Abstract
Magnetic force microscopy (MFM) is a scanning microscopy technique that is commonly employed to probe the sample’s magnetostatic stray fields via their interaction with a magnetic probe tip. In this work, a quantitative, singlepass MFM technique is presented that maps one magnetic strayfield component and its spatial derivative at the same time. This technique uses a special cantilever design and a special highaspectratio magnetic interaction tip that approximates a monopolelike moment. Experimental details, such as the control scheme, the sensor design, which enables simultaneous force and force gradient measurements, as well as the potential and limits of the monopole description of the tip moment are thoroughly discussed. To demonstrate the merit of this technique for studying complex magnetic samples it is applied to the examination of polycrystalline MnNiGa bulk samples. In these experiments, the focus lies on mapping and analyzing the strayfield distribution of individual bubblelike magnetization patterns in a centrosymmetric [001] MnNiGa phase. The experimental data is compared to calculated and simulated strayfield distributions of 3D magnetization textures, and, furthermore, bubble dimensions including diameters are evaluated. The results indicate that the magnetic bubbles have a significant spatial extent in depth and a buried bubble top base.
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Introduction
Current developments in the area of nanomagnetism, including studies on very small noncollinear magnetic textures, call for more advanced, quantitative magnetic microscopy solutions. Nanoscopic magnetization patterns can be accessed indirectly via their emanating strayfield distributions by a variety of microscopy techniques such as scanning probe microscopy based on single nitrogenvacancy defects in diamond^{1}, superconducting quantum interference devices^{2} and force sensors equipped with ferromagnetic tips. The latter, magnetic force microscopy (MFM), provides spatially resolved maps of magnetostatic probesample interactions at the nanometer scale^{3,4,5,6}. Basic MFM is a relatively straightforward method being widely available in labs worldwide. Therefore, it is extensively used for nanomagnetic characterization^{7} and, in particular, for skyrmion research^{8,9}. Here, MFM is not only beneficial for providing strayfield maps; the magnetic tip may also act as a tool for localized skyrmion creation and manipulation^{10}. Sophisticated quantitative MFM (qMFM) data evaluation methods even enable the analysis of the skyrmion type (Néel vs. Bloch), helicity and size of skyrmions, and the force required to move individual skyrmions^{11,12,13}.
The capabilities of MFM can be further enhanced by multifrequency^{14} cantilever detection including bimodal^{15,16,17,18,19}, intermodulation^{20} and side band detection^{21}.
So far, detailed approaches for qMFM were developed including: (i) transfer function approaches in Fourier space in which calibration procedures based on high resolution MFM maps of appropriate reference samples provide effective magnetic charge distributions of the probe tip or tip strayfield derivatives;^{22,23,24,25,26} (ii) description of the probe’s magnetic response by magnetic point poles^{27} (i.e. dipole^{28,29,30}, monopole^{31,32,33}, or “pseudopole“^{34}); and (iii) calibration schemes related to specific sample types, e.g. spherical dipole type particles^{35}.
The majority of commercially available MFM probes is based on Si cantilevers with magnetic thinfilm coated pyramidal or conical tips. It turns out that their magnetostatic behavior differs from that of a magnetic point dipole or monopole^{27}.
A number of reports describe the application of magnetic nanowires as major components of MFM probes. Such nanowire probes offer a variety of promising features including high spatial topographic and magnetic resolution^{36}, predictable single domain magnetization stabilized by shape anisotropy^{37,38,39}, superior mechanical stability in case of magnetic nanowires encapsulated in multiwall carbon nanotubes^{40,41}, and in addition, monopoletype strayfield characteristics for high aspect ratio singledomain magnetic nanowires^{31,32,33,42,43}. Furthermore, in addition to their magnetic functionalities, nanowires may directly constitute mechanical force transducers for MFM operation^{43,44,45}. Mattiat et al.^{43} used a nanowirebased MFM approach to quantitatively measure the amplitude of an oscillating magnetic field with nT/√Hz sensitivity at low temperatures.
Here, we report on a single pass multifrequency MFM approach that provides immediately accessible quantitative maps of both the magnetic field component B_{z} and its spatial derivative \({B}_{z}^{\prime}=\partial {B}_{z}/\partial z\). \({B}_{z}^{\prime}\) maps show much sharper magnetic features compared to B_{z} maps. This is not a consequence of spatial resolution or sensitivity, but it is rather a demonstration that field derivatives simply amplify higher spatial frequencies compared to their respective field distributions^{22}. Our simultaneously measured B_{z} and \({B}_{z}^{\prime}\) maps immediately illustrate these characteristic differences. Furthermore, recording both quantities may provide information on the magnetization distribution in the depth of the sample in specific cases as shown herein.
Our magnetostatically active probe element is an ironfilled carbon nanotube (FeCNT), containing a single domain iron nanowire (FeNW) with an aspect ratio as high as 250. We simultaneously measure the deflection of a tailored lowstiffness cantilever and the frequency shift of one of its flexural vibration modes in fully noncontact operation. Snapin events are avoided by electrostatic distance control employing a higher order flexural vibration mode^{46}. The monopoletype characteristics of the FeNW tip allows for a straightforward translation of the cantilever deflection and the frequency shift into B_{z} and \({B}_{z}^{\prime}\), respectively. The validity of the pointmonopole description with respect to particular spatial wavelength intervals of magnetic strayfield patterns will be thoroughly discussed herein.
Skyrmionic spin textures constitute a promising class of nanomagnetic configurations with particlelike properties. Magnetic skyrmions are associated with topological protection and attract a lot of interest both in fundamental magnetism research and spintronics such as advanced memory technologies and neuromorphic computing^{47}. There are various types of magnetic skyrmions with different topological charge Q_{t} including Bloch skyrmions (Q_{t} = 1), Néel skyrmions (Q_{t} = 1), and biskyrmions (Q_{t }= 2). Biskyrmions, i.e. bound pairs of Bloch skyrmions, have been observed by Lorentz transmission electron microscopy in a centrosymmetric bilayered manganite^{48} and in hexagonal MnNiGa^{49,50,51,52}. Skyrmions in MnNiGa were observed at various temperatures including room temperature and were reported to survive even at zero external field^{50}. In other studies, however, it was shown that Lorentz transmission electron microscopy data of magnetic hard bubbles, i.e., typeII bubbles with Q_{t} = 0, might be misinterpreted under certain conditions as Q_{t} = 2 biskyrmions^{53,54}. Also, a recent soft Xray holography investigation found singlecore typeII bubbles in MnNiGa^{55}. Nevertheless, theoretical studies present the prospect of a biskyrmion formation in centrosymmetric materials with uniaxial magnetic anisotropy^{56,57,58}. The above mentioned experimental MnNiGa studies were limited to thin lamella samples. Here, we investigate magnetic nanodomains in bulk MnNiGa.
After detailing the fundamentals of the approach for simultaneous B_{z} and \({B}_{z}^{\prime}\) measurements in the “Concept of combined field and field gradient microscopy” section of the Results and Discussion part, we report on the experimental realization of a corresponding MFM probe design in “Sensor design and fabrication” in the Results and Discussion. Finally, we apply our technique to the study of magnetic bubblelike domains in bulk [001] MnNiGa (“Quantitative B_{z} and \({B}_{z}^{\prime}\) measurements on bulk [001] MnNiGa” in the Results and Discussion). We focus on the specific question of how the investigated bubbles continue along the third dimension, as discussed for other materials^{59,60}, and determine their diameters ranging from 120 nm to 200 nm. Our results indicate that the magnetic bubbles have a significant spatial extent in depth and a buried bubble top base.
The presented method will boost the capabilities of commonly used magnetic force microscopy equipment towards quantitative nanomagnetic microscopy.
Results and discussion
Concept of combined field and field gradient microscopy
Our MFM measurement concept allows for combined quantitative magnetic field B_{z} and field derivative \({B}_{z}^{\prime}\) scanning with simultaneous sample topography tracking in a singlepass, true noncontact operation mode (see Supplementary Note 1 and Supplementary Table S1). The presented microscopy modes can be performed in standard AFM/MFM setups but require a specific probe design as introduced in “Sensor design and fabrication” in the Results and Discussion. Such probes are not yet commercially available.
