Tailoring Microwave Absorption via Ferromagnetic Resonance and Quarter-Wave Effects in Carbonaceous Ternary FeCoCr Alloy/PVDF Polymer Composites

Abstract

In this study, we investigate the dominant electromagnetic wave absorption mechanism–ferromagnetic resonance (FMR) loss versus quarter-wave cancellation in a novel PVDF-based polymer composite embedded with carbonaceous nanostructures incorporating FeCoCr ternary alloy. The majority of the nanoparticles are embedded at the terminal ends of the carbon nanotubes, while a small fraction exists as isolated core–shell, carbon-coated spherical particles. Overall, the synthesized material predominantly exhibits a nanotubular carbon morphology. High-resolution transmission electron microscopy (HRTEM) confirms that the encapsulated nanoparticles are quasi-spherical in shape, with an average size ranging from approximately 25 to 40 nm. The polymeric composite was synthesized via solution casting, ensuring homogenous dispersion of filler constituent. Electromagnetic interference (EMI) shielding performance and reflection loss characteristics were evaluated in the X-band frequency range. Experimental results reveal a significant reflection loss exceeding −20 dB at a matching thickness of 2.5 mm, with peak absorption shifting across frequencies with thickness variation. The comparative analysis, supported by quarter-wave theory and FMR resonance conditions, indicates that the absorption mechanism transitions between magnetic resonance and interference-based cancellation depending on the material configuration and thickness. This work provides experimental validation of loss mechanism dominance in magnetic alloy/polymer composites and proposes design principles for tailoring broadband microwave absorbers.

1. Introduction

The rapid expansion of electronic devices, radio communication systems, and radar systems has significantly increased the level of electromagnetic radiation in the environment. This rise in electromagnetic (EM) pollution or electromagnetic interference (EMI) poses a serious threat to the performance, reliability, and safety of electronic equipment [1,2,3,4,5]. Furthermore, prolonged exposure to electromagnetic radiation is found to be harmful to human health [6], which has attracted public and governmental concern. Thus, there is an urgent need for high-performance electromagnetic interference shield materials to absorb unwanted radiation through absorption or reflection mechanisms.

EMI shielding has historically been achieved with the assistance of bulk metallic enclosures, which rely on the principle of reflecting incoming EM waves according to their high electrical conductance. However, metals are usually heavy and corrosion-prone and have poor absorption characteristics. These limitations have prompted intense research into light, flexible, and corrosion-resistant materials that not only shield but also absorb EM radiation over broad frequency bands. Such materials for these applications that display high magnetic and dielectric loss are of considerable interest because they enable absorption-dominant mechanisms for secondary reflection suppression and increased shielding effectiveness through conversion of the incident electromagnetic energy into heat energy.

Ferromagnetic substances, especially transition metal-based ones, such as Fe, Co, and Ni, are known to possess intrinsic magnetic loss properties due to natural ferromagnetic resonance, eddy current loss, and domain wall resonance [7]. These substances possess high permeability and can be designed to possess attractive impedance matching with free space, which is one of the most important requirements for maximizing microwave absorption. However, the ferromagnetic materials themselves are often afflicted with high density and restricted bandwidths of absorption. In addition, the magnetic permeability of pure metals also declines significantly at high frequencies (above a few GHz), reducing their effectiveness in the X-band (8–12 GHz) and K_u-band (12–18 GHz) frequency bands of concern for radar and communications technologies.

In attempting to overcome these restrictions, the integration of ferromagnetic materials with carbon matrices has been a successful strategy [4,8,9,10,11,12,13,14]. Carbon-based materials, such as carbon nanotubes (CNTs), graphene, reduced graphene oxide (rGO), carbon nanofibers (CNFs), and porous carbon architectures, exhibit excellent electrical conductivity, large surface area, tunable dielectric response, and structural flexibility. All of these attributes render them best suited for composing hybrid materials with synergistic electromagnetic loss properties. Specifically, the conjunction of ferromagnetic nanoparticles in carbon networks is capable of imparting high magnetic and dielectric loss tangents to materials [15], improved impedance matching, multiple scattering, and interface polarization effects, resulting in enhanced microwave attenuation.

