Bidirectional quantitative force gradient microscopy

Dynamic operation modes of scanning force microscopy based on probe resonance frequency detection are very successful methods to study force-related properties of surfaces with high spatial resolution. There are well-recognized approaches to measure vertical force components as well as setups sensitive to lateral force components. Here, we report on a concept of bidirectional force gradient microscopy that enables a direct, fast, and quantitative real space mapping of force component derivatives in both the perpendicular and a lateral direction. It relies solely on multiple-mode flexural cantilever oscillations related to vertical probe excitation and vertical deflection sensing. Exploring this concept we present a cantilever-based sensor setup and corresponding quantitative measurements employing magnetostatic interactions with emphasis on the calculation of mode-dependent spring constants that are the foundation of quantitative force gradient studies.


Introduction
Dynamic scanning force microscopy (dSFM) is a versatile sub-nanometer resolution tool for examining the surface topography and many more force-related properties of a huge variety of materials [1,2]. A basic dSFM setup requires a dSFM probe, which is typically a microstructured cantilever beam with a sharp tip. Also, it requires the means to excite and detect flexural oscillations of the cantilever, as probe-sample interactions result in changes of the resonance behavior of the probe. Appropriate measurement variables include the resonance frequency, the oscillation amplitude, and the phase shift of the cantilever oscillation. Scanning the probe above a sampleʼs surface allows for recording spatially resolved maps of the measurement variables. In many cases, dSFM can be sufficiently described by considering a simple harmonic motion of the dSFM probe at a single mechanical frequency. In more advanced approaches, however, the outcome of dSFM measurements can be significantly improved by either detecting additional frequency components of the dSFM signal or exciting the dSFM probe at more than one frequency. In this regard our paper presents an approach that enables either simultaneous or subsequent measurements of two orthogonal force gradients at two different resonant frequencies, while using only a one-dimensional excitation and detection scheme. Before describing this novel variant of dSFM, the advantages of multi-frequency versus mono-frequency measurements will be briefly discussed in the following.

The basic principles of dSFM
In mechanics, a free harmonic oscillator vibrates at its eigenfrequency, which is defined by its spring constant and its effective mass. The more complex oscillation of a damped harmonic oscillator driven by a timedependent sinusoidal force is described by a superposition of two frequencies: the frequency of the driving force and the eigenfrequency of the free oscillator during the transient response. In the steady state, the amplitude of the latter is suppressed by damping, but nevertheless the oscillatorʼs movement is still heavily influenced by the eigenfrequency. The eigenfrequency affects both the amplitude and the oscillationʼs phase with respect to the Any further distribution of this work must maintain attribution to the author (s) and the title of the work, journal citation and DOI. periodic driving force. A single frequency model in the steady state can be applied to dSFM if a single frequency excitation is used and if anharmonic contributions to the probe-sample interaction potential are neglected.
In order to understand the basics of signal formation in dSFM the potential energy of both the cantilever deformation and the probe-sample interaction can be considered. The potential energy of a free cantilever oscillation corresponds to the deformation energy of the cantilever beam for the flexural oscillation mode in consideration. For a sufficiently small oscillation amplitude it can be approximated quite well by a harmonic potential = U cu 2 2 . Here c is the effective spring constant, and u the corresponding deflection at a given point of reference. This potential when oscillating at the fundamental flexural mode deviates slightly from that of a static cantilever deflection induced by a point-like force acting on the cantileverʼs free end. Therefore, the effective force constants for the static deflection and for the fundamental flexural oscillation differ by a small amount. Mathematically, a description of this difference is given by the equivalence of the inverse static spring constant and the sum of all inverse dynamic spring constants [3]. The conservative part of the dSFM probe-sample interaction can be described by a local probe-sample potential. In case of small oscillation amplitudes the change of the probeʼs eigenfrequency is proportional to the second derivative of the probe-sample potential with respect to the spatial coordinate pointing along the tip oscillation direction, i.e. the probe-sample force gradient. In general, at large probe oscillation amplitudes the probe-sample force gradient might not be constant along the tip trajectory. Here, a weighted average of the gradient can be used [4]. The dependence of the eigenfrequency on the probe-sample interaction is employed in frequency modulation dSFM [5].