As a key feature, once the sensor is calibrated, our MFM approach provides direct B_{z} and \({B}_{z}^{\prime}\) data at any location in real space without any postprocessing and without the need for carrying out additional data evaluation in Fourier space.
The concept map (Fig. 1) illustrates an example on how qMFM for both magnetic fields and magnetic field gradients can be achieved. A magnetic point charge exposed to a magnetic field or field gradient is subject to a force or a force gradient, respectively. Forces and force gradients can easily be measured via deflections and frequency shifts of microcantilevers. In terms of magnetostatics, the end of an ideal infinitely long and infinitesimally thin ferromagnetic wire, i.e. a Dirac string, behaves like a point monopole (probe nanowire shown in the upper part of Fig. 2a) and generates a radially symmetric magnetostatic strayfield. Thus, in our approach, highaspect ratio iron nanowires are used to approximate such magnetic point monopole probes. A quantitative evaluation of the validity and the limits of the point monopole model when describing such magnetic nanowire probes is discussed in what follows (see Fig. 2b–f).
To enable sensitive static force measurements, low spring constant cantilevers are necessary. Generally, they work well for contact mode measurements or in pendulum geometry. However, they are not easily usable for dynamic mode measurements in conventional cantilever geometry since attractive contributions to the overall tipsample interaction may lead to snapin events that can damage or change the tip’s characteristics. Thus, intermittent contact or tapping mode is very difficult to implement especially when using soft cantilevers. To enable the use of low static spring constant k_{stat} cantilevers in dynamic operation modes, an independent tipsample distance control is required that prevents any direct tipsample contact. Therefore, we employ an electrostatic distance control^{17} that ensures a safe tipsample distance for true noncontact operation.
Once an appropriate tipsample distance z_{ts} is set and maintained and, additionally, the tip monopole moment q_{t} is known, measurements of the cantilever deflection ∆z and of the ith order flexural mode resonance frequency shift ∆f_{i} yield \({B}_{z}\left({\hat{z}}^{* }\right)\) and \({B}_{z}^{\prime}\left({\hat{z}}^{* }\right)\), respectively (see Fig. 1). Here, \({\hat{z}}^{* }\) refers to the zposition of the probe’s magnetic pole q_{t} that is closest to the sample (Fig. 2a).
Prospects of simultaneous quantitative field and gradient measurements
Simultaneous quantitative magnetic field \({B}_{z}\left(x,y,{\hat{z}}^{* }\right)\) and field derivative \({B}_{z}^{\prime}\left(x,y,{\hat{z}}^{* }\right)\) mapping allow for direct linear Taylor approximations of B_{z} at zlocations in the vicinity of the measurement location \((x,y,{\hat{z}}^{* })\), for which:
We illustrate the potential of simultaneous B_{z} and \({B}_{z}^{\prime}\) measurements for the case of three simple and idealized strayfield distributions (Fig. 3). In case of buried pointlike strayfield sources, both their depth \(\check{z}\)* below the sample surface and their magnetic monopole moment (i.e., magnetic charge) q_{s} or dipole moment m_{s} can be directly inferred from simultaneously available B_{z} and \({B}_{z}^{\prime}\) data (Fig. 3a, b, and Supplementary Note 2). Furthermore, in case of a sample that generates a B_{z} distribution that is sinusoidal along inplane directions with wavenumber \({{{{{{\bf{k}}}}}}}_{{xy}}=k_{{xy}}=\sqrt{{k}_{x}^{2}+{k}_{y}^{2}}\), the ratio of \({B}_{z}^{\prime}\) to B_{z} immediately provides k_{xy} without the necessity of performing a full 2D surface MFM scan (Fig. 3c and Supplementary Note 2). \({{{{{{\bf{k}}}}}}}_{{xy}}=\left({k}_{x},{k}_{y}\right)\) is the corresponding inplane wavevector.
Figure 3 sketches the potential of a simultaneous B_{z} and \({B}_{z}^{\prime}\) measurement. Inferred quantities like depth or width of the magnetization textures can be directly accessed, if the measured magnetization textures are well approximated by idealized monopoles or dipoles, or if being periodic in a lateral direction (i.e. parallel to the surface).
B_{z} and \({B}_{z}^{\prime}\) complement each other. On the one hand, B_{z} might be considered as more fundamental data. On the other hand, as mentioned in the Introduction, \({B}_{z}^{\prime}\) maps generally show sharper features since higher spatial frequencies are amplified by a factor of k_{xy} as compared to B_{z} mapping^{22}.
A perfect magnetic point monopole probe enables us to measure B_{z} and \({B}_{z}^{\prime}\). A single domain magnetic nanowire exhibits several aspects that lead to deviations from the magnetic behaviour of a magnetic point monopole: (i) a finite nanowire (NW) length, (ii) a nonzero NW diameter, and deviations from a homogenous NW magnetization parallel to the NW axis due to (iii) demagnetization effects, and (iv) external magnetic fields (Fig. 2b–f).
Validity of the point monopole approximation: finite length and nonzero diameter effects of nanowire probes
In the experimental part: “Sensor design and fabrication”, our MFM sensor is presented which features a \({d}_{{{{{{\rm{Fe}}}}}}}=20\) nm and \({l}_{{{{{{\rm{Fe}}}}}}}=5\) µm iron NW as magnetic probe. Compared to the hypothetical singlepoint monopole probe, this real NW comprises two magnetic poles with the magnetic charge of each pole distributed over the NW’s cross section. We quantitatively evaluate the effects of nonzero diameter and finite length that constitute deviations from a point monopole probe behaviour. According to Hug et al.^{22}, the relation between the force at a magnetic NW probe and the field can be described by a force transfer function (FTF) that depends on the wavenumbers k_{x}, k_{y} of the field distribution.
For a NW probe with squareshaped crosssection of side length d, the FTF reads^{22,38}:
in which \({M}_{{{{{{\rm{S}}}}}}}\) and \(l\) are the NW probe saturation magnetization and the NW length, respectively. Note that for l → ∞ and \({k}_{x},{k}_{y}\ll \frac{2}{d}\) the \({{{{{{\rm{FTF}}}}}}}_{{{{{{\rm{NW}}}}}}}\) approaches the NW’s magnetic charge \({q}_{{{{{{\rm{t}}}}}}}={d}^{2} {M}_{{{{{{\rm{S}}}}}}}\). Under these conditions, the probe shows the same characteristics as a point monopole.
Analogously, the FTF of a NW probe with circular NW crosssection can be derived by taking advantage of an expression for the Fourier transformation of a unity circle, described by spherical Bessel functions of the first kind. After scale conversion and normalization with respect to the magnetic NW diameter d, we obtain:
with \({k}_{{xy}}=\sqrt{{k}_{x}^{2}+{k}_{y}^{2}}\). Furthermore, for simplicity, here we use the same symbol d as above for the side length of a squareshaped crosssection. Again, for l → ∞ and \({k}_{x},{k}_{y}\ll \frac{2}{d}\) the \({{{{{{\rm{FTF}}}}}}}_{{{{{{\rm{NW}}}}}}}\) approaches the magnetic charge \({q}_{{{{{{\rm{t}}}}}}}={\frac{{{{{{\rm{\pi }}}}}}}{4}d}^{2}{M}_{{{{{{\rm{S}}}}}}}\).
In Fig. 4, calculated FTF(λ) normalized by the FTF of a point monopole probe, i.e., by q_{t}, are shown. Instead of k_{xy} we now use the wavelength \(\lambda =2{{{{{\rm{\pi }}}}}}/{k}_{{xy}}\) to describe the spatial variation of the sample’s strayfield. The \({{{{{\rm{FTF}}}}}}(\lambda )/{q}_{{{{{{\rm{t}}}}}}}\) ratio for a magnetic probe with NW dimensions similar to that of our fabricated MFM probe introduced below is shown in Fig. 4a. For a broad band of λ values the FTF(λ) almost perfectly matches q_{t}. It implies that in this range the simple point monopole description with magnetic charge q_{t} is fully sufficient for high precision B_{z} and \({B}_{z}^{\prime}\) mapping. The FTF(λ) reduction for very small and very large λ can be attributed to the finite dimensions of d and l, respectively. Figure 4b–e reveal corresponding calculations for a variety of d  l pairs.
