An acoustic teaching model illustrating the principles of dynamic mode magnetic force microscopy

1 Introduction

Nanotechnology has not only become an immensely important interdisciplinary enabling technology along the complete value chain as well as an ever-expanding research topic, but also more and more found its way to day-to-day life. In order to enable the general public to understand and participate in current scientific research, it is necessary to facilitate the core principles of nanotechnology, e.g. for the secondary-school level. One of the most versatile and technologically mature nanotechnological methods is atomic force microscopy (AFM), which was first developed in 1985 by Binnig et al. [1]. It represents a high-resolution type of scanning probe microscopy (SPM) [2] with resolution of the order of fractions of a nanometer, clearly outperforming optical resolution. Since its invention, AFM has been refined in order to provide access to different physical properties. Part of this refinement is the development of different operational modes and detection methods to suit varying measurement variables and conditions. As such, magnetic force microscopy (MFM) [3] allows for the detection of magnetic force gradients at a local scale and has become an important analytical tool, e.g. for the examination of magnetic storage media.

In this paper, we present a brick-based versatile macroscopic MFM teaching model enabling teachers or instructors in nanotechnology education to replicate the model for their classrooms easily. The model visualizes the key principles of dynamic MFM, i.e. raster-scanning a ferromagnetic oscillating probe operating at its resonance frequency over a magnetic surface. The occurring shifts in the resonance frequency correspond to magnetic force gradients being directly measured and visualized with the model system. Furthermore, the model makes the named frequency shifts of the dynamic mode in MFM as well as in AFM acoustically tangible, allowing for a more concrete understanding of the operational mode.

In this way, the presented dynamic-mode acoustic MFM model extends the functionality of existing macroscopic models (e.g. [4], [5], [6], [7]), which are mostly limited to visualizing the so-called static or contact mode of AFM and MFM. Moreover, it allows visualizing another detection method of the dynamic mode in comparison with an existing dynamic mode model [8]. The model of Planinšič and Kovač operates in the so-called slope detection method. The cantilever is driven at a constant excitation frequency close to its resonance frequency detecting changes of the oscillation amplitude for measuring the variation of the magnetic force gradient. However, in this operational mode, it is typically the phase shift between the oscillation of the cantilever and the driving force that represents a suitable measure of the interactive force gradient, as it is a more sensitive measure. In contrast, the newly presented MFM model operates in frequency modulation (FM) mode. In this mode, the measured variable is the resonance frequency of the cantilever, which is lower/higher for attractive/repulsive forces as the resonance frequency of the uninfluenced cantilever. Although one has to incorporate a positive feedback circuit into the model setup for this operational mode, it is perfectly suited for teaching the dynamic mode of AFM and MFM, as the frequency shift can be made audible and thus allows for a unique sensory perception.

Moreover, the model allows analyzing and discussing the underlying physics of MFM in a variety of ways. Using the teaching model, the theoretical description of magnetic forces necessary to explain MFM can be accessed visually, acoustically, and numerically in an experimental framework. Along these lines, the MFM model provides multiple representations for the underlying physics concepts, representing an important tool to raise students’ motivation and conceptual knowledge (e.g. [9], [10]). Additionally, interpretation methods of visualizations and limitations of models can be discussed (e.g. [11]).

In the following, core ideas of MFM (Section 2), the experimental configuration of the model (Section 3), and some exemplary measurements (Section 4) will be explained, and the model is compared with the scales of a typical microscopic MFM (Section 5). Finally, an educational classification is presented (Section 6).

2 Core ideas of MFM

This section gives insight into the core ideas of MFM as a specific operational mode of AFM. Moreover, as the model represents the FM detection method, a special emphasis lies on its explanation. Thorough background knowledge about AFM, MFM, modes, and other detection methods can be obtained, for instance, from Binnig et al., Meyer et al., or Hartmann [1], [2], [12].

