Directional power loss in magnetic thin films guided by magnetic anisotropy from magnetic nanoparticle chains

To date, there are many papers confirming Kittel’s 1948 theory. These papers take into account different types of magnetic crystal anisotropies and elastic deformations of the magnetic material, which can affect the ferromagnetic resonance condition. It is important to note that the single-domain magnetic nanoparticles with uniaxial magnetic anisotropy show an analogous shift in the resonance value of the applied $H → DC$ field as the one caused by the demagnetization field in Kittel’s model. The example can be found in paper⁶ concerning ferromagnetic resonance experiment (FMR) in magnetic thin films filled by spatially ordered nanowires composed of magnetic nanoparticles, which possess uniaxial anisotropy. In this case, for the ferromagnetic resonance condition to be satisfied for a given microwave frequency $f$⁠, the resonance value $( H DC res ) ⊥$ of the static magnetic field $H → DC$ perpendicular to the anisotropy axis must overcome the effect of the presence of the magnetic uniaxial anisotropy field $H → a$ of the magnetic nanoparticles and $( H DC res ) ⊥> ( H DC res ) ∥$⁠. In the same paper, the ferromagnetic resonance measured in thin films with magnetic nanowires placed in random directions did not show this effect.

Depending on target applications, the magnetic thin films are prepared by various vapor deposition techniques such as epitaxial method, magnetron sputtering, electrodeposition, and pulsed laser deposition. These methods are complemented by spin coating and sol–gel synthesis, which opened the possibility for new strongly anisotropic magnetic films based on magnetic nanowires. A particular example can be the method of self-assembling the colloidal magnetic nanoparticles under the external magnetic field into a structure of nanowires.^6,7 Self-assembling of colloidal magnetic nanoparticles can also be forced by substrate asymmetry.⁸ Some other approaches use templates for nanowire synthesis, e.g., iron-oxide coated carbon nanotubes,⁹ or they use the possibility of directional chemical bonding between polymer coatings of the coated magnetic nanoparticles, e.g., silica coated magnetic nanoparticle nanowires.¹⁰ The last few decades have seen the rapid development of stretchable electronics and stretchable magnetoelectronics,^11–13 which include the magnetic thin films and which introduce additional challenges for electronic systems to enable control of the magnetic properties when changing the shape of the thin film or applying elastic deformation. It is the case of flexible sensors based on nanomaterials,¹³ soft robotics,¹⁴ and still increasing number of other applications. In all these applications, it is important to limit power dissipation in the magnetic materials used to prevent uncontrolled heat release. Magnetic thin films are considered also in electromagnetic shielding applications, such as microwave shielding.¹⁵ Their task is to absorb the microwave power. In the latter case, the most common applications concern the protection of medical or military equipment components from the incident microwave radiation, but they also include the health aspect and the protection from information leakage and influence.

In the current work, we focus on the role of magnetic anisotropy in 2D arrays of magnetic nanoparticle chains to control the power loss in them. In discussion, we refer to Kittel’s theory of ferromagnetic resonance to explain some of the results. The experimental findings were confirmed by computer simulations.

The magnetic materials considered in this work come from the magnetic nanoparticle colloid based on $Fe 3 O 4$ nanoparticles stabilized with APTES (3-aminopropyltriethoxysilane), where the pH value of the colloid solution is approximately 4. In the solution, the average hydrodynamic size of the coated nanoparticles is approximately 37 nm (measured by Malvern Zetasizer Ultra). The bare $Fe 3 O 4$ nanoparticles have an average size of 12 nm. Transmission electron microscopy (TEM) with the energy-dispersive x-ray spectrometer (EDX) was used to confirm the size of the coated magnetic nanoparticles and their chemical composition. Raman spectroscopy with the Renishaw InVia Qontor spectrometer was used to confirm the kind of magnetic iron oxide nanoparticles. The $Fe 3 O 4$ nanoparticles were obtained by coprecipitation method according to the chemical protocols published in papers.^8,16 The same applies to the preparation of the colloid of the magnetic nanoparticles, where these nanoparticles are stabilized with APTES. Such colloid, in a form of the 5  $μ$l magnetic colloid droplets, was placed on an elastic hydrophobic substrate to dry, which was the low density polyethylene (LDPE) with the thickness of 20  $μ$m.

In the experiments on power loss in magnetic films, two types of magnetic nanoparticle self-assemblies were considered. The first one concerns magnetic nanoparticle stains on the virgin LDPE, which take the shape of round spots. In the second case, the substrate is an asymmetric LDPE, where such asymmetry is introduced by stretching the film in one selected direction, and the multi-ring nanoparticle structures appear instead of the round spots. In the current work, 1 cm of the film has been stretched to 2 cm (1–5 cm in papers^8,16).