Multi-frequency dSFM
To date, many force microscopy approaches have been developed that include at their heart the excitation or measurement of more than one frequency. Even if the cantilever is excited at its fundamental flexural eigenfrequency, higher harmonics contribute to the periodic motion if the probe-sample force includes nonlinear components. The amplitudes of these higher harmonics contain valuable information on the probesample interaction. These amplitudes can be used to reconstruct the probe-sample force field [6]. They are related to higher order force gradients that decay very fast. Therefore, the detection of the amplitudes of higher harmonics can be used to enhance the spatial resolution of dSFM [7]. Depending on the cantilever geometry specific higher harmonics of the cantileverʼs fundamental flexural mode may coincide with a higher order flexural mode [8] leading to a resonant amplification of those higher harmonics. External excitation of two flexural eigenmodes of the cantilever, e.g. the fundamental mode and a higher order mode, allows for a simultaneous tracing of both the topography using the interaction response on one mode and compositional information contained in the higher order mode response [9,10]. Transferred to magnetic force microscopy (MFM), a descendant of dSFM, such an approach based on the excitation of two flexural eigenmodes enables a parallel measurement of the sample topography and the long range magnetostatic interactions of the ferromagnetic tip with the sample stray field [11][12][13]. By this means some disadvantages of the widely used twopass MFM technique can be avoided. A general overview of multi-frequency dSFM has been given by Garcia und Herruzo [14].

Bidirectional dSFM
Single-directional force or force gradient measurements that only probe a lateral rather than the usual vertical direction are well-established. Rotation of the force sensor by 90°leads to the pendulum or shear force configuration. Giessibl et al used this geometry to perform lateral force gradient and friction measurements at the atomic scale [15]. MFM based on the pendulum geometry is sensitive to lateral force derivatives of magnetostatic interactions [16,17]. This setup, however, cannot easily be switched to perpendicular sensitivity. It is limited to in-plane measurements. In contrast, torsional resonance mode dSFM [18][19][20] that is sensitive to in-plane forces can be combined with perpendicular modes based on flexural oscillations.
Recently, we published a concept and a corresponding experimental implementation of a cantilever-based sensor in which the excitation of two different eigenmodes allowed for a successive or simultaneous resonant oscillation of the sensorʼs tip along two orthogonal directions [21]. A high aspect ratio single-domain iron nanowire was employed to interact with the magnetic stray field to be measured. For practical reasons we used Fe nanowires contained in and stabilized by multiwalled carbon nanotubes (FeCNT). A selected FeCNT was attached to a cantilever beam via a spacer element. Exciting the cantileverʼs fundamental flexural oscillation led to the obvious vertical oscillation of the free end of the probeʼs magnetic nanowire. In contrast the free end oscillated along a horizontal direction, if the cantileverʼs second flexural mode was excited. Such a bidirectional sensor, equipped with a magnetic monopole-like tip, is sensitive to magnetic field derivatives along these directions. This sensitivity is based on the measurement of two force derivatives, e.g. ∂ ∂ F z z and ∂ ∂ F x x , if z denotes the vertical and x denotes the horizontal direction. Since then, Stirling has suggested a similar concept of a multipurpose vertical and lateral force microscopy sensor: here, a wire representing both the spacer and the probe tip was proposed to be attached to the center of a two-side clamped quartz beam [22]. Again the fundamental and second mode flexural beam oscillations give rise to vertical and horizontal tip oscillations, respectively.

Outline of this work
After this introduction, we discuss the concept of bidirectional quantitative force gradient microscopy and present appropriate cantilever-based sensors. We build on our previous results [21] and generalize them in terms of force gradients. We calculate effective dynamic spring constants for two resonant oscillation modes of the cantilever, which correspond to two orthogonal oscillation directions of the probe tip. As an experimental force field we utilize the magnetostatic interaction between a magnetic tip and the well-defined magnetic stray field of a multilayer sample, so in essence this is an MFM experiment. Based on the calculated spring constants we convert resonance frequency shift measurements of probe-sample interactions for the fundamental flexural mode and the second order mode into perpendicular and in-plane force derivative maps, respectively. Then, a two-dimensional map of measured magnetostatic vertical force derivative data is used to calculate the corresponding in-plane force derivative; the latter is compared to the measured in-plane data. Finally, we confirm the capability to extract quantitative data from our bidirectional force gradient measurements.

Sensor concept and spring constant calculation
Knowledge of the dynamic spring constants of an dSFM probe is vital to extract quantitative force gradient data from dSFM measurements. Therefore, in addition to presenting our novel sensor concept we provide a detailed description of the calculation of appropriate dynamic spring constants. Figure 1 shows the measurement principle of bidirectional force gradient microscopy, the corresponding sensor structure, and the coordinate system in use. The probe tip is attached to the cantilever via a spacer element at the nodal point of the cantileverʼs second flexural oscillation mode. Of course any modifications on cantilevers may alter the location of nodal points. However, this can be neglected if the mass and the moments of inertia of the spacer element are much smaller than those of the cantilever. Furthermore, the spacer element needs to be stiff, i.e. its eigenfrequencies should exceed the frequencies used when operating the sensor. Exciting the cantilever at the fundamental or second order flexural mode leads to an oscillation of the probe tip along the z-or x-direction, respectively. In dSFM the resonance frequency shift is proportional to the weighted average of ∂ ∂ F s s . Here, F s is the projection of the interaction force on the tip oscillation direction and s corresponds to the spatial coordinate along the mode dependent oscillation direction. If we neglect a spatial dependence of ∂ ∂ F s s , the frequency shift Δf reads:

Sensor concept
n s dyn f n denotes the resonance frequency of the cantilever and c dyn the effective dynamic spring constant; f n and c dyn are mode-dependent. Thus, dSFM measurements according to figures 1(b) and 1(c) provide Δf data that are proportional to ∂ ∂ F z z and ∂ ∂ F x x , respectively.

Static spring constant
In order to extract quantitative force gradients we need to know the dynamic spring constants of the probe with regard to the tip location and the particular vibration mode. Starting from the basic static spring constant of a cantilever beam we shall develop equations for the dynamic spring constants with respect to the probe tip position. All external force gradients are regarded as point-like and acting on the probe tip only. This condition is usually well satisfied in dSFM in general and with our long probe tip in particular.
To make the following calculations as simple as possible we use the special case of a constant cantilever cross section. We apply a dimensional approach [23] to calculate the static spring constant c L stat of a cantilever with a point load acting on its free end: L stat 3 with E being Youngʼs modulus, I the second moment of area of the cross section, and L the length of the cantilever beam. I can be calculated from the cantileverʼs cross section geometry data which can be derived from scanning electron microscopy (SEM) measurements, for example. This approach, however, has the disadvantage of forwarding the uncertainties of E as well as I with the latter being proportional to the cube of the height of the cantilever h 3 . To circumvent this source of inaccuracy we use the measured resonance frequency f 1 of the fundamental flexural mode of the cantilever, which is related to EI by with A being the cantileverʼs cross sectional area. Now c L stat has only a linear dependence on all geometrical quantities of the cantilever; no knowledge of E is required. Instead, we need data on ρ, which are easily available.

Dynamic spring constants
As indicated in the introduction, the momentary deformation shape, or mode shape, of an oscillating cantilever and the related potential energy is different compared to the case of static bending. The dynamic spring constants of the fundamental and the second mode flexural oscillation, c L,1 dyn and c L,2 dyn , respectively, are related to their static counterpart in the following way   Next we calculate the dynamic spring constant of the second mode corresponding to an in-plane oscillation of the probe tip. Assuming a rigid spacer element, the horizontal displacement Δx of the tip in the second mode is proportional to the slope of the cantilever at the position of the spacer element in a small angle approximation: dist 2 t i p l dist denotes the distance between the probe tip and the neutral fiber of the cantilever. Equation (6) can be adapted for the horizontal displacement Δx with the same argumentation as given above: x max elastic ,2 dyn 2 tip By combining equations (6), (10) and (11) this results in

Experimental verification of the quantitative sensor concept using magnetostatic force gradients
In the following we experimentally validate our sensor model as introduced in the previous section. We measure and simulate force derivatives resulting from magnetostatic interactions of a ferromagnetic nanowire probe and a ferromagnetic multilayer sample, i.e. we perform MFM-type experiments in order to confirm the more general measurement principle of bidirectional force gradient microscopy by comparing measured to calculated results.

Probe preparation
The basis of our sensor is a tipless silicon cantilever. We remove the triangular free end by focused ion beam (FIB) milling in order to obtain an approximately constant cross section along the length of the cantilever. A constant cross section is not mandatory but it simplifies the nodal point calculation of the second flexural vibration mode. At the calculated position of this nodal point (x tip ) a pillar-like spacer element is grown onto the cantilever by FIB-assisted deposition of carbon. Usual dSFM or MFM instruments require the cantilever to be introduced at a certain angle deviating from parallel to the sample plane. This causes the oscillation direction to be slightly off-axis. For the second mode such a deviation can be taken into account when defining the orientation of the spacer element to make sure the tip oscillation is precisely parallel to the sample surface. The FeCNTs used as magnetostatic interaction tips of our probes were grown by chemical vapor deposition [26]. The iron fillings are high aspect ratio nanowires having lengths of several microns and diameters in the range of 15 nm to 50 nm. It has been shown that the iron nanowires have a magnetic single domain configuration with the easy axis parallel to the long wire axis. With the help of a micro-manipulator, a single FeCNT is attached to the end of the spacer element by electron beam induced carbon deposition. Finally, we apply electron beam assisted etching in water vapor environment to remove unfilled parts of the FeCNT at its free end [27,28]. Figure 2 shows SEM images of the bidirectional probe that has been used for the measurements described below. Its properties include the cantilever length μ = ± L (217.  figure 2 is the°10 inclination angle of the spacer element with respect to the cantileverʼs surface normal to ensure that the tip movement in the second mode is parallel to the sample surface. As mentioned before, this is necessary to compensate for the cantilever tilting that is required by our MFM instrument. Please note such compensation has not been considered in the probe shown in the SEM images of figure 1.