In Fig. 4c, e, λl and λd phase diagrams depict calculated areas with \({{{{{\rm{FTF}}}}}}(\lambda )/{q}_{{{{{{\rm{t}}}}}}}\) ratios exceeding 99%, 95%, and 90%. These calculated areas can be approximated by triangles even though there are no sharp vertices on the lefthand side of Fig. 4c and the righthand side of Fig. 4e. The upper area boundaries are mainly predefined by l. For large l (in Fig. 4c) and small d (in Fig. 4e), these upper boundaries λ_{max} are proportional to the NW length l.
They can be approximated by equating the \((1{{{{{{\rm{e}}}}}}}^{{k}_{{xy}}l})\) term of equation 3 with 99%, 95% or 90% and substituting k_{xy} by \(\lambda =2{{{{{\rm{\pi }}}}}}/{k}_{{xy}}\). For the 99% case \({\lambda }_{{{\max }}}\approx 1.36l\) and for the 90% case \({\lambda }_{{{\max }}}\approx 2.73l\). As an example, an l=1 µm NW MFM probe can be used to map B_{z} and \({B}_{z}^{\prime}\) of a periodic λ ≤ 1.36 µm strayfield distribution with ≥99% accuracy using the simple point monopole model.
The lower boundaries λ_{min} of the triangles in Fig. 4c, e are mainly defined by the NW diameter d. For large l (in Fig. 4c) and small d (in Fig. 4e) we get \({\lambda }_{{{\min }}}\approx 11.0d\) (or \({\lambda }_{{{\min }}}\approx 3.44d\)) corresponding to 99% (or 90%). As an example, a NW MFM probe with d = 10 nm can be used to map B_{z} and \({B}_{z}^{\prime}\) of a periodic λ ≥ 110 nm strayfield distribution with ≥ 99% precision using the simple point monopole approach.
Validity of the point monopole approximation: nanowire probes with nonhomogeneous and nonaligned magnetization
Both nonhomogeneous magnetization of the NW and a NW magnetization direction deviating from the long NW axis lead to deviations of a real NW’s behaviour from that of a magnetic point monopole. Apart from nontrivial internal magnetization effects deviations can additionally be promoted by local sample strayfields or an external applied field.
Therefore, the internal magnetization structure of a simple l = 5 µm Fe cylinder with its axis oriented along the zdirection was modelled using the MuMax3 micromagnetic simulation software (version 3.9.1c)^{61,62}. For an individual cell, the spatial extent in zdirection amounts to 2.4 nm, not exceeding the exchange length of iron of approximately 3.5 nm. For the exchange constant, we assume 22 pJ/m being within the reported range of values for pure iron^{63,64,65,66}.
For the saturation magnetization, \({M}_{{{{{{\rm{S}}}}}}}({{{{{\rm{Fe}}}}}})=1710\,{{{{{\rm{kA}}}}}}/{{{{{\rm{m}}}}}}\) was used^{67}. Here, we neglected the effect of magnetocrystalline anisotropy as the overall magnetic anisotropy is mainly governed by the highaspect ratio NW’s shape anisotropy^{37}.
In the simulation results of Fig. 5, M_{z}/M_{S} along the long NW axis (zaxis) is shown. For a d = 21 nm NW, \({M}_{z}/{M}_{{{{{{\rm{s}}}}}}}\, > \,0.97\) at the free NW end even in case of a 300 mT (−200 mT) applied field nearly parallel (antiparallel) to the zaxis.
Here, we assumed that the applied fields do not exceed the switching field of the NW. For similar FeNWs and external fields parallel to the NW axis, switching fields between 100 mT and 400 mT are reported^{68,69,70}. Small deviations from \({M}_{z}/{M}_{{{{{\rm{S}}}}}}=1\) are present for the last 10 nm close to the NW tip apex. The application of a perpendicular 100 mT field leads to a small overall reduction of M_{z}, i.e., \({M}_{z}/{M}_{{{{{\rm{S}}}}}}\approx 0.99\) along the entire NW and a further decrease towards the NW end. For larger NW diameters, vortex states form close to the NW free end which are accompanied by a local reduction of \({M}_{z}/{M}_{{{{{\rm{S}}}}}}\). Altogether, small NW diameters and external fields much smaller than both NW anisotropy and switching fields support the application of a point monopole model for the magnetostatic behaviour of magnetic NWs.
Sensor design and fabrication
We developed a cantilever design based on the following demands:

(i)
low static spring constant \({k}_{{{{{{\rm{stat}}}}}}}\);

(ii)
higher order flexural modes with high resonant frequencies, i.e., the frequency ratios f_{i}/f_{1} should be much larger as compared to the case of a simple EulerBernoulli cantilever;

(iii)
efficient fabrication routine by focused ion beam (FIB) preparation and micromanipulation.
Note that requirement (ii) is related to the ith mode’s settling time \({\tau }_{i}\), which represents the bandwidthlimiting factor for a distance control relying on the oscillation amplitude \({A}_{i}\) as a feedback loop input parameter. Further details are introduced in section: “Magnetic force microscopy setup and modes of operation”. The sensor preparation according to our MFM sensor concept is based on a commercial picketshaped cantilever (Nanoworld TL1) consisting of highly doped (001) singlecrystalline Si with nominal \({k}_{{{{{{\rm{stat}}}}}}}=30\,{{{{{\rm{mN}}}}}}/{{{{{\rm{m}}}}}}\). The dimensions of the cantilever geometry as provided by the manufacturer read: length \(l=500\,{{{{{\rm{\mu m}}}}}}\), thickness \(t=1\,{{{{{\rm{\mu m}}}}}}\), width w = 100 µm. To support low spring constants, we selected a cantilever with a particularly low thickness. The cantilever dimensions, as measured by scanning electron microscopy (SEM) read: total cantilever length l = (503.8 ± 2.2) µm, the length of the triangular part \({l}_{{{{{\rm{t}}}}}}=\left(93.0\pm 1.7\right)\,{{{{{\rm{\mu m}}}}}}\), the total width w = (93.6 ± 1.4) µm, and the thickness t = (0.795 ± 0.015) µm.
To fulfil the requirements mentioned above, we applied FIBbased line cutting along a predefined Πshape using a combined SEMFIB (ZEISS 1540 XB). The result is a doublecantilever beam structure and an inner paddle as shown in the SEM image of Fig. 6a. The desired low k_{stat} is mainly determined by the combined width 2 · b of the two remaining high aspect ratio beams. Those beams with b = (10 ± 1.0) µm connect the paddle structure to the carrier chip as shown in Fig. 6a. The width b is much smaller than the pristine cantilever width w. In fact, the rectangular inner paddle has a huge impact on the cantilever dynamics but negligible influence on the static spring constant.
For optimizing the sensor design and to characterize its mechanical properties, we employed simulations based on finite element analysis (Fig. 6d) with the structural mechanics module of COMSOL Multiphysics® v. 5.4 (COMSOL AB, Stockholm, Sweden). For the simulations, we neglected nonlinearities in our study since the cantilever deflection is small compared to the total sensor length. The material properties of the simulated cantilever read: Silicon mass density ρ = 2.329 g/cm³, elastic modulus E = 170 GPa and Poisson ratio ν = 0.28. Similar values for E and ν are reported in literature^{71}.
Ironfilled carbon nanotubes (FeCNT) provide welldefined nanosized magnetic moments for MFM sensors^{31,32,33,42,68,69,70,72,73,74,75,76}. We selected an appropriate FeCNT according to the following criteria: (i) Fe diameter d_{Fe} as small as possible, (ii) Fe length \({l}_{{{{{{\rm{Fe}}}}}}}\) as large as possible, (iii) Fe crosssection as homogeneous as possible, and (iv) carbon cap width t_{cap} as small as possible.
The selected FeCNT with \({d}_{{{{{{\rm{Fe}}}}}}}=\left(20\pm 2\right){{{{{\rm{nm}}}}}}\) was attached to the cantilever end using a Kleindiek nanomanipulator by electronbeam assisted deposition of carbon (Fig. 6b, c). To experimentally confirm the monopoletype magnetic strayfield near the apex of FeCNTs, we performed transmission electron microscopy (TEM) based offaxis electron holography^{77} to an about 8 µm long FeCNT stemming from the same sample as the FeCNT that was used for assembling our sensor tip (see Supplementary Note 6, Supplementary Fig. S4). Figure 6e depicts the reconstructed phase image of an FeCNT tip in which the magnetic phase shift in vacuum is emphasized by red isolines. The latter are proportional to the field lines of the projected magnetic induction revealing indeed a monopolelike behaviour except at the lower side. These perturbations can be explained by stray fields generated by contamination parts attached to the FeCNT in the vicinity of the field of view (see Supplementary Note 6, Supplementary Fig. S5). We also supported these TEM observations by magnetostatic and image simulations in combination with tomographic methods (see Supplementary Note 6, Supplementary Fig. S5).