MFM as a specific mode of operation of AFM allows for the measurement of magnetic force gradients of a sample. As in AFM, the sharp tip raster-scans the surface in order to detect the interaction between the tip and the sample. While in AFM, depending on the operational mode, several short-range forces (mechanical contact, van der Waals force, capillary force, chemical forces, electrostatic force, and ionic repulsion) influence the deflection of the flexible cantilever, MFM is supposed to register only the long-range magnetostatic interaction of the magnetized tip and the ferromagnetic sample. To this end, the typical operational modes are altered.

In contact or static mode operation in AFM, the abovementioned forces bend the cantilever with its sharp tip at each lateral position, and the sample topography is registered using a laser beam reflected on the top of the non-fastened ending of the cantilever to a photo detector (beam-deflection method). In non-contact or dynamic mode operation, the same forces interact with the cantilever, which is attached to a mechanical oscillator (typically a bimorph piezoelectric plate). Here, the interaction alters the oscillation of the cantilever in frequency, amplitude, and phase [12]. Each of these can be used as the measured variable – then keeping one of the others constant – depending on the detection method. In static as well as in dynamic mode, the registered deflection of the cantilever needs to be calibrated with a well-defined sample for absolute readings of the measured variable.

To investigate the microscopic magnetic structure of a sample via dynamic MFM, typically two scans are conducted. At first, the topography of the sample is measured using dynamic AFM detection methods. Subsequently, the cantilever follows the surface profile at constant height z₀ above the sample surface investigating the magnetic structure. For this purpose, the chosen measured variable of the cantilever oscillation provides information about the magnetic interaction between cantilever and sample. Furthermore, the cantilever exhibits a fixed magnetization to measure the strength and direction of magnetic interaction. The basic MFM principle is shown in Figure 1A.

Figure 1:

(A) Basic MFM principle using a permanently magnetized oscillating cantilever driven by a bimorph piezoelectric plate to investigate microscopic magnetic structures. The oscillation is analyzed in amplitude, phase, or frequency via a reflected laser beam applying a photo detector. (B) Acoustic MFM model principle using an electromagnet to cause permanent oscillation of a non-ferromagnetic flat spring to analyze magnetic samples investigating the force gradient dependent frequency shift applying a microphone.

In comparison, our acoustic MFM model uses a magnetizable soft ferromagnetic flat spring to investigate macroscopic magnetic structures. While an electromagnet causes constant oscillation of the flat spring, magnetic forces alter the oscillation frequency. Consequently, the frequency shift provides detailed information about the magnetic structure of the investigated sample. The measurement of the oscillation frequency can be obtained with a microphone. As we assume an approximately constant height of the macroscopic magnetic samples, the scanning process is a single scan of the surface in dynamic mode for a fixed height z₀. The principle of the acoustic MFM model is shown in Figure 1B.

For real MFM, this height is typically z₀=50–250 nm from the sample surface [3], [12], [13]. Thus, short-range forces are eliminated from the measurement process, leaving only the long-range magnetic forces. In FM detection method, the cantilever with the magnetized tip is driven at its momentary resonance frequency. For a magnetic sample with permanent magnetization M→ and consequently magnetic field B→s, attractive or repulsive magnetic forces F→mag(Bs,z) act on the ferromagnetic tip. The force gradient ∂F_mag(B_s,_z)/∂z along the vertical axis z leads to a shift of the resonance frequency Δf due to a changed effective spring constant k′ of the cantilever. While repulsive interaction leads to a higher effective spring constant and thus to a shift towards higher frequencies, attractive interaction leads to a shift to lower frequencies (see Figure 2). A positive feedback circuit keeps the cantilever oscillating at its changed resonance frequency, and the shift in frequency Δf between the free f₀ and the present resonance frequency f′0 is the measured variable to determine the magnetic structure of the sample. In FM, a variety of methods are realized to measure the oscillation frequency with very high precision, e.g. the beam-deflection method, optical interferometry, or piezoresistive detection. In our macroscopic acoustic MFM teaching model, the frequency can simply be registered by a microphone and measured via a LabVIEW (National Instruments, Austin, TX, USA) program on a PC (see Section 3.1).