Ferromagnetic test sample
Our perpendicular anisotropy test samples, which provide well defined magnetic stray fields are Co/Pt multilayer thin films with the following architecture: Pt(5nm) [Pt(0.9 nm)Co(0.4 nm)] Pt(2 nm) 100 . In the zero field state, where the sample shows a band domain configuration the domain structure, as shown in figure 3(a), it is usually not affected by MFM probe stray-fields and therefore constitutes an ideal reference as shown for similar multilayers in previous work [29]. Vibrating sample magnetometry measurements revealed an uniaxial anisotropy constant of

Magnetostatic force gradient measurements
The measurements of the fundamental and second mode cantilever resonance frequency were conducted under high vacuum conditions in a NanoScan AG hr-MFM employing frequency modulation. Tip oscillation amplitudes of around10 nm were used in both modes. All measurements were performed at a tip-sample distance of = z 80 scan nm. Figures 3(a) and (b) show MFM force gradient images corresponding to the perpendicular and the in-plane tip oscillation direction, respectively. The primary MFM signal, i.e. Δf , is already converted into force gradient data using the calculated spring constants and equation (1). The software WSxM [30] was used to apply a plane subtraction to the data and, in order to take small x-y difts into consideration, extract overlapping image parts.

Magnetostatic force gradient calculation
The force gradient as displayed in figures 3(a) and (b) is caused by magnetostatic interactions between the stray field of the sample and the magnetization of the FeCNT. Using two-dimensional Fourier transforms of the magnetic field distribution, k k z H ( , , ) x y , where the spatial in-plane coordinates (x, y) are replaced by the spatial frequencies k k ( , ) x y , enables the calculation of the x and y components of the magnetic stray field from the zcomponent in the x-y plane, except for the average value corresponding to the point = = k k ( 0, 0) x y [31]: x y z x y k denotes the absolute value of the in-plane wave vector ( = + k k k x y 2 2 ). The tilde sign ( ∼ ) denotes the Fourier transform of the corresponding quantity. The nabla operator in Fourier space is given by = −  ik ik k ( , , ) x y . Equation (15) leads to a relation between the x-component and the z-component of the field gradient: Taking the probe-sample interaction into account, this equation can be converted into a corresponding force gradient relation. In the simple case of a magnetic point charge tip, a multiplication of equation (16) by the tip charge−q and by μ 0 leads to: For arbitrarily shaped MFM tips the force gradient in Fourier space results from the multiplication of the Fourier transform of ∂ ∂ H z z with a so called tip transfer function k k (TTF)( , ) x y , which can be fully determined by a calibration measurement of an appropriate reference sample [29,31]. Although the TTF now depends on the spatial frequencies k k ( , ) x y , it is independent of the oscillation direction of the sensing tip, as long as the oscillation amplitude is smaller than the length scale of force gradient changes, which ensures an amplitude independent force gradient. The condition cited in connection with equation (1) is thus sufficient to ensure that equation (17) applies for an arbitrary MFM sensor.
After applying equation (17) to the experimental ∂ ∂ F z z data, the inverse Fourier transform provides calculated ∂ ∂ F x x data, as shown in figure 3(c), which can be compared with the corresponding experimental lateral force gradient ∂ ∂ F x x data shown in figure 3(b). The excellent quantitative agreement is further illustrated  Figure 4 shows a line scan through the obtained symmetrical tip field gradient. It is compared to the stray field characteristics of an ideal point charge or monopole-like tip, which has been demonstrated to be a good description for high aspect ratio single-domain nanowire MFM tips [32,33]. The stray field gradient of a monopole q along a line at a distance δ + z ( ) scan is given by [34, p 28]:

Conclusions
In this paper, we introduced a new measurement principle of quantitative bidirectional force gradient microscopy. Quantitative force gradient measurements rely on the knowledge of the corresponding dynamic spring constants. We calculated the latter for the case of a constant cantilever cross section. Force gradient measurements exploiting magnetostatic interactions of a ferromagnetic nanowire tip with a well-characterized magnetic multilayer film agreed very well with calculations confirming the presented quantitative approach. These results prove the applicability of the bidirectional force gradient microscopy for a direct, fast and quantitative real space analysis of force gradients in two dimensions. Our presented measurement principle has a strong practical advantage, because it relies solely on flexural vibrations, i.e. all microscopy modes are based on vertical probe excitation and vertical deflection measurement only.