Using a piezodriven oscillation stage within the SEMFIB, we comprehensively investigated the fundamental and higher order flexural oscillation modes. Imaging the oscillation envelopes allows for a visual analysis of flexural mode shapes. Moreover, amplitudefrequency curves of the sensor were measured and evaluated using the model of a damped harmonic oscillator^{78} providing modedependent resonant frequencies and Qfactors. These and all further MFM calibration and application measurements described in the next section were performed in our highvacuum MFM (NanoScan AG hrMFM) supported by a lockin amplifier (Zurich Instruments HF2LI). Our experimental setup is sketched in Fig. 7. All modes of operation presented in this work can also be performed in ambient conditions.
Magnetic force microscopy setup and modes of operation
The positionsensitive detector (PSD) signal, which contains information about the measurement quantities \({f}_{i}\), \({A}_{i}\), and \(\triangle z\), is fed into and demodulated by both the MFM main controller and the additional lockin amplifier. The former provides the static cantilever deflection readout and controls the second flexural mode with the first phaselocked loop (PLL 1) tracking changes of the resonance frequency \({f}_{2}\). Moreover, an integrated proportionalintegral(derivative) controller (PID 1) ensures a constant second mode vibration amplitude A_{2}. The second flexural mode is selected to measure force gradients via frequency shifts as it is a stable low order vibration mode associated with low k_{2}.
Obviously, the fundamental mode comes with an even lower spring constant, but its low resonant frequency was not compatible with our MFM setup. The second mode vibration is excited by piezodriven mechanical actuation of the cantilever base.
An additional phaselocked loop (PLL 2) provided by the external lockin amplifier tracks the resonance frequency f_{5} of the 5th flexural mode and drives this mode by applying a tipsample AC bias voltage at \({f}_{{{{{{\rm{AC}}}}}}}={f}_{5}/2\), i.e., at half of the mode’s resonance frequency. A_{5} is proportional to \({V}_{{{{{{\rm{AC}}}}}}}^{2}\), Q_{5}, and to the tipsample capacitance gradient which is a measure of the tipsample distance. Keeping A_{5} constant by a properly set PI(D)regulated tipsample distance control enables tracking of the sample surface topography. Furthermore, a welldefined tipsample distance prevents snapin events that may arise from a large sample surface roughness and the usage of lowk probes. Selecting the 5th mode provides a sufficiently small settling time τ_{5} (see Supplementary Note 3) which guarantees an acceptable bandwidth for the distance control feedback loop.
Spring constant calibration
The static spring constant k_{stat} of our sensor shown in Fig. 6 was determined by means of the above mentioned COMSOL simulations by virtually applying a small force (e.g., 100 pN) to the sensor tip, with the force vector pointing perpendicular to the cantilever basal plane, resulting in \({k}_{{{{{{\rm{stat}}}}}}}=3.3\,{{{{{\rm{mN}}}}}}/{{{{{\rm{m}}}}}}\). We compared this simulation result with an analytical approximation describing an arrowshape cantilever with a shaft of width 2 b and a triangle of base width w and height (i.e. length of triangular free end) \({l}_{{{{{\rm{t}}}}}}\)^{79}:
resulting in \({k}_{{{{{{\rm{stat}}}}}}}=\left(3.4\pm 0.6\right)\,{{{{{\rm{mN}}}}}}/{{{{{\rm{m}}}}}}\), which is consistent with the simulation. The dynamic spring constants \({k}_{i}\) were determined by COMSOL simulations by virtually attaching a small spring (\(\triangle k\ll {k}_{i}\), e.g., \(\triangle k=1\,{{{{{\rm{\mu N}}}}}}/{{{{{\rm{m}}}}}}\)) to the sensor tip resulting in a small eigenfrequency alteration \(\triangle {f}_{i}\) of the respective mode. Based on the harmonic oscillator model, the dynamic spring constant k_{i} can be calculated according to:
As far as the important second flexural mode is concerned, we obtain \({k}_{2}=0.47\) N/m. Further k_{i} results are presented in Supplementary Note 3 (Supplementary Table S2).
To experimentally confirm the simulation results, the second order dynamic spring constant k_{2} was calibrated by comparing \({F}_{z}({z}_{{{{{{\rm{ts}}}}}}})\) and \(\triangle {f}_{2}({z}_{{{{{{\rm{ts}}}}}}})\) sweeps (Fig. 8). For these sweeps a sample location was selected that provides attractive magnetostatic interactions and thus monotonic tipsample forces. The \({F}_{z}({z}_{{{{{{\rm{ts}}}}}}})\) sweep was taken without any excitation of dynamic modes.
Considering a nonlinear \({F}_{z}({z}_{{{{{{\rm{ts}}}}}}})\), the following relation based on an analysis of F. J. Giessibl^{80} is used:
Solving this equation by numerical integration yields \({k}_{2}=\left(0.45\pm 0.07\right)\,{{{{{\rm{N}}}}}}/{{{{{\rm{m}}}}}}\). Note, that for the calibration measurement shown in Fig. 8, all z_{ts} data are based on the height control piezo data corrected by the static cantilever deflection. Hence, z_{ts} reflects the real distance between the tip apex and the sample surface.
Calibration of the magnetic tip moment
The knowledge of k_{stat}, k_{i}, and f_{i} allows us to quantitatively measure magnetostatic forces and force gradients, i.e., F_{z} and \({F}_{z}^{\prime}\), respectively. These quantities can be translated into B_{z} and \({B}_{z}^{\prime}\) using the magnetic charge \({q}_{{{{{{\rm{t}}}}}}}\) according to the point monopole description of our NW tip. The calibration procedure is based on MFM scans of a known magnetic reference sample^{33,81} (see “Magnetic reference sample” in the “Methods” section) and a quantitative analysis of the MFM data. Furthermore, the results are compared with a simple \({q}_{{{{{{\rm{t}}}}}}}\) determination using the NW’s geometry and saturation magnetization \({M}_{{{{{{\rm{s}}}}}}}({{{{{\rm{Fe}}}}}})\).
By calculating an effective magnetic surface charge map of the reference sample based on its materials parameters (given in the “Magnetic reference sample” section) and the known sample thickness of 130 nm, the MFM tip’s transfer function is derived and fitted to a point monopole model. The details of this calibration procedure are described elsewhere^{22,33,81}. The calibration results in the effective monopole moment \({{q}_{{{{{{\rm{t}}}}}}}=\left(0.50\pm 0.10\right)\cdot {10}^{9}\,{{{{{\rm{Am}}}}}}}\) and the position of the effective point monopole with respect to the physical tip apex. This distance is mainly defined by the FeCNT carbon cap layer thickness \({t}_{{{{{{\rm{cap}}}}}}}\), even if, technically, both are not the same. For clarity, we use the term \({t}_{{{{{{\rm{cap}}}}}}}\) for both quantities and obtain \({t}_{{{{{{\rm{cap}}}}}}}=33\,{{{{{\rm{nm}}}}}}\) in agreement with SEM analysis. This value is also used for the evaluation of the MFM scan images shown in section: “Quantitative \({B}_{z}\) and \({B}_{z}^{\prime}\) measurements on bulk [001] MnNiGa”. Note that due to electronbeam induced local contamination during repeated SEM inspection of our MFM probe, the thickness of the nonmagnetic cap layer increased gradually by a factor of two and finally reached \({t}_{{{{{{\rm{cap}}}}}}}\approx 70\,{{{{{\rm{nm}}}}}}\) (Fig. 6).