Figure 2:

Shift of the resonance frequency f₀ due to attractive (blue) and repulsive (green) interaction between the ferromagnetic sample with magnetization M→ and the permanently magnetized probe. The amplitude of the oscillation is kept at a constant level by the positive feedback circuit. Damping is not considered in this figure.

As mentioned before, FM-MFM allows for the detection of magnetic force gradients along the vertical z-axis. One assumes that the tip-sample interaction potential can be approximated by a Taylor expansion up to second order for small deflections, leading to a parabolic potential [14]. In this approximation, the interaction force is proportional to z. The effective spring constant k′ is then given by

Under the influence of, for instance, attractive magnetic forces, the effective spring constant of the cantilever is smaller compared to the original spring constant k. The deflection of the cantilever needs more time as compared to the case without magnetic field. This results in a shift of the resonance frequency [15] according to

where m is the effective mass of the cantilever, ω=2πf=2πk/m the angular frequency, and 1−x≈1−x/2, which consequently depends on the magnetic field gradient. Here, it becomes obvious that a uniform force causes no frequency shift in harmonic approximation. Measuring the frequency shift and its sign allows to determine whether attractive or repulsive forces are exerted. Further interpretation of the experimental data with respect to absolute magnetic forces is quite complex. This stems from the unknown and most often complicated magnetization vector field of the approximately pyramidal or cone-shaped tip leading to a dipole-dipole interaction between the ferromagnetic tip and the magnetic field of a sample. The point-probe approximation by Hartmann [12] introduces a great simplification by assuming a probe of infinitesimal size located at a suitable distance away from the surface sample. Nevertheless, magnetic contrast formation requires thorough knowledge of the experimental situation.

MFM is widely used as analytical tool for the stray-field variation of a magnetic sample, e.g. magnetic storage media as hard disk drives. Here, knowledge of the magnetization of the sample is needed to correctly interpret experimental data. In-plane magnetized storage media (longitudinal magnetic fields) lead to dips and rises of the resonance frequency below and above a constant frequency at bit boundaries of opposite magnetization [16]. In comparison, out-of-plane magnetized storage media (transversal magnetic fields with respect to the storage media surface) lead to areas of constantly higher or lower resonance frequency for the area of one bit (see Figure 6).

3 Experimental configuration of the macroscopic acoustic MFM model

The development of the presented macroscopic acoustic MFM teaching model should be considered within the framework of pre-existing approaches visualizing the key principles and properties of AFM via use of teaching models [4], [5], [8], [17], [18]. Out of these existing models, only one focuses on the dynamic or non-contact AFM mode instead of the static mode [8]. In the current contribution, the focus lies on MFM in dynamic mode instead. To the best of our knowledge, this is realized for the first time in a macroscopic teaching model. The principle of our acoustic MFM model is shown in Figure 1 and compared to the basic MFM principle.

3.1 Cantilever unit

In Figures 3–5, we show a photograph of the total experimental configuration and two sketches of specific components of the macroscopic model, respectively. In the dynamic mode of original MFM, the magnetic dipole-dipole interaction between the ferromagnetic tip of a cantilever and the magnetic field of a magnetized sample causes a frequency shift of the resonance frequency of the oscillating cantilever. This interaction is mimicked in our model. The investigated samples of the MFM model are different arrangements of neodymium magnets, while the macroscopic cantilever is a metallic leaf spring. Due to the direction and strength of their magnetization, these neodymium magnets are suitable to realize varying magnetic field configurations of the “samples” (see Figures 7 and 8).

Figure 3:

Photograph of the experimental configuration consisting of microscope and sample unit of the macroscopic acoustic MFM model.

Figure 4:

Sketch of microscope unit and feedback circuit as part of the macroscopic MFM model (blue=3D-printed material, orange=brass, light gray=LEGO).

Figure 5:

Sketch of sample unit in front and side view of the macroscopic MFM model.