The power loss properties of the prepared magnetic films were investigated with the help of the FMR experiment, where the X-band EPR spectrometer was used (Radiopan SE/X-2013 type) with the alternating magnetic field frequency $f=9.32$ GHz.

To find the numerical solution of the LLG equation, the Heun method for solving ordinary differential equations with a given initial condition was used, which is a modified Euler method providing a predictor-corrector method. The selected time step is defined as $Δt=τ/ N div$⁠, where $N div$ is the number of divisions of $τ$⁠. In particular, in the case of microwave magnetic field frequency $f=9.3151$ GHz and amplitude $μ 0 H AC 0=0.5$ mT applied to a single magnetic nanoparticle, $N div=16384$ and $Δt≈6.5522× 10 − 15$ s. The integral in Eq. (8) was solved numerically by applying the fast Fourier transform to a stable magnetization cycle ${ M →(0), M →(Δt),…, M →(( N div−1)Δt)}$ obtained from the numerical solution of LLG.

Modeling the magnetization precession around an effective local field for each nanoparticle, $i=1,2,…,N$⁠, using a system of LLG equations for $N$ nanoparticles requires the calculation of the magnetic dipole interactions between nanoparticles for each time step dt. Obtaining stable solutions of the LLG equations, where, with a full cycle of the $H AC$ field change, the magnetization values $M i(t)$ for each nanoparticle also change cyclically requires performing at least several dozen cycles of the $H AC$ field. In this work, only nanoparticle systems with maximum N = 24 nanoparticles are considered. The numerical results which were obtained for determining the ferromagnetic resonance of a single nanoparticle are qualitatively consistent with the results for a larger number of nanoparticles and the results of the FMR experiment.

In the paper¹⁸ for modeling elastic deformation, taking into account the LLG equations for nanoparticle magnetization, a Monte Carlo approximation for elastic deformation was used, and in each Monte Carlo step, the LLG equations were used solely to calculate a stable magnetization cycle.

Single-domain magnetic nanoparticles with an uniaxial magnetic anisotropy have a special ability to absorb the power of a microwave magnetic field and release it to their surroundings as a heat. This nanoparticle ability is exploited, for example, in medical applications for magnetic hyperthermia.^19,20 The experimental results presented in the current work and those derived from theoretical models are of more general interest and apply both to magnetic nanoparticles and to any type of ferromagnetic thin films with magnetic anisotropy. Such a general feature of the experimental results is shown in Fig. 1. It demonstrates an effect of directional resonant absorption of microwave magnetic field in thin magnetic films. In this case, the thin films of magnetic $Fe 3 O 4$ nanoparticles stabilized with APTES are placed on a LDPE foil, where the foil takes the form of a roll. The FMR experiment is performed on two such samples to compare their FMR signals. The samples differ mainly in the way they have been rolled. The magnetic nanoparticles appear on LDPE in the form of round spots visible in panel (a) of Fig. 1, where these spots are the stains after drying of 5  $μ$l colloidal droplets taken from a suspension of such nanoparticles in an aqueous solution of pH = 4. The average thickness of the spots is about 100–300 nm and due to this small value, the in-plane magnetic anisotropy field dominates the effective intrinsic magnetic field in them. The demagnetization field in the spots is negligible. In both samples, the spots appear as the flat circular structures. Panel (b) in Fig. 1 shows the FMR signals measured for two rolled LDPE samples, sample 1 and sample 2, shown in the upper right corner of the panel (b). The signals suggest the presence of the more than one resonance value of the DC magnetic field. During the measurements, each of the samples is placed in the microwave cavity of the EPR spectrometer so that the DC magnetic field is perpendicular to the long axis of the LDPE roll. This means that for each sample, the DC magnetic field is directed into the spots at a wide range of angles from 0 $°$ to 90 $°$⁠. Thus, the in-plane anisotropy of the spots is responsible for the complexity of the FMR spectra presented in panel (b). To demonstrate the role of magnetic in-plane anisotropy, the LDPE film of the sample 1 was cut into square pieces as shown in panel (a) and the pieces were collected into a new cube-shaped sample. For such a sample, depending on its arrangement with respect to the direction of the DC field, $H → DC= ( H → DC ) ∥$ or $H → DC= ( H → DC ) ⊥$⁠, the FMR signal was measured. It confirms the presence of two different values of the resonant DC magnetic fields, as can be seen in panel (c), and $( H DC res ) ⊥> ( H DC res ) ∥$⁠.