For comparison, a \({q}_{{{{{{\rm{t}}}}}}}\) determination via direct measurement of the FeNW diameter \({d}_{{{{{{\rm{Fe}}}}}}}=\left(20\pm 2\right)\,{{{{{\rm{nm}}}}}}\) and taking account of the saturation magnetization \({M}_{{{{{{\rm{s}}}}}}}({{{{{\rm{Fe}}}}}})=1710\,{{{{{\rm{kA}}}}}}/{{{{{\rm{m}}}}}}\) would result in a tip moment \({q}_{{{{{{\rm{t}}}}}}}=\frac{{{{{{\rm{\pi }}}}}}}{4}{d}_{{{{{{\rm{Fe}}}}}}}^{2}{M}_{{{{{{\rm{s}}}}}}}=\left(0.54\pm 0.11\right)\cdot {10}^{9}{{{{{\rm{Am}}}}}}\).
Detection sensitivities
In all types of cantileverbased scanning force microscopy measurements, including those presented herein, thermal noise and other noise sources limit the detection sensitivity. The minimal detectable force limited by thermal noise \({F}_{{{\min }}}/\!\sqrt{{{{{{\rm{BW}}}}}}}\) is given by^{82}:
in which \({{{{{\rm{BW}}}}}}\) denotes the detection bandwidth.
Considering a force that is periodic with the fundamental mode frequency \({f}_{1}\), the minimal detectable \({F}_{{{\min }}}/\sqrt{{{{{{\rm{BW}}}}}}}\) of our sensor as introduced above is \(0.93\,{{{{{\rm{fN}}}}}}/\sqrt{{{{{{\rm{Hz}}}}}}}\) at \(T=300\,{{{{{\rm{K}}}}}}\) and \(2.72\,{{{{{\rm{fN}}}}}}/\sqrt{{{{{{\rm{Hz}}}}}}}\) for \({f}_{2}\). Given the monopolelike moment \({q}_{{{{{{\rm{t}}}}}}}=5\cdot {10}^{10}\,{{{{{\rm{Am}}}}}}\), the minimal detectable magnetic fields at \(T=300\,{{{{{\rm{K}}}}}}\) read \({B}_{z}=1.9\,{{{{{\rm{\mu }}}}}}{{{{{\rm{T}}}}}}/\sqrt{{{{{{\rm{Hz}}}}}}}\) and \({B}_{z}=5.44\,{{{{{\rm{\mu }}}}}}{{{{{\rm{T}}}}}}/\sqrt{{{{{{\rm{Hz}}}}}}}\) for \({f}_{1}\) and \({f}_{2}\), respectively.
The minimal detectable force gradient limited by thermal noise \({F}_{{{\min }}}^{\prime}/\sqrt{{{{{{\rm{BW}}}}}}}\) is given by^{83}:
In the present case, \({F}_{{{\min }}}^{\prime}/\sqrt{{{{{{\rm{BW}}}}}}}\) at \({f}_{2}=34130\,{{{{{\rm{Hz}}}}}}\) is \(270\,\frac{{{{{{\rm{aN}}}}}}}{{{{{{\rm{nm}}}}}}}/\sqrt{{{{{{\rm{Hz}}}}}}}\) at \(T=300\,{{{{{\rm{K}}}}}}\) when using the second flexural mode. Considering \({q}_{{{{{{\rm{t}}}}}}}=5\cdot {10}^{10}\,{{{{{\rm{Am}}}}}}\), the minimal detectable field derivative \({B}_{z}^{\prime}\) at \(T=300\,{{{{{\rm{K}}}}}}\) is \(545\,\frac{{{{{{\rm{nT}}}}}}}{{{{{{\rm{nm}}}}}}}/\sqrt{{{{{{\rm{Hz}}}}}}}\). For these calculations, spring constants \({k}_{i}\), quality factors \({Q}_{i}\), and frequencies \({f}_{i}\) for the fundamental and second flexural modes are applied according to Supplementary Note 3 (see Supplementary Table S2). The vibration amplitude is \({A}_{2}=10\,{{{{{\rm{nm}}}}}}\).
Please note that the mean square deflection induced by thermal noise of an EulerBernoulli cantilever appears to be enhanced by a factor of \(\sqrt{4/3}\) when the inclination at the cantilever free end is measured instead of a direct deflection measurement by, e.g., interferometry^{84}. As an example, the optical beam deflection method measures the cantilever inclination rather than the deflection. If the cantilever geometry deviates from that of an EulerBernoulli beam as in our case, this factor needs to be calculated individually.
Please note that the considerations above address thermal noise limitations only. The investigation of other contributions to the total noise was beyond the scope of this study.
MnNiGa sample fabrication and surface characterization
A polycrystalline MnNiGa sample was fabricated using the arc melting technique (for details see “MnNiGa sample” in the Methods section). To identify a suitable homogeneous sample location with the desired [001] crystal orientation (Fig. 9a, b) we performed electron backscatter diffraction (EBSD) at 15 kV acceleration voltage (Fig. 9c). Furthermore, the chemical composition and the crystal homogeneity were investigated by surfacesensitive energy dispersive Xray spectroscopy (EDX) L–edge mapping at 5 kV and by employing backscatter electron (BSE) micrographs, respectively. For MFM investigations, we selected a region with homogeneous (redcolored) EBSD [001] surface pattern, located next to a region with [110] crystal orientation, see Fig. 9c. This ensures that the easy magnetization direction in the studied [001] crystallite is perpendicular to the sample surface.
Quantitative \({{{{{{\boldsymbol{B}}}}}}}_{{{{{{\boldsymbol{z}}}}}}}\) and \({{{{{{\boldsymbol{B}}}}}}}_{{{{{{\boldsymbol{z}}}}}}}^{\prime}\) measurements on bulk [001] MnNiGa
The MnNiGa sample was introduced into the MFM vacuum chamber 48 h prior to the measurement to ensure stable high vacuum conditions and to approach thermal equilibrium at room temperature. After calibration measurements and compensation of the contactpotential difference V_{cpd} by application of an appropriate constant tipsample DC bias, the main measurement was performed as a singlepass \({(10\,{{{{{\rm{\mu m}}}}}})}^{2}\)/\({(512)}^{2}\) pixel scan at a controlled scanning distance \({z}_{{{{{{\rm{ts}}}}}}}=70\,{{{{{\rm{nm}}}}}}\) and a tip velocity of \(1.67\,{{{{{\rm{\mu m}}}}}}/{{{{{\rm{s}}}}}}\), resulting in a full frame completion time of 103 min. To guarantee a welldefined magnetic state, the MnNiGa sample and the magnetic probe tip were initially magnetized by a perpendicular \(400\,{{{{{\rm{mT}}}}}}\) field. Note that the applied field is larger than the switching field of the magnetic nanowire probe (\(275\,{{{{{\rm{mT}}}}}}\pm 10\,{{{{{\rm{mT}}}}}}\)).
Figure 10a–d show simultaneously recorded maps of the [001] oriented uniaxial MnNiGa crystallite in an outofplane field of \(270\,{{{{{\rm{mT}}}}}}\) pointing along the magnetic easy axis. Included are the raw zpiezo position data (Fig. 10a), the derived topography map (Fig. 10b; representing the sum of the zpiezo output and the static cantilever deflection \(\triangle z\) data), the magnetic \({B}_{z}\) map (Fig. 10c), and the \({B}_{z}^{\prime}\) map (Fig. 10d). Note, \({B}_{z}\propto \triangle z\) and \({B}_{z}^{\prime}\) \(\propto \triangle {f}_{2}\).
Several bubblelike features can be observed, accompanied by other types of magnetization patterns including magnetic stripe domains and field background generated by adjacent crystallites. The spatial resolution of the probesample interaction is identical for the \({B}_{z}\) and the \({B}_{z}^{\prime}\) channel but in \({B}_{z}^{\prime}\) data higher spatial frequencies are amplified compared to corresponding \({B}_{z}\) data^{22}. As illustrated in Fig. 4a, our MFM approach is not sensitive to homogeneous field contributions, since \({{{{{\rm{FTF}}}}}}\left(\lambda \right)\) approaches zero for large \(\lambda\). Nevertheless, the finite size of the studied crystallite causes a small but measurable field variation associated with a large \(\lambda\). Such contributions, however, only constitute a field background that is nearly constant within the length scale of the bubble diameters. Therefore, it does not affect our detailed bubble analysis discussed in the next section.