Nitrogen-doped carbon structures are one among many types of carbon that have recently gained specific interest due to their modified electronic structures, which contribute to higher charge carrier density and polarization sites [16]. This improves interfacial polarization and dipole relaxation, further improving dielectric loss. Defect and heteroatom incorporation into carbon frameworks also improves impedance matching by converting incident electromagnetic waves to heat energy. Moreover, hierarchical architectures of carbon frameworks as hollow spheres, porous foams, or tubular networks can be tailored to trap and dissipate electromagnetic waves through internal reflection and resonance effects.

One of the main challenges to the development of high-performance EMI shielding and absorbing materials is to achieve balanced magnetic and dielectric loss interaction, low weight, and broadband characteristics. Synergism between the ferromagnetic components and conductive networks of carbon is one of the promising means of development. In such ferromagnetic/carbon hybrid composites, magnetic losses may be induced by natural and exchange resonance effects caused by ferromagnetic nanoparticles, while carbon matrices create conducting pathways and enable dielectric losses through polarization and conduction mechanisms. Additionally, interfacial areas between magnetic and carbon phases act as active areas for interfacial polarization and scattering that improve the attenuation capacity of the composite. Control of particle size, morphology, distribution, and the nature of the carbon matrix is crucial to tailor the electromagnetic features of the composites.

The recent advances in material synthesis have made it possible to engineer a wide range of magnetic–carbon hybrid architectures, including core–shell nanoparticles [17,18,19,20,21], heterostructured nanocomposites [22,23], and self-assembled structures [24,25,26]. Techniques like sol–gel processing, hydrothermal synthesis, chemical vapor deposition (CVD), metal–organic precursor pyrolysis, and templated carbonization have been utilized to fabricate these materials with engineered morphology and composition. For example, pyrolytic synthesis allows for the simultaneous synthesis of magnetic nanoparticles and nitrogen-doped carbon nanotubes in one step to produce materials with hierarchical structures and optimal EM wave dissipation characteristics.

Characterizing the electromagnetic behavior of such materials often calls for the measurement of complex permittivity and permeability, followed by the calculation of reflection loss (RL) and shielding effectiveness (SE). A material is typically a good microwave absorber if it has a reflection loss of less than −10 dB (better than 90% absorption) and excellent if less than −20 dB. Shielding effectiveness does encompass reflection and absorption contributions, and ratings of greater than 30 dB are usually preferred for practical applications. Thickness, filler loading, and microstructure optimization is necessary to attain desired electromagnetic properties over a target frequency range. The perfect absorbers of EMI need to be light in weight, flexible, and effective for wide-spectrum attenuation. Polymer-based nanocomposites are promising candidates due to the ease of production, mechanical tunability, and ability to couple multifunctional nanofillers. In heterogeneous composites, electromagnetic wave attenuation typically arises from multiple mechanisms. Alongside true absorption processes, such as dielectric relaxation, magnetic resonance, and conductive losses, structural inhomogeneities and interfacial mismatches also contribute significantly through scattering effects. Consequently, the overall shielding or attenuation behavior is best described as electromagnetic energy dissipation, encompassing both absorption and scattering contributions.

With the above points in view, the present research aims at the synthesis, characterization, and evaluation of ferromagnetic–carbon-based nanocomposites for electromagnetic interference shielding and microwave absorption. We aim to synthesize nanostructured hybrids with tailored magnetic and dielectric properties using transition metal-based precursors and nitrogen-enriched organic precursors. The novelty of our approach is the incorporation at a strategic level of ferromagnetic alloys (FeCoCr in this case) and N-doped carbon nanostructures for their enhanced attenuation mechanisms, like ferromagnetic resonance, dielectric polarization, and multiscale interfacial scattering. Microwave absorption in such composites typically stems from a combination of magnetic loss, dielectric loss, and internal multiple reflections. Among these, ferromagnetic resonance (FMR) and quarter-wave cancellation are the most significant mechanisms. FMR is resonance absorption by magnetic dipoles in a material when subjected to an alternating magnetic field [27,28], whereas quarter-wave cancellation results from destructive interference when the thickness is a quarter wavelength [29].