In order to clarify the role of the magnetic anisotropy introduced by single-domain $Fe 3 O 4$ nanoparticles for the possibility of directional absorption of microwave magnetic field power, a comparative FMR experiment was performed. However, in contrast to the previous case, a ring-shaped chains of $Fe 3 O 4$ nanoparticles were used instead of the circular flat structures of magnetic nanoparticles from Fig. 1. To prepare the arrangement of magnetic nanoparticles into nanoparticle chains, the method of self-assembly of magnetic nanoparticles into the chains during drying of the magnetic colloid droplet was used.⁸ Such an effect occurs on the LDPE film, on the surface of which directional asymmetry is introduced, which is obtained by stretching it in a predetermined direction, let us say the $x$-direction. In the case of such asymmetric deformation, the root mean square roughness of the height deviations of the LDPE surface measured from their mean value in the stretching direction can be smaller by a few nanometers compared to the transverse direction. As a result, during the drying of a colloid droplet that shrinks on the hydrophobic LDPE, magnetic nanoparticles that agglomerate near the contact line of the droplet are able to slow down this shrinking and pin the contact line to the substrate by counteracting the capillary forces. Further evaporation of the droplet decreases the contact angle of the droplet, and at some values, the contact line jumps to a new position with a smaller radius, leaving a chain of magnetic nanoparticles. The process is repeated, leaving ring-shaped chains of nanoparticles on the LDPE surface. In contrast, in virgin LDPE, as in Fig. 1, the magnetic nanoparticles are unable to counteract the capillary forces and the magnetic chains, which start to form at the contact line of the droplet become broken before they pin to the substrate permanently. Figure 2 shows microscopy images of the obtained structures. Panel (a) shows the optical microscopy image of the magnetic nanoparticle stains after drying a 5  $μ$l droplet of magnetic colloid (same parameters as in the previous experiment in Fig. 1) with the magnetic nanoparticles self-assembled in the form of the multi-ring structure. A typical fragment with magnetic chains is shown in panel (b) using optical microscopy and in panel (c) using AFM microscopy. In the fragment shown in the AFM image, the height of the chains is about 60 nm defined by single nanoparticles and small agglomerates. The red arrows indicate the direction of stretching of the LDPE film (⁠ $x$-direction). It should be added that the optical microscopy image is possible due to the phenomenon of light scattering on the nanoparticle chains. Panel (e) of Fig. 2 shows a schematic of the preparation of a cube-shaped sample for FMR measurements. It contains 300 layers of LDPE with ring structures as in panel (a) of Fig. 2. Usually, colloid droplets leave an additional stain of nanoparticles in the center of the dried droplet but the process of deposition of nanoparticles in the center of the dried drop can be controlled by removing the excess amount of colloid just before it dries. Under appropriate conditions of temperature, humidity, and sonication time of the colloid with nanoparticles before taking a droplet to be laid on LDPE, it is possible to obtain rings without nanoparticles in the center.¹⁶ In the case of panel (e), still small amounts of nanoparticles remained unremoved.

Figure 3(a) shows the results of the FMR experiment for the cube-shaped sample in Fig. 2(e) composed of 300 LDPE layers with the ring-shaped chains of magnetic nanoparticles. Since the nanoparticle rings at the contact line of the drying droplet were formed diffusely in the presence of magnetic dipole–dipole interaction between the nanoparticles, the directions of the magnetic anisotropy axes of the magnetic nanoparticles can be assumed to correlate and not to differ much from the tangential directions to the nanoparticle rings. The FMR results indicate the existence of two resonance values for the DC field, parallel and perpendicular to the plane containing the rings, respectively, and $( H DC res ) ⊥> ( H DC res ) ∥$⁠.

To see how the direction of the nanoparticle anisotropy axis affects the FMR resonance result, the complementary computer simulations were performed for a system of 24 nanoparticles with the help of the LLG equations in Eq. (9). The nanoparticles are arranged in the shape of a 1/4 ring, as in the inset in Fig. 3(b). In the simulation, the magnetic dipole–dipole interactions between the nanoparticles were taken into account and to mimic the presence of the APTES coating of the magnetic nanoparticles; the distances between the nanoparticle cores represented by spheres were set to 2 nm. The anisotropy axes of the magnetic nanoparticles are tangent to the ring of nanoparticles. Figure 3(b) shows the plot of the derivative of the imaginary part of the calculated dynamic susceptibility $χ$ in Eq. (8). To calculate $χ$⁠, the magnetization values $M → i$ (⁠ $i=1,2,…,24$⁠) for the stable magnetization cycle were obtained from the LLG equations in Eq. (9). The method is described in Sec. III. Two directions of the DC field are considered, when it is an in-plane magnetic field in the $x$-direction, $( H DC ∥ ) x$⁠, and when it is perpendicular to the plane of the ring, (⁠ $H DC ⊥$⁠). The single resonance for the $H DC ⊥$ field is qualitatively consistent with the experimental result in Fig. 3(a). Surprising can be the presence of the more than one resonance value for the $( H DC ∥ ) x$ field, whose effect is not observed in the experiment in Fig. 3(a). The explanation for this difference is that the sample used in the experiment contains a large number of ring-shaped chains and their radii are different. In addition, there are many individual agglomerates of nanoparticles of different sizes in the sample. This causes the FMR signal to give an averaged image. However, in the case of computer simulation for the $( H DC ∥ ) x$ field, as can be seen from the inset to Fig. 3(b), some nanoparticles have anisotropy axes that are approximately parallel to the DC field, a comparable number of nanoparticles have anisotropy axes approximately perpendicular to the DC field and about 1/3 of them have anisotropy axis directions changing from parallel to perpendicular. This is the reason for the appearance of a multimodal feature of the FMR spectrum obtained from computer simulation. This theoretical result can be of great importance for the design of magnetoelectronic systems for directional power loss.