Evaluation of individual magnetic nanodomains
Finally, we analyze the observed circular nanodomains. Typical magnetic bubble domains form in a material with perpendicular anisotropy as minority domains in an applied external magnetic field. To determine the diameters \({d}_{{{{{{\rm{qs}}}}}}}\) and the vertical extent of the considered magnetic bubbles, we evaluate angularly averaged \({B}_{z}\) and \({B}_{z}^{\prime}\) profiles. An investigation of further properties such as depthdependent helicity^{85}, bubble tube bending and merging^{59} is not reported in this paper.
We limit our detailed evaluation to radially symmetric bubbles. In our analysis, we compare the measured and angularly averaged strayfield distributions of ten individual MnNiGa bubbles with strayfield characteristics of four idealized magnetization distributions:

(a)
a magnetic point dipole pointing in the direction perpendicular to the sample surface and exhibiting two degrees of freedom: the magnetic moment \({m}_{{{\rm{s}}}}\) and its \({\check{z}}^{* }\) position, i.e., its depth below the surface;

(b)
a high aspect ratio magnetic typeI bubble consisting of a homogenously magnetized core that is separated from the magnetic environment by a Bloch wall, again with two degrees of freedom: diameter \({d}_{{{{{{\rm{qs}}}}}}}\) and \({\check{z}}^{* }\) position of the bubble’s top base;

(c)
a double monopole arrangement (Dirac string) with three degrees of freedom: the top monopole with moment \({q}_{{{{{{\rm{s}}}}}}}\) and its \({\check{z}}^{* }\) position and the second monopole \({q}_{{{{{{\rm{s}}}}}}}\) buried at a depth of \({\check{z}}^{* }+h\);

(d)
a single magnetic point monopole with two degrees of freedom: the magnetic monopole moment \({q}_{{{{{{\rm{s}}}}}}}\) and its \({\check{z}}^{* }\) position.
In all cases, the considered magnetic structures are embedded in a homogenously magnetized sample with perpendicular magnetization. The magnetization direction of the considered magnetic structures is opposite to that of the homogeneously magnetized environment.
To assess the applicability of the respective magnetostatic models to our measured MFM data, we make use of the coefficient of determination R^{2} (see Supplementary Note 5 and Supplementary Fig. S1 for illustration). For the simulations and fitting functions, a MnNiGa saturation magnetization of \({M}_{{{{{{\rm{S}}}}}}}({{{{{\rm{MnNiGa}}}}}})=470\,{{{{{\rm{kA}}}}}}/{{{{{\rm{m}}}}}}\) was used which was determined by SQUID magnetometry measurements at room temperature. For the Bloch domain wall width, a value of 47 nm, as reported in literature^{54}, was assumed.
Magnified images of an individual bubble (in Supplementary Fig. S2 named bubble #3) are shown in the insets of Fig. 10c, d. In Table 1, the angularly averaged \({B}_{z}\) profile of bubble #3 (Fig. 10e) is compared to the strayfield distributions of the above introduced models. The fitting functions are given in Supplementary Note 4 (Supplementary Equations S13, S14, S16, and S17). The data evaluation indicates that the dipolelike description is inferior to the bubble domain and monopole models. The significance of the resulting R^{2} differences is proven in Supplementary Note 5 by comparing to FEMMsimulations of related nanomagnetic configurations^{86}. As far as the mutual, vertical pole distance \(h\) is concerned, the double pole fitting results in \(h\ge 10\,{{{{{\rm{\mu m}}}}}}\), which essentially corresponds to the monopole model with identical fitting results. For the bubble and monopoletype models, \({q}_{{{{{{\rm{s}}}}}}}\), \({d}_{{{{{{\rm{qs}}}}}}},\)and \({\check{z}}^{* }\) of bubble #3 are given in Table 1.
Although both the magnetic bubble and the monopole models describe the measured magnetic field distribution well, there is no microscopic magnetization configuration with true pointlike character. In all cases a nonvanishing Heisenberg exchange interaction forces a magnetization structure to be spatially extended. This implies that the measured \({B}_{z}\) profiles of the localized domains in [001] MnNiGa are well described by circular bubble domains.
An interesting aspect of the analysis is the determination of a nonzero depth \({\check{z}}^{* }\) of the bubble’s top surface. This can be explained by a possible formation of flux closure domains near the sample surface. An additional source of indeterminate error in the specification of \({\check{z}}^{* }\) might be the possible existence of a magnetic dead layer on the sample surface.
The magnetic point pole evaluation of the remaining bubbles based on angularly averaged \({B}_{z}\) and \({B}_{z}^{\prime}\) MFM data is included in Supplementary Note 5 (Supplementary Fig. S2 and Supplementary Table S3). In any case, the fitting procedures using \({B}_{z}^{\prime}\) data confirm magnetic bubbles with monopolelike strayfield characteristics and a significant vertical extent, presumably proliferating through the depth of the entire crystallite in agreement with a smallangle neutron scattering study^{52}. Furthermore, all bubbles are buried as described by a nonzero \({\check{z}}^{* }\). The simple monopole fits result in bubble diameters \({d}_{{{{{{\rm{qs}}}}}}}\) and depths \({\check{z}}^{* }\) covering ranges from \(120\) \({{{{{\rm{nm}}}}}}\) to \(200\) \({{{{{\rm{nm}}}}}}\) and \(70\) \({{{{{\rm{nm}}}}}}\) to \(110\) \({{{{{\rm{nm}}}}}}\), respectively. Thus, our study indicates that the bubble diameters of bulk MnNiGa bubbles are somewhat larger than the diameters of the assumed biskyrmions in thin MnNiGa lamellae prepared for transmission electron microscopy investigations^{49,50}.
Conclusions
In this work we demonstrated a unique magnetic force microscopy approach that employs the combination of specially structured lowstiffness cantilevers, highaspectratio magnetic nanowire tips with excitation and control of multiple vibration modes. This technique enables simultaneous singlepass distancecontrolled noncontact quantitative measurements of both a magnetic sample’s strayfield and its strayfield gradient. As proof of principle we used this technique to investigate magnetic bubbles of MnNiGa bulk samples. We presented data that suggest that the magnetic bubbles have significant vertical extent and a buried bubble top base.
The presented method is well suited to study complex magnetic structures and can be easily implemented to enhance the capabilities of commonly used magnetic force microscopy equipment. This further extends the set of available microscopy methods for studying magnetic systems and may be applied to contribute to the characterization and understanding of a large variety of nano and micronsized magnetic strayfield patterns generated by ferromagnetic, superparamagnetic, and superconducting samples as well as electric current distributions.
Methods
Magnetic reference sample
A Pt(5 nm)[Pt(0.9 nm)/Co(0.4 nm)]_{100}Pt(2 nm) multilayer with perpendicular (easyaxis) anisotropy is used as magnetic reference sample for the tip calibration procedure. Vibrating sample magnetometry reveals an effective multilayer magnetization \({M}_{{{{{{\rm{s}}}}}}}({{{{{\rm{CoPt}}}}}})=510\) \({{{{{\rm{kA}}}}}}/{{{{{\rm{m}}}}}}\) and a perpendicular anisotropy constant \({K}_{{{{{{\rm{u}}}}}}}=0.52\) \({{{{{\rm{MJ}}}}}}/{{{{{{\rm{m}}}}}}}^{3}\). The large ratio \({K}_{{{{{{\rm{u}}}}}}}/{K}_{{{{{{\rm{d}}}}}}}=3.2\) with the shape anisotropy constant \({K}_{{{{{{\rm{d}}}}}}}=0.5\) \({{{{{{\rm{\mu }}}}}}}_{0}{M}_{{{{{{\rm{s}}}}}}}^{2}\) promotes the required perpendicular magnetization orientation within individual domains^{33,81}.
MnNiGa sample
A polycrystalline MnNiGa sample was fabricated using arc melting under Ar atmosphere using an ingot with 99.99% nominal composition of its constituent elements Ni, Mn, and Ga. During the preparation process, the ingot was molten and recrystallized several times to ensure good homogeneity. The synthesis was accompanied by a final weight loss of less than 1%. Afterwards, the ascast button shaped ingot was annealed at a temperature of 1073 K for 6 days in a sealed quartz tube under vacuum to further improve its homogeneity. Finally, the ingot was quenched in ice water to sustain the hexagonal phase. For the finished MnNiGa alloy sample \({M}_{{{{{{\rm{s}}}}}}}({{{{{\rm{MnNiGa}}}}}})=470\) \({{{{{\rm{kA}}}}}}/{{{{{\rm{m}}}}}}\) was determined by SQUID magnetometry.
