Over the past few decades, the magnetic properties of iron oxide nanoparticles have been widely explored mainly due to their potential applications in various fields, including storage media, environmental remediation, and bio-medicine.^[1–10] For scientific research, a precise characterization of the magnetic properties of individual magnetic nanoparticles (MNPs) is essential. In the last decade, the introduction of new experimental techniques, such as the anomalous Hall-effect,^[11] superconducting quantum interference device,^[12,13] spin-polarized scanning tunneling microscopy,^[14] transmission x-ray microscopy,^[15] and nitrogen-vacancy magnetometer^[16], have helped us to obtain insight into the magnetic properties of individual and isolated MNPs. However, these techniques require a highly complex experimental apparatus and a dedicated sample preparation process. Although a lot of magnetic microscopy techniques have been developed over the past decades, imaging magnetism at a sub-10 nm nanoscale remains a challenging task because it requires a combination of high spatial resolution and sensitivity.^[17]

Another remarkable technique that allows the direct imaging of magnetic nanostructures is magnetic force microscopy (MFM). It is an ideal tool to investigate the magnetic behaviors of nanosized magnets due to its high lateral resolution, which can provide microscopic information about the magnetic behavior of an individual nanomagnet.^[18–26] The ability of MFM to detect superparamagnetic and low-coercivity MNPs, and the interpretation of the resulting MFM images, are subjects of ongoing research. Agarwal^[22] reported the use of MFM to detect superparamagnetic nanoparticles (SPNs) with a dextran shell (giving a final diameter of 30–50 nm) in ambient atmospheric conditions. Sievers described the quantitative characterization of the magnetization of individual MNPs (about 19 nm in diameter) using MFM.^[23] Park investigated the magnetic interactions of aggregated MNPs with a size of approximately 20–30 nm by using MFM with a Ferritin-based MFM probe.^[24] Recently, Moya reported on the experimental characterization of the magnetic domain configurations in cubic isolated Fe_3−xO₄ nanoparticles with a lateral size of 25–30 nm.^[25] Although successful attempts to characterize individual MNPs have been made,^[22–26] the capability of MFM to detect a signal from nanoscale superparamagnetic particles with a grain size less than 10 nm has not been fully explored. A systematic study of the applicability of MFM for characterizing SPNs with a grain size less than 10 nm in ambient air is still lacking.

In addition, extensive experiments and theoretical studies show that the magnetization reversal process of MNPs is very complicated, and sometimes the results of these studies are not consistent with each other.^[27–29] For instance, in some cases, the accurate value of reversal fields of MNPs was different from the value measured by conventional magnetometer instruments, or showed a different dependence on the diameters. Furthermore, when the nanoparticles become more and more dense, the magnetization reversal processes become affected by interparticle interactions.^[30] On this basis, it is paramount to study the magnetization properties of individual nanoparticles as well as their mutual interactions.

In previous studies, we successfully carried out a direct imaging of magnetic domains and quantitative measurements of the magnetic information from individual superparamagnetic Fe₃O₄ nanoparticles by using MFM.^[3,4] In this work, in order to better understand the magnetization reversal process of 10 nm-sized Fe₃O₄ MNPs, the frequency-modulated MFM^[31] with a variable applied magnetic field is adopted to investigate the dynamic magnetization behavior of the Fe₃O₄ nanoparticles. A vibrating sample magnetometer is employed to measure the hysteresis loops of the Fe₃O₄ nanoparticles at room temperature to study the coercive field (H_c) and saturation magnetization (M_s) under a maximum magnetic field of ±10 kOe. In addition, micromagnetic calculations are important for interpreting the experimental MFM images and providing some insight into the phenomenon taking place during the magnetization reversal of the Fe₃O₄ nanoparticles.

The frequency modulation detection method is used in the field-variable MFM technique. Figure 1(a) shows the schematic diagram of the MFM system with a variable magnetic field. The ferrite core generates a homogeneous alternating magnetic field at the sample surface with an amplitude from −12 Oe to 12 Oe. A detailed description of the frequency-modulated MFM system has been given in Ref. [31]. It is possible to perform the measurement of the magnetic field gradient for the magnetic samples. Here, the sign (upward and downward) of the magnetic field reflects the polarity (N or S pole) of the surface magnetic pole. Thus, it is possible to directly detect the sign of the magnetic field to reflect the polarity of the surface magnetic pole of the magnetic samples.