The leaf spring cantilever is set into oscillation using an electromagnet, which is placed within a short distance above the spring (cf. Figure 4 for detailed setup) [19]. This excitation method is similar to a technique being used in the abovementioned model visualizing dynamic mode AFM [8]. Without the presence of a sample, the cantilever oscillates in its resonance frequency f₀. If the sample is placed at a distance z0 below the cantilever, the resonance frequency of oscillation changes according to equation (2). Note that the frequency shift occurring in the model can be derived along the same lines as for original MFM. For the steel ruler used during the presented measurements, the parameters are determined experimentally as f₀=(302±1) Hz and k=(3184±284) N/m for a free length of 35 mm, a thickness of 0.5 mm and a width of 13 mm (cf. Figure 4) [20].

The choice of a possible leaf spring is determined experimentally. Under the influence of the electromagnet, the leaf spring should oscillate after being struck within audio range. Under the additional influence of neodymium magnets serving as samples, the changed resonance frequency should vary significantly enough to generate audible changes. Thus, different types of (soft ferromagnetic) leaf springs can be deployed to serve as cantilevers. In particular, semi-flexible steel rules or feeler gauges are suitable to implement the specific oscillation and magnetic properties of a cantilever being frequently used in physics education in schools or universities. As all cantilevers of our measurements reveal similar results, the experimental data of the investigation of real-life samples in Section 4 are obtained with the same steel ruler being described above.

The frequency shift can be interpreted as an image of the magnetic properties B_s of the sample. To ensure a continuous oscillation of the cantilever, a positive feedback circuit is implemented. The circuit is based on a microphone being located just above the cantilever (cf. Figures 1 and 4) turning the oscillation frequency to a voltage signal. This signal is amplified and triggers the power supply of the electromagnet. In this way, the continuous oscillation of the cantilever is preserved and instantly follows any shift of the resonance frequency. Furthermore, a low-pass filter is incorporated to suppress any overtone oscillations (see left section of Figure 4).

It is important to note that in contrast to original MFM, the interaction in our model takes place between the sample and the overall cantilever. For a genuine MFM cantilever the interaction would only occur between the sample and the tip of the cantilever. However, the frequency displacement follows analogous rules.

The construction of the apparatus follows the superior principle that all materials being used, except the one of the cantilever, should be free of any oscillation and possibly damp external oscillations ensuring the frequency displacement origins only from the sample magnetization. The leaf springs are attached with the help of two brass plates, and 3D printing is used to realize the stand equipment (see Figures 3 and 4). These materials are non-ferromagnetic so that the interaction between the neodymium magnets and the attachment device can be neglected. Based on the high mass density of brass, the heavy attachment device damps the whole experimental setup.

4 Investigating real-life probes

A typical ferromagnetic domain structure with relevance for students’ daily lives, representing an important issue to gain motivation in physics education [22], [23], can be found in hard disk drives (HDDs) or comparable magnetic storage media. While classical audiotapes utilize longitudinal magnetic recording (LMR), current HDDs apply perpendicular magnetic recording (PMR) and thus out-of-plane magnetization, optimizing the areal density [24]. Both LMR and PMR facilitate the storage of a defined quantity of bits, either 1 (+M) or 0 (−M) (see Figure 6), depending on the size and areal density currently exceeding 1 Tbit/in² [25].

Figure 6:

Out-of-plane magnetized storage medium displaying stray field and idealized magnetic force F→mag between cantilever and sample.

MFM imaging uses the interaction between this magnetized domain structure and the ferromagnetic cantilever revealing a fixed perpendicular magnetization to a color-coded image as an example of a line scan (see Figure 6). According to the alignment of the HDDs’ magnetic moments being either parallel or antiparallel to the cantilever’s magnetization, an attractive or repulsive magnetic force F→mag appears.

In the MFM model, the gradient of the magnetic force F→mag influences the initial resonance frequency of the cantilever, as shown in equation (2). However, in contrast to real MFM measurements, our model is solely able to detect attractive forces between sample and cantilever, always leading to lower frequencies. Thus, the teaching model is insensitive to the alignment of the sample’s magnetization. This stems from the use of soft magnetic leaf spring cantilevers, being always attracted by the sample’s strong neodymium magnets due to a reset of magnetization. A direction-dependent measurement could be realized with a hard magnetic cantilever. In this case, a phase shifter has also to be integrated in order to maintain the oscillation of the cantilever at resonance frequency.