Data availability
The data analyzed and presented in this study are available from the corresponding authors upon reasonable request.
Code availability
All code developed in this study is available from the corresponding authors upon reasonable request.
References
Tetienne, J.P. et al. The nature of domain walls in ultrathin ferromagnets revealed by scanning nanomagnetometry. Nat. Commun. 6, 6733 (2015).
Vasyukov, D. et al. A scanning superconducting quantum interference device with single electron spin sensitivity. Nat. Nanotechnol. 8, 639 (2013).
Martin, Y. & Wickramasinghe, H. K. Magnetic imaging by “force microscopy” with 1000 Å resolution. Appl. Phys. Lett. 50, 1455 (1987).
Saenz, J. J. et al. Observation of magnetic forces by the atomic force microscope. J. Appl. Phys. 62, 4293 (1987).
Schwarz, A. & Wiesendanger, R. Magnetic sensitive force microscopy. Nanotoday 3, 28 (2008).
Giles, R. et al. Noncontact force microscopy in liquids. Appl. Phys. Lett. 63, 617 (1993).
Kazakova, O. et al. Frontiers of magnetic force microscopy. J. Appl. Phys. 125, 060901 (2019).
Hrabec, A. et al. Currentinduced skyrmion generation and dynamics in symmetric bilayers. Nat. Commun. 8, 15765 (2017).
Legrand, W. et al. Roomtemperature currentinduced generation and motion of sub100 nm skyrmions. Nano Lett. 17, 2703 (2017).
Zhang, S. et al. Direct writing of room temperature and zero field skyrmion lattices by a scanning local magnetic field. Appl. Phys. Lett. 112, 132405 (2018).
Casiraghi, A. et al. Individual skyrmion manipulation by local magnetic field gradients. Commun. Phys. 2, 145 (2019).
Meng, K. Y. et al. Observation of nanoscale skyrmions in SrIrO_{3}/SrRuO_{3} bilayers. Nano Lett. 19, 3169 (2019).
Yagil, A. et al. Stray field signatures of Néel textured skyrmions in Ir/Fe/Co/Pt multilayer films. Appl. Phys. Lett. 112, 192403 (2018).
Garcia, R. & Herruzo, E. T. The emergence of multifrequency force microscopy. Nat. Nanotechnol. 7, 217–226 (2012).
Li, J. W., Cleveland, J. P. & Proksch, R. Bimodal magnetic force microscopy: Separation of short and long range forces. Appl. Phys. Lett. 94, 163118 (2009).
Schwenk, J., Marioni, M., Romer, S., Joshi, N. R. & Hug, H. J. Noncontact bimodal magnetic force microscopy. Appl. Phys. Lett. 104, 112412 (2014).
Schwenk, J. et al. Bimodal magnetic force microscopy with capacitive tipsample distance control. Appl. Phys. Lett. 107, 132407 (2015).
Dietz, C., Herruzo, E. T., Lozano, J. R. & Garcia, R. Nanomechanical coupling enables detection and imaging of 5 nm superparamagnetic particles in liquid. Nanotechnol. 22, 125708 (2011).
Gisbert, V. G. et al. Quantitative mapping of magnetic properties at the nanoscale with bimodal AFM. Nanoscale 13, 2026 (2021).
Forchheimer, D., Platz, D., Tholén, E. A. & Haviland, D. B. Simultaneous imaging of surface and magnetic forces. Appl. Phys. Lett. 103, 013114 (2013).
Zhao, X. et al. Magnetic force microscopy with frequencymodulated capacitive tip–sample distance control. New J. Phys. 20, 013018 (2018).
Hug, H. J. et al. Quantitative magnetic force microscopy on perpendicularly magnetized samples. J. Appl. Phys. 83, 5609 (1998).
Van Schendel, P. J. A., Hug, H. J., Stiefel, B., Martin, S. & Güntherodt, H. J. A method for the calibration of magnetic force microscopy tips. J. Appl. Phys. 88, 435–445 (2000).
Vock, S. et al. Quantitative magnetic force microscopy study of the diameter evolution of bubble domains in a (Co/Pd)_{80} multilayer. IEEE Trans. Magn. 47, 2352–2355 (2011).
Hu, X. et al. Round robin comparison on quantitative nanometer scale magnetic field measurements by magnetic force microscopy. JMMM 511, 166947 (2020).
Corte‐León, H. et al. Comparison and validation of different magnetic force microscopy calibration schemes. Small 16, 1906144 (2020).
Lohau, J., Kirsch, S., Carl, A., Dumpich, G. & Wassermann, E. F. Quantitative determination of effective dipole and monopole moments of magnetic force microscopy tips. J. Appl. Phys. 86, 3410 (1999).
Hartmann, U. The point dipole approximation in magnetic force microscopy. Phys. Lett. A. 137, 475 (1989).
Lee, I. et al. Magnetic force microscopy in the presence of a strong probe field. Appl. Phys. Lett. 99, 162514 (2011).
Uhlig, T., Wiedwald, U., Seidenstücker, A., Ziemann, P. & Eng, L. M. Single core–shell nanoparticle probes for noninvasive magnetic force microscopy. Nanotechnol. 25, 255501 (2014).
Wolny, F. et al. Iron filled carbon nanotubes as novel monopolelike sensors for quantitative magnetic force microscopy. Nanotechnol. 21, 435501 (2010).
Vock, S. et al. Monopolelike probes for quantitative magnetic force microscopy: Calibration and application. Appl. Phys. Lett. 97, 252505 (2010).
Reiche, C. F. et al. Bidirectional quantitative force gradient microscopy. New J. Phys. 17, 013014 (2015).
Häberle, T. et al. Towards quantitative magnetic force microscopy: theory and experiment. New J. Phys. 14, 043044 (2012).
Sievers, S. et al. Quantitative measurement of the magnetic moment of individual magnetic nanoparticles by magnetic force microscopy. Small 8, 2675 (2012).
Jaafar, M. et al. Customized MFM probes based on magnetic nanorods. Nanoscale 12, 10090 (2020).
Wolny, F. et al. Magnetic force microscopy measurements in external magnetic fields—comparison between coated probes and an iron filled carbon nanotube probe. J. Appl. Phys. 108, 013908 (2010).
Porthun, S., Abelmann, L., Vellekoop, S. J. L., Lodder, J. C. & Hug, H. J. Optimization of lateral resolution in magnetic force microscopy. Appl. Phys. A 66, 1185–1189 (1998).
Phillips, G. N., Siekman, M., Abelmann, L. & Lodder, J. C. High resolution magnetic force microscopy using focused ion beam modified tips. Appl. Phys. Lett. 81, 865 (2002).
Wolny, F. et al. Ironfilled carbon nanotubes as probes for magnetic force microscopy. J. Appl. Phys. 104, 064908 (2008).
Winkler, A. et al. Magnetic force microscopy sensors using ironfilled carbon nanotubes. J. Appl. Phys. 99, 104905 (2006).
Mühl, T. et al. Magnetic force microscopy sensors providing inplane and perpendicular sensitivity. Appl. Phys. Lett. 101, 112401 (2012).
Mattiat, H. et al. Nanowire magnetic force sensors fabricated by focusedelectronbeaminduced deposition. Phys. Rev. Appl. 13, 044043 (2020).
Rossi, N., Gross, B., Dirnberger, F., Bougeard, D. & Poggio, M. Magnetic force sensing using a selfassembled nanowire. Nano Lett. 19, 930 (2019).
Reiche, C. F., Körner, J., Büchner, B., Mühl, T. Bidirectional scanning force microscopy probes with coresonant sensitivity enhancement. 2015 IEEE 15th International Conference on Nanotechnology (IEEENANO), 1222 (2015).
Puwenberg, N. et al. Magnetization reversal and local switching fields of ferromagnetic Co/Pd microtubes with radial magnetization. Phys. Rev. B. 99, 094438 (2019).
Tokura, Y. & Kanazawa, N. Magnetic skyrmion materials. Chem. Rev. 121, 2857 (2021).
Yu, X. Z. et al. Biskyrmion states and their currentdriven motion in a layered manganite. Nat. Commun. 5, 3198 (2014).
