martina Schmid and visiting scientist Toyoaki Eguchi worked hard on the first implementations of qPlus sensors at low temperatures in Augsburg. In Regensburg, the first PhD students Elisabeth and Thorsten Wutscher enabled first steps to AFM in ambient conditions using the qPlus sensors as well as the discovery of the phantom force. Joachim Welker achieved the first precise low temperature measurements in Regensburg, Florian Pielmeier achieved spin resolution by exchange force microscopy, and Maximilian Schneiderbauer and Matthias Emmrich built a custom designed low temperature microscope and produced wonderful data with it. Thomas Hofmann obtained very nice results on graphene, lateral force microscopy, and COFI data. Daniel Wastl was a huge driving force in ambient measurements, and the work in ambient conditions is continued forcefully with graduate students Dominik Kirpal, Korbinian Pürckhauer, and Hiroshi Ooe, guest from Kanazawa University. Ferdinand Huber perfected the science of force detection as well as subatomic resolution. Julian Berwanger worked on topological insulators and helped to get the NIST 10 mK STM/AFM of the group of Joseph Stroscio up and running. A big thank you also to Postdoctoral Fellows Sonia Matencio from Spain, Jay Weymouth from Canada, and Angelo Peronio from Italy. Sonia has performed beautiful experiments on clusters and on imaging molecules at room temperature. Jay arrived with an excellent knowledge about STM and surface science and has pushed forward ambient AFM, lateral force microscopy, and the phantom force studies, culminating in an award winning habilitation. Angelo has provided great progress in the combination of STS with AFM, together with our frequent guest, co-author, and visiting professor Norio Okabayashi from Japan. Anja Merkel and Florian Griesbeck have provided expert technical assistance in designing and machining mechanical and electrical parts and components. A big thank you for proofreading goes to Korbinian Pürckhauer (chapter I), Ferdinand Huber (chapter II), Julian Berwanger (chapter III), Angelo Peronio (chapter IV), Dominik Kirpal, Alexander Liebig, and Jay Weymouth (chapter V). Thanks also to Petra Wild for obtaining all the reprint permissions. One of the benefits of working at a university is the constant flow of young people into our group. Many of these young bachelor and master students have provided important impact by bringing their skills and curiosity into our group. I wish to thank the groups at IBM Almaden (Andreas Heinrich, Christopher Lutz, Don Eigler) and IBM Rüschlikon (Gerhard Meyer and Leo Gross) who were early adopters of the qPlus sensor and invited me to several visits and joint projects. Gerhard Meyer is a coinventor of the beam deflection technique 159 and demonstrated high resolution AFM in vacuum 160 as well as biaxial AFM 161 with it early on. The beam deflection technique is by far the most popular method to measure the deflection of cantilevers in an estimated 10 000 ambient AFMs around the globe. Usually, the "not-invented-heresyndrome" 177 would prevent a highly successful inventor to even try a competing technique, but Gerhard had the greatness to see some benefits, enabling his and Leo Gross' breakthroughs in imaging organic molecules with the qPlus sensor that opened a new field. Joseph Stroscio at NIST invited me to spend a sabbatical in his group where we just started to explore the possibilities of qPlus based STM/AFM at ultralow temperatures and high magnetic fields. Also, a big thanks to Jascha Repp who runs a phantastic group on atomic scale science on insulating films using STM/AFM next door here in Regensburg and provides continuous physical insights into surface science and related topics in our joint group seminar and discussions.

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scitation.org/journal/rsi further in ambient AFM at that time, we adapted the FM box to our vacuum AFM. The design of the vacuum AFM proceeded quickly-one of the blessings of working in silicon valley was the quick machine shops who would pick up a drawing in the evening and return the machined parts in the next morning. The piezoresistive cantilevers worked well in FM-AFM, and in November 1993, we submitted our first paper on its applications showing atomic steps and a lateral resolution below 3 nm 70 and atomic rows on Si. 69 In the spring and summer of 1994, a long standing challenge in atomic force microscopy was finally solved: obtaining atomic resolution on the silicon (111)-(7 × 7) surface. 71 At that point, I thought that my career in science is over-big goal achieved, nothing else on the horizon. As I had been hired by PSI to design a commercial UHV microscope, and the product achieved an outstanding goal, I found it hard to understand why the microscope did not sell by itself. To find that out, I joined the Munich office of McKinsey&Company, Inc., to explore a different realm of hard work. 189 However, there was one more thing about successfully obtaining atomic resolution on Si(111)-(7 × 7) that bothered me deeply: the oscillation amplitude of the cantilever had to be very large. Incredulously about its implications, I even showed a scale representation of that in Fig. 4 of Ref. 71. In my little spare time between those long consulting hours (and paid as well as unpaid vacation-thanks again to McKinsey), I finally found a reason why large amplitudes were required: stability. A much stiffer cantilever was required. I had learned through my diploma work with Gerhard Abstreiter at the Technical University of Munich that a single covalent Si-Si bond has a stiffness of 170 N/m. The cantilever I had used at PSI had only 10% of that stiffness. So, if one would manage to create levers with 1700 N/m, one would still achieve a frequency shift of 5% and be able to oscillate at small amplitudes. Furthermore, the new vocation at McKinsey encompassed several functional benchmarking studies. The idea of benchmarking is to identify a weak spot in a business operation and model whoever is best in that particular operation to get to an excellent performance level. 189 It dawned on me that, when musing about improvements of frequency modulation atomic force microscopy, we never studied the techniques elaborated by the champions of frequency measurements. Atomic clocks, of course. But quartz watches come next and are really simple. Most quartz watches use a quartz tuning fork that oscillates at f 0 = 2 15 Hz = 32 768 Hz. The tines of the tuning forks are typically 2-3 mm long and have a stiffness of about 2-10 kN/m. The two tines form two coupled oscillators that oscillate in an antiparallel mode where they reach Q-values of 100 000 when placed in an evacuated metal can. While piezoelectric sensors had been used in scanning probe microscopy with normal forces 13 and shear forces before, 45,136,137 the systematic use of them based on a mathematical analysis of the required sensor properties was new.
When I learned about quartz tuning forks, two questions came to mind: (a) can one make them to oscillate with amplitudes of an atomic diameter instead of their typical amplitudes of A ≈ 5 µm and (b) can one still obtain reasonably high Qvalues when one of the tines is tied down and the other one carries a tip as needed when turning it into a quartz cantilever. Management consultants typically do not have ready access to a well stocked laboratory, so I purchased an oscilloscope, a frequency counter, and an oscillator and borrowed an ambient STM from Theodor Hänsch at Ludwig Maximilians University Munich who had hosted our IBM Research Group Munich in his premises at the University before. Early experiments in the home laboratory confirmed that high Q values can be obtained with one fixed prong, and oscillation amplitudes can be reduced to at least nanometers. The combination of past experience, theoretical background, and promising new trials 130 gave me confidence for another change in career. I dubbed the new device "qPlus sensor," filed for and later obtained patents 72-76 and a registered trademark qPlus ® , 77 and searched for a way to turn its high promise into reality. Many people helped me in this effort; hopefully most are listed in the acknowledgments, but I am sure I forgot some. So thank you as well!