Analogous to the MFM measurement of an original HDD, we build different magnetic domain line structures inserting cylindrical neodymium magnets into the 3D-printed sample sledges. As our model is unable to identify the alignment of the neodymium magnets, the “bits” were coded as 0 (no magnet) and 1 (magnet).

One of our samples represents a “01101”-bit pattern (see Figure 7), making use of three uniform cylindrical neodymium magnets (radius r=7.5 mm, height h=8 mm, N42). The measurement of the frequency against the position of the cantilever exhibits characteristic decreases in the oscillation frequency compared to the initial resonance frequency f₀≈302 Hz. These correspond to the positions of the neodymium magnets (cf. Figure 7). For each magnet, the shift arises continuously. Furthermore, interactions of the magnetic fields obviously superpose because even if no neodymium magnet is present at 5.5 cm, the frequency shift does not vanish. In addition, the frequency shift for the first and second neodymium magnets significantly differs from the third magnet, a consequence of interfering magnetic fields caused by the spacing d≈10 mm between the neodymium magnets and displayed by real HDDs as well. However, the MFM model is clearly able to detect the positions of the neodymium magnets according to a shift in the oscillation frequency. Interestingly, both initial and altered oscillation frequencies are placed within the human hearing range so that the frequency shift, which is about |Δf|=20–30 Hz, cannot only be measured but also instantly heard.

Figure 7:

Analysis of a “01101”-bit pattern referring to a PMR medium (uniform magnets).

As the frequency shift depends on the magnetic field-dependent magnetic force gradient, see equation (2), another feature of our MFM model is the sensitivity for the magnetic field strength of the sample. This feature was investigated using another sledge containing cylindrical neodymium magnets (r=7.5 mm) of varying strength (cf. Figure 8). While the first and third magnets (h_1,3=8 mm, residual flux density B_r=1.30 T) are equal, the second (h₂=3 mm, B_r=1.28 T), fourth (h₄=5 mm, B_r=1.35 T), and fifth (h₅=2 mm, B_r=1.30 T) magnets differ in strength of their magnetic fields. The measurement of the resonance frequency, see Figure 8, reveals once more that the MFM model is clearly able to detect the positions of the neodymium magnets being placed underneath the cantilever. Again, continuous transitions of frequency shift are measured. Likewise, the frequency shift for the first and third magnets, exhibiting the same magnetic field strength, deviates according to a total magnetic field of the sample being composed of the superposition of the individual magnetic fields. Nevertheless, the order of the strength of the neodymium magnets is correctly represented by the amount of frequency shift for all magnets. This means, besides being able to detect the position of each neodymium magnet, that the MFM model is also able to identify the relative strength. Two videos of the scanning process of the acoustic MFM model for an exemplary sample are provided with this manuscript [21].

Figure 8:

Analysis of a magnetic domain sample with neodymium magnets of varying strength.

With the help of equation (2), the measurement allows calculating the magnetic force gradients ∂F_mag/∂z for each neodymium magnet. The quantitative experimental results are presented in Table 1 and support the qualitative interpretations being described in the previous passage.

In comparison to a real MFM measurement, it should be noticed that both samples have approximately constant height surfaces, allowing a single scan instead of conducting a prior topography scan [26].

5 Scales in the teaching model

Precisely spoken, the presented MFM model reproduces a magnetic force microscope taking a line scan of a plane sample with out-of-plane magnetization in FM detection mode. Besides the obvious analogy of the line scan of a plane sample, the system is completely automated with a program including an easy-to-use user interface as in real MFM. Thus, the functionality of the model resembles that of a real magnetic force microscope under the same restraints. It exhibits a shift in the resonance frequency of an oscillating magnetized probe under the influence of a ferromagnetic sample. Furthermore, the teaching model features the interchangeability of the probe as well as of the sample. Even the low-pass filter resembles the band-pass filter utilized to filter unwanted frequencies in MFM.