From Eq. (12), it follows that the resonant frequency can also be expected for $H DC=0$⁠. To show this, a computer simulation of the zero-field ferromagnetic resonance for a single $Fe 3 O 4$ magnetic nanoparticle was carried out using the LLG equation [Eq. (9)] for the $H AC$ field being perpendicular to the magnetic anisotropy field $H a$⁠. The obtained dependence of the SAR coefficient on the frequency $f$ has been shown in Fig. 5(a). The resonant frequency has a value of about 2.2 GHz. It can be seen that the value of the SAR coefficient at this frequency is an order of magnitude smaller than the value of the SAR coefficient in Fig. 4 at the resonant value of the DC magnetic field. Figure 5(a) includes also simulation results for small $H DC$ static fields with values of 0.01, 0.025, 0.075, and 0.1 T. They suggest the possibility of the resonant power absorption of the $H A C$ field over a wide frequency range using the $H D C$ field with small values as a control parameter. This provides a potential opportunity for applications in electronics with microwave materials.^25,26

In Fig. 5(b), it has been shown the SAR coefficient dependence at zero DC field for a magnetic nanowire represented by a chain of ten magnetic nanoparticles interacting through magnetic dipole interactions. The surface-to-surface distance between the nanoparticles is 1 nm. A strong dependence of the SAR coefficient on the angle $α$ of the nanowire with respect to the $x$-axis can be seen. The value of SAR decreases to zero as the value of $α$ increases. The largest SAR value is for $α=0$ when the $H AC$ field is perpendicular to the nanowire. The plots for microwave frequencies $f=2.2$ and $f=2.3$ GHz, which are close to the resonant value, suggest that the absorption of the $H AC$ field can take on a step-like character in a certain range of angles for $f$ near the resonant value. Note that if magnetic anisotropy is neglected, the dipole–dipole magnetic interactions themselves also cause the angle dependence of the SAR coefficient with respect to the $H AC$ field but the observed effect is much weaker. In the case of the randomly oriented magnetic anisotropy, axes in the nanowire SAR $(α)≈ const$⁠.

It should be noted that the dependence of power loss on the direction of the incident microwave in thin-film nanowire systems can also have great potential for electromagnetic shielding applications in magnetoelectronics.

The presented results discuss the possibility of using thin magnetic films composed of single-domain magnetic nanoparticles with uniaxial anisotropy for directional absorption of microwave radiation. In this work, thin magnetic films are 2D printed systems with a thickness of several dozen nanometers. The films are placed on an LDPE foil. The presented results are more general and may concern thin anisotropic magnetic layers obtained by other methods and not necessarily using magnetic nanoparticles. In this work, the experimental results with stabilized magnetic nanoparticles have been shown. Still, the presented technique of creating nanowires by self-assembling magnetic nanoparticles into chains also works for bare nanoparticles. The latter would mean the presence of exchange interactions at the surface of the contacting magnetic nanoparticles. The issues of directional absorption of microwave radiation in thin films are currently very relevant for numerous applications for shielding purposes. The work also indicates the potential of magnetic nanoparticle materials in the so-called zero-field ferromagnetic resonance.

M.M., W.W.W, A.D, and M.R.D acknowledge funding by the Minister of Science under the “Regional Excellence Initiative” program, Project No. RID/SP/0050/2024/1.

The authors have no conflicts to disclose.

M. Marć: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Investigation (equal); Writing – review & editing (equal). A. Drzewiński: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Writing – review & editing (equal). W. W. Wolak: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Writing – review & editing (equal). A. Drzewiecki: Conceptualization (supporting); Data curation (equal); Formal analysis (supporting); Writing – review & editing (equal). S. Mudry: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Writing – review & editing (equal). I. Shtablavyi: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Writing – review & editing (equal). M. R. Dudek: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Investigation (equal); Supervision (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal).

The data that support the findings of this study are available from the corresponding author upon reasonable request.