Fig. 1.

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	Fig. 1. Schematic diagrams of (a) the frequency-modulated MFM system and (b) sample status.

The experiment was done in air atmosphere. The resonant frequency of the cantilever was about 256 kHz, and the Q value was about 500. The Fe₃O₄ MNPs were randomly distributed on the silicon substrate. After that, a thin layer (about 3 nm) of gold particles was sputtered on the MNPs, which were then covered and immobilized on the substrate, as shown in Fig. 1(b). The gold particles were obtained by using an ion sputter coater of a scanning electron microscope. The sputtering power, air pressure, and sputtering time were held at 15 W, 3 Pa, and 12 s, respectively. Tapping and lift mode AFM/MFM scans were carried out using a high-coercivity FePt tip ( ), which was coated with a thin FePt film with a radius of 10 nm. The lift height (tip-sample distance) was kept at 5–10 nm. The magnetization direction of the tip (which was magnetized before use) was perpendicular to the sample surface.

Figure 2(a) shows the transmission electron microscopy (TEM) image of the as-synthesized Fe₃O₄ nanoparticles. The TEM image indicates the presence of small Fe₃O₄ particles of nanometric size. The diameters of most nanoparticles are within 20 nm, with the minority less than 10 nm, and some overlapping. Figure 2(b) shows the magnetic hysteresis loop of the Fe₃O₄ nanoparticles. The hysteresis loop shows that the as-synthesized Fe₃O₄ nanoparticles exhibit a saturation magnetization of ∼60 emu/g, which is in accordance with that of iron oxide nanoparticles with a similar size (typically in range of 40–70 emu/g). A magnification of the field range where these interceptions take place is shown in the inset of Fig. 2(b). The obtained coercive fields are about 6.8 Oe. This very small value of H_c for the studied MNPs suggests that the majority of the particles in this sample are superparamagnetic at room temperature.

Fig. 2.

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	Fig. 2. (a) The TEM image of the as-synthesized Fe3O4 nanoparticles. (b) Magnetic hysteresis loop of the Fe3O4 nanoparticles (inset: zoom-in of the low-field region).

In Figs. 3(a) and 3(b), we give an example of the AFM/MFM images of the Fe₃O₄ nanoparticles without an external applied magnetic field (corresponding to the magnetic fields of 0 Oe). It can be seen that the topography (Fig. 3(a)) and phase (Fig. 3(b)) images are indeed well separated. The results show that the investigated surface has perfectly particle-like features, which presents the surface information of the uniform-sized gold particles. However, the magnetic signals observed from the MFM phase image are very weak (dark MFM contrasts, with a lift height of 5 nm), indicating a weak attractive force between the MFM tip and SPNs.

Fig. 3.

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	Fig. 3. MFM images in the same area of Fe3O4 samples. (a) Topography, (b)–(h) phase images with the various applied magnetic fields, (i) spectrum line profiles of particles 1#–4# shown in (g), and (j) 3-dimensional MFM image of the white line box area shown in (h).

Figure 3(c)–3(h) show the field-variable MFM images of the Fe₃O₄ nanoparticles with the in-plane driven magnetic field H ranging from −10 Oe to 10 Oe in order to gain deeper insight into the magnetic nature of these isolated particles. As the external field varies, we observe no change in the topographic images but only in the magnetic phase images, thus confirming that the topographic images are not appreciably affected by the applied magnetic field. In the MFM images, almost all of the MNPs present a characteristic with bright–dark dipolar contrast. The bright area and dark area indicate a phase difference of ∼180°. It is obvious that the magnetic particles remain distinguishable from the background. Moreover, it can be found that the magnetic contrast of the Fe₃O₄ MNPs changes uniformly with the applied magnetic fields. Similar magnetic domain structures with opposite polarities are seen when comparing MFM images at opposite values of the applied magnetic field. The MNPs show a typical dipolar contrast (bright and dark) after the application of various external magnetic fields, thus providing support for the single-domain structure in the samples. Considering that the individual particles are of superparamagnetism at zero field, the observed changes in magnetic polarization can be understood as just being induced by the action of the variable external magnetic field.