The key characteristics of both the microscopic and the macroscopic magnetic force microscope are a further point of interest. Minimal detectable force gradients in microscopic MFM are of the order of 10⁻⁴ N/m [27] under ambient conditions and lead to a frequency shift range of |Δf|=0–50 Hz. In the presented teaching model, force gradients have to be comparably higher to influence the oscillation of the cantilever. The force gradients of the magnet in our sledges acting on the steel ruler are of the order of 10² N/m, which is found by measuring the force of the magnets on the ruler in varying distance. As can be seen from Figures 7 and 8, these higher force gradients lead to comparable frequency shifts, as the resonance frequency and the spring constant of the steel ruler are correspondingly higher.

Another interesting key feature is the spatial resolution of the microscopes. Here, the magnetized part of the probe, which interacts with the sample’s stray field, and the probe-sample distance are relevant variables [12]. Commonly used cone-shaped probes have diameters of approximately d=100–120 mm and are operated at z₀=50–250 nm lift height (ratio r_mic=d/z₀≈0.4–2.4). The spatial resolution that can be reached with these probes then amounts to below 100 nm [12]. For the so-called supertips with diameters down to 40 nm, resolutions down to 50 nm or even 10 nm have been reported [12], [28], [29]. For the macroscopic model, the width of the steel ruler is w=13 mm operating at z₀=6 mm above the sample (ratio r_mac=w/z₀=2.2), and Figures 7 and 8 show that the teaching model easily resolves the spacing of the magnets (a=25 mm). Obviously, the ratios r_mic and r_mac of probe diameter (probe width) and lift height are in good agreement between microscopic MFM and the teaching model. Here, one can deduce a general feature of scalable physical quantities: For the scaled lift height and comparable ratio r, the spatial resolution is also comparable – here, this means that it is in the same magnitude as the diameter/width of the probe.

The comparison between the qualitative and quantitative key features shows a striking resemblance between the macroscopic teaching model and original microscopic MFM in functionality as well as in the resonance shift and the spatial resolution. This can be attributed to the physical properties of MFM. The interaction between the sample and the probe is of long-range magnetostatic nature [30]. Therefore, in contrast to most SPM types, no quantum effects have to be considered. MFM uses magnetic force gradients that can be described by classical electromagnetism. Therefore, the underlying theory of microscopic and macroscopic MFM is the same, as the classical magnetic forces are scalable. Even the point-probe approximation introduced for microscopic MFM [12] holds true for the macroscopic teaching model.

However, the teaching model varies in a single major characteristic: while microscopic MFM can detect attractive and repulsive force gradients, the presented teaching model is only suited to register attractive interaction. This is partly owed to the steel ruler serving as a cantilever. During a scanning procedure, its magnetization is not constant but is influenced by the neodymium magnets as well as by the driving electromagnet. Moreover, no separate tip is applied to the steel ruler so that the magnetic interaction takes place between the complete magnets and the area of the ruler above them.

6 Introduction of MFM model and learning objectives

As the presented MFM teaching model includes comparable elements to all major parts of a real magnetic force microscope, the working principle of MFM can be explained solely with the acoustic teaching model. Furthermore, several key principles of the functionality of the teaching model are related to more general topics of physics education and, therefore, pave the way for a thorough involvement. The following topics are strongly connected to the acoustic MFM model:

• ferromagnetism (magnetic attraction and repulsion, magnetic field lines, magnetic dipole character, magnetic forces, etc.),

• electromagnetism (construction of electromagnets, magnetic field lines, relation between electric current and magnetic fields, etc.),

• waves and oscillations (free and driven harmonic oscillations, resonance frequency, harmonic oscillations of springs and leaf springs, etc.), and

• acoustics (human hearing range, standing waves, sound propagation, etc.).