Wang, W. et al. A centrosymmetric hexagonal magnet with superstable biskyrmion magnetic nanodomains in a wide temperature range of 100–340 K. Adv. Mater. 28, 6887–6893 (2016).
Peng, L. et al. Realspace observation of nonvolatile zerofield biskyrmion lattice generation in MnNiGa magnet. Nano Lett. 17, 7075–7079 (2017).
Peng, L. et al. Multiple tuning of magnetic biskyrmions using in situ LTEM in centrosymmetric MnNiGa alloy. J. Phys.: Condens. Matter 30, 065803 (2018).
Li, X. et al. Oriented 3D Magnetic Biskyrmions in MnNiGa Bulk Crystals. Adv. Mater. 31, 1900264 (2019).
Yao, Y. et al. Magnetic hard nanobubble: a possible magnetization structure behind the biskyrmion. Appl. Phys. Lett. 114, 102404 (2019).
Loudon, J. C. et al. Do images of biskyrmions show typeII bubbles? Adv. Mater. 31, 1806598 (2019).
Turnbull, L. A. et al. Tilted Xray holography of magnetic bubbles in MnNiGa lamellae. ACS Nano 15, 387 (2021).
Göbel, B., Henk, J. & Mertig, I. Forming individual magnetic biskyrmions by merging two skyrmions in a centrosymmetric nanodisk. Sci. Rep. 9, 9521 (2019).
Capic, D., Garanin, D. A. & Chudnovsky, E. M. Stabilty of biskyrmions in centrosymmetric magnetic films. Phys. Rev. B. 100, 014432 (2019).
Capic, D., Garanin, D. A. & Chudnovsky, E. M. Biskyrmion lattices in centrosymmetric magnetic films. Phys. Rev. Res. 1, 033011 (2019).
Göbel, B., Mertig, I. & Tretiakov, O. A. Beyond skyrmions: review and perspectives of alternative magnetic quasiparticles. Phys. Rep. 895, 1–28 (2021).
Birch, M. T. et al. Realspace imaging of confined magnetic skyrmion tubes. Nat. Commun. 11, 1726 (2020).
Vansteenkiste, A. & Van de Wiele, B. (2011). MuMax: a new highperformance micromagnetic simulation tool. JMMM 323, 2585–2591 (2011).
Vansteenkiste, A. et al. The design and verification of MuMax3. AIP Adv. 4, 107133 (2014).
Gao, R. W. et al. Exchangecoupling interaction, effective anisotropy and coercivity in nanocomposite permanent materials. J. Appl. Phys. 94, 664 (2003).
Vavassori, P. et al. Interplay between magnetocrystalline and configurational anisotropies in Fe (001) square nanostructures. Phys. Rev. B. 72, 054405 (2005).
Antoniak, C. et al. Composition dependence of exchange stiffness in Fe_{x}Pt_{1−x} alloys. Phys. Rev. B. 82, 064403 (2010).
Metlov, K. L., Suzuki, K., Honecker, D. & Michels, A. Experimental observation of thirdorder effect in magnetic smallangle neutron scattering. Phys. Rev. B. 101, 214410 (2020).
Skomski, R. & Coey, J. M. D. Nucleation field and energy product of aligned twophase magnetics – progress towards the ‘1 MJ/m^{3}’ magnet. IEEE Trans. Magn. 29, 2860 (1993).
Weissker, U., Hampel, S., Leonhardt, A. & Büchner, B. Carbon nanotubes filled with ferromagnetic materials. Materials 3, 4387–4427 (2010).
Lipert, K. An individual iron nanowirefilled carbon nanotube probed by microHall magnetometry. Appl. Phys. Lett. 97, 212503 (2010).
Schwarz, T. et al. Lownoise YBa_{2}Cu_{3}O_{7} nanoSQUIDs for performing magnetizationreversal measurements on magnetic nanoparticles. Phys. Rev. Appl. 3, 44011 (2015).
Hopcroft, M. A., Nix, W. D. & Kenny, T. W. What is the young’s modulus of silicon? J. Microelectromech. Syst. 19, 229 (2010).
Grobert, N. et al. Enhanced magnetic coercivities in Fe nanowires. Appl. Phys. Lett. 75, 3363 (1999).
Mühl, T. et al. Magnetic properties of aligned Fefilled carbon nanotubes. J. Appl. Phys. 93, 7894 (2003).
Kozhuharova, R. et al. Synthesis and characterization of aligned Fefilled carbon nanotubes on silicon substrates. J. Mater. Sci.: Mater. Electron. 14, 789 (2003).
Golberg, D. et al. Atomic structures of ironbased singlecrystalline nanowires crystallized inside multiwalled carbon nanotubes as revealed by analytical electron microscopy. Acta Mater. 54, 2567 (2006).
Lutz, M. U. et al. Magnetic properties of αFe and Fe_{3}C nanowires. J. Phys., Conf. Ser. 200, 72062 (2010).
Simon, P. et al. Synthesis and threedimensional magnetic field mapping of Co_{2}FeGa Heusler nanowires at 5 nm resolution. Nano Lett. 16, 114–120 (2015).
Cappella, B. & Dietler, G. Forcedistance curves by atomic force microscopy. Surf. Sci. Rep. 34, 1–104 (1999).
Hähner, G. Normal spring constants of cantilever plates for different load distributions and static deflection with applications to atomic force microscopy. J. Appl. Phys. 104, 084902 (2008).
Giessibl, F. J. A direct method to calculate tip–sample forces from frequency shifts in frequencymodulation atomic force microscopy. Appl. Phys. Lett. 78, 123 (2001).
Vock, S. et al. Magnetic vortex observation in FeCo nanowires by quantitative magnetic force microscopy. Appl. Phys. Lett. 105, 172409 (2014).
Stowe, T. D. et al. Attonewton force detection using ultrathin silicon cantilevers. Appl. Phys. Lett. 71, 288 (1997).
Albrecht, T. R., Grütter, P., Horne, D. & Rugar, D. Frequency modulation detection using high‐Q cantilevers for enhanced force microscope sensitivity. J. Appl. Phys. 69, 668 (1991).
Butt, H. J. & Jaschke, M. Calculation of thermal noise in atomic force microscopy. Nanotechnol. 6, 1–7 (1995).
Van der Laan, G., Zhang, S. L. & Hesjedal, T. Depth profiling of 3D skyrmion lattices in a chiral magnet—a story with a twist. AIP Adv. 11, 015108 (2021).
Meeker, D. C. Finite Element Method Magnetics: User’s Manual. v.4.2 [Online] available at http://www.femm.info (2015).
Acknowledgements
We hereby thank Julia Körner and Thomas Wiek for supporting our FIB work, Uhland Weissker for preparing the FeCNT sample, and Ulrich Rößler for sharing his theoretical expertise in vivid discussions. Moreover, we want to thank Maneesha Sharma and Mykhailo Flaks for proofreading the manuscript. Furthermore, we express our gratitude to Natascha Freitag for her support in editorial work and final proof corrections. This research project was supported by the Deutsche Forschungsgemeinschaft (DFG) (Grant No. MU 1794/131) and the European Research Council (ERC) under the Horizon 2020 research and innovation program of the European Union (grant agreement no. 715620).
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N.H.F.: conceptualization, formal analysis, investigation, methodology, software, visualization, writing—original draft. C.F.R.: conceptualization, validation, writing—review & editing. V.N.: formal analysis, investigation, software, validation, writing—review & editing. P.D.: formal analysis, investigation, writing—original draft. U.B.: formal analysis, investigation, writing—review & editing. C.F.: project administration, resources, supervision. D.W.: investigation, validation, visualization, writing—original draft. A.L.: investigation, validation, visualization, writing—review & editing. B.B.: Project administration, resources, supervision. T.M.: conceptualization, funding acquisition, investigation, methodology, project administration, supervision, validation, visualization, writing—original draft.
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Freitag, N.H., Reiche, C.F., Neu, V. et al. Simultaneous magnetic field and field gradient mapping of hexagonal MnNiGa by quantitative magnetic force microscopy. Commun Phys 6, 11 (2023). https://doi.org/10.1038/s42005022011193
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DOI: https://doi.org/10.1038/s42005022011193
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Communications Physics (2023)
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