Figure 3(i) shows the MFM signal profiles across the center of the selected nanoparticles (1#–4#) in Fig. 3(g). The spatial resolution of the MFM measurement in the current study is estimated from the spectrum line profiles of the MFM phase signals. The resolution is determined as the full width at half maximum of the peak signals. As shown in Fig. 3(i), the determined resolution is around 7.6 nm under the ambient condition with a lift height of 10 nm. Figure 3(j) clearly reveals the fine magnetic domain configurations of the individual MNPs corresponding to the four isolated nanoparticles with typical domain structures in the white line box of Fig. 3(h). The results imply that the four nanoparticles are all in a single domain state with a domain size of about 22 nm (estimated from the line profile), which is larger than the result determined from TEM experiments (∼12 nm). This mainly resulted from the magnetic stray field at the boundary of MNPs. For all the magnetic domains of nanoparticles, left is bright (repulsive force), while right is dark (attractive force). This domain configuration is a typical dipolar response and can be considered as an in-plane magnetic domain. Summing up the magnetic properties of the Fe₃O₄ nanoparticles, we can conclude that they can be described as consisting of single-domain particles. The above results reveal that a stable magnetic dipole can be induced in SPNs by an external magnetic field of a few Oe at room temperature. Our MFM experiments reveal that the presence of an external magnetic field and a magnetic probe is essential to detect and distinguish the MFM signal from the SPNs. By applying a magnetic field to the sample, we could detect the in-plane dipole moment of SPNs as a combination of positive and negative phase contrasts. The field-dependent magnetic domains of the dipole moment in individual MNPs help confirm the presence of magnetic interaction between the MFM probe and the nanoparticles and also distinguish it from nonmagnetic contaminants present on the surface.

The recorded reversal of the spin configuration from one direction (for negative fields) to another one (for positive fields) is experimental evidence of the switching process being performed by a coherent rotation mechanism. In order to verify this presumption, micromagnetic calculations are performed to simulate the magnetization distribution in the Fe₃O₄ MNPs. In micromagnetic modeling, the total energies in each grid include the Zeeman energy, crystalline anisotropy energy, shape anisotropy energy, exchange interaction, and magnetostatic interaction. The simulation parameters are selected as follows: a saturation magnetization of 311 emu/cm³ (at room temperature), and a magnetocrystalline anisotropy constant of 5.0×10⁶ ergs/cm³. The calculation of the magnetization reversals is accomplished based on the Landau–Lifshitz–Gilbert equation. The simulated MFM images are obtained by working out the first-order derivative of the magnetic field ( ) at the sample surface.

The calculated in-plane hysteresis loop of a single MNP (10 nm) is shown in Fig. 4. This curve represents the magnetization reversal of the nanoparticle. The calculation suggests that the reversal mechanism of an individual Fe₃O₄ nanoparticle follows a coherent rotation of the spins, as indicated in Figs. 4(a)–4(c). Since the Fe₃O₄ nanoparticle is very small, the exchange energy may lead to the reversal process. The micromagnetic calculations suggest that a coherent rotation mechanism should be correlated with the magnetization reversal process observed in the MFM images, rather than with the nucleation and propagation of domain walls.

Fig. 4.

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	Fig. 4. Micromagnetic calculations. (a)–(c) Distribution of the spin configurations for the three situations highlighted by red arrows in the calculated hysteresis loop in (e). Panels (d) and (f) show the calculated MFM contrasts for the configurations shown in (a) and (c), respectively. (e) Calculated hysteresis loop for a single Fe3O4 nanoparticle.

In summary, the local detection of magnetic domains of isolated 10 nm-sized Fe₃O₄ MNPs has been investigated with field-variable MFM. The field-variable magnetic domain structures and magnetization process of Fe₃O₄ MNPs were determined locally in the nanometer scale with high spatial resolution. The present results suggest that the field-variable frequency-modulated MFM indeed provides a means to characterize the magnetization behaviors of individual MNPs with very small coercive force, thus enabling us to separately estimate the distributions of the dipolar fields and the local switching fields of the MNPs. Therefore, this technique has great potential to provide information on magnetic features of magnetic nanostructures, such as nanoparticles, nanodots, skyrmions, and vortex, with high spatial resolution. Such information is crucial for the development and application of magnetic nanostructures and devices.