In addition to these general topics of physics education that can be considered in an educational setting related to the acoustic MFM model, the following more specific learning objectives should be aspired to during working with students on the MFM model:

• understanding the need for nanotechnology tools for surface analysis;

• knowing the difference between the two basic operational modes (static and dynamic mode);

• understanding the basics of the dynamic mode AFM / MFM principle: function of raster scanning, influence of sample forces on probe movement, image formation of data, data analysis, and drawing conclusions from measured data; and

• understanding the limitations of the model and transferring model characteristics to real MFM functionality.

As mentioned above, the working principle of MFM can be explained solely with the acoustic teaching model. To this end, the listed general topics can be involved by the repetition of those topics taught in middle school (e.g. acoustics, basics of magnetism) and by the introduction of those topics typically taught in (senior) high school. Working with the acoustic MFM teaching model then establishes the concept of MFM and allows for a short overview over the more specific learning objectives.

A different approach can be chosen when a specific course or closed teaching unit on AFM and/or MFM is held. Here, when at the educational level of senior high school or even at university level, the more general topics can either be assumed as known or be revised shortly in, e.g. marketplace learning with students’ experiments. On the other hand, the more specific learning objectives should take up a major part of the allotted course time. Such an educational setting on MFM either can stand for itself or can even be part of an introductory course into nanotechnology that includes analytical tools.

As stated in Section 4, relevance for students’ daily lives is important to gain motivation in physics education. Therefore, comparison of provided samples with magnetic storage media or other well-known ferromagnetic domain structures is an indispensable part of any introduction to MFM with the teaching model. If an original magnetic force microscope is available, investigation of a storage medium and calculation of its storage size are possible beneficial extensions on MFM teaching units. A guideline for a learning module is presented, e.g. by Vandervoort et al. [31].

Moreover, the critical reflection of possible (mis-)interpretation of data is an important competency in science education. The teaching model invites to a critical discussion of the interpretation of the frequency shift in real MFM. Due to its scalability, the macroscopic magnetic structure of the sample can directly be compared to the measured frequency shifts, as shown in Figures 7 and 8. In real MFM, only measured frequency shifts are given as numerical data. The possible origins of frequency shifts cannot easily be deduced without knowledge of the topography of the sample. But even if the topography is known, as the frequency shift depends in a non-trivial way on the magnetic field gradient [equation (2)], the data will not provide a unique solution for the magnetic field distribution without further approximations or simplifications. Here, our model reveals a novel method to gain insight into the relation of magnetic structure and measured frequency shift. Furthermore, the model supplies multiple representations (visual, acoustic, and numerical information) to discuss the magnetic force dependency of the frequency shift, in particular, and the magnetic field’s gradient dependency of magnetic forces, in general, which represents an important aspect in the understanding of magnetic interactions.

7 Summary and conclusions

The presented macroscopic MFM teaching model allows for a thorough understanding of MFM. It permits studying the measurement process of a magnetic force microscope taking a scan of a magnetic surface and allows for visual and acoustic perception of the FM detection mode. Moreover, it features interchangeability of probe and sample as well as complete automatization. Comparable frequency shifts and a similar scaled spatial resolution participate in generating an in-depth resemblance of the teaching model to the microscopic nanotechnology tool.

The model can additionally serve as a basis for the revision or introduction of contents of a typical school curriculum for middle and high schools. It provides insight into nanotechnology and therefore constitutes educational content at the cutting edge of actual scientific topics. Overall, the model therefore provides a great opportunity for any teaching unit about MFM. The easy understandable LEGO setup invites pupils and students to experiment with the model, which faithfully reproduces the functionality of a widely used analytical tool in nanotechnology, the magnetic force microscope. Even critical reflection of possible (mis-)interpretation of data, which is an important competency in science education, can be integrated into a teaching unit.

The authors plan on improving the model in several points. A hard ferromagnetic tip is to be added to the cantilever, so that magnetic interaction takes place between the sample and the tip, instead of the cantilever. This consequently allows for measurement of attractive and repulsive forces but needs some adjustment in the electrical circuit. Other nice extensions of the model would be an aerial instead of a line scan as well as a controllable measurement area instead of tactile sensors to raise the resemblance to microscopic MFM even higher.