The goal of this work is to determine the dominant absorption mechanism in carbonaceous architecture-based ternary FeCoCr alloy and PVDF composites. While the permeability and soft ferromagnetic behavior of FeCoCr are sufficiently large to trigger FMR, the structure of the composite layer also meets quarter-wave requirements. Isolation of these mechanisms provides valuable information to attain optimal material design. Our findings aim to guide the development of future-generation EMI shielding materials that are not only high-performance but also light, thermally stable, and scalable for industrial applications.

2. Experimental Details

To prepare nitrogen-doped carbon nanotubes embedded with FeCoCr alloy nanoparticles, a pyrolysis method was adopted using transition metal acetylacetonate precursors [30,31]. Iron(III) acetylacetonate (92 mg), cobalt(II) acetylacetonate (67 mg), and chromium(III) acetylacetonate (91 mg) were employed in 1:1:1 molar proportion. Besides the organometallic precursors, the source of nitrogen was melamine (100 mg), and the additional source of carbon was toluene (1 mL). These were placed in a quartz tube reactor. In order to maintain an inert atmosphere during synthesis, the nitrogen gas was introduced through a rubber balloon at one end of the quartz tube. The tube was positioned inside a homemade tube furnace such that the precursor mixture was approximately 30 cm from the furnace’s mouth. This offset the perceived temperature experienced by the precursors to be approximately 650 °C, a few degrees lower than the central furnace temperature at 700 °C due to the axial thermal gradient. The furnace’s temperature was increased at a constant rate of 20 °C min⁻¹ to 700 °C and kept constant at that value for 30 min. Once the heating was done, the system cooled naturally to room temperature. The precipitated carbonaceous product formed along the inner quartz tube wall was collected and hand-ground into a very fine powder by using a mortar and a pestle. This resultant sample was designated as FeCoCr@CN.

The morphological and structural characteristics of the as-synthesized sample were studied using various analytical techniques. X-ray diffraction (XRD) patterns were recorded on a diffractometer (PANalytical, Malvern, UK) over a 2θ range of 10–90°, which established the existence of amorphous as well as crystalline carbon and the alloy phase. Scanning electron micrographs (SEM) were obtained using a scanning electron microscope (Ultra55 FE-SEM, Carl Zeiss India Pvt. Ltd, Bengaluru, India). Transmission electron microscopy (TEM) and high-resolution TEM (HRTEM) micrographs were recorded using a 300 kV FETEM instrument (Jeol, Peabody, MA, USA). The Raman spectrum of the powder sample was measured using a LabRam HR instrument (HORIBA, Kyoto, Japan) with a 532 nm laser. Thermogravimetric analysis (TGA) measurement was carried out using a TQ-50 instrument in air atmosphere. A EMX spectrometer (Bruker, Karlsruhe, Germany) recorded the EPR (specifically, FMR in this instance) spectra at a nominal X-band frequency of 9.4 GHz. Magnetic field modulation at 100 kHz was employed alongside phase sensitive detection to capture the field derivative dP/dH of the power absorbed, P, in relation to changes in the magnetic field. A small amount of DPPH (2,2-diphenyl-1-picrylhydrazyl) served as the field marker.

For polymer composite synthesis, the sample (FeCoCr@CN) was dispersed in a polyvinylidene fluoride (PVDF) matrix, similarly to our previous reports [13,14]. PVDF powder was first dissolved in dimethylformamide (DMF) to form a uniform solution. The solution was then mixed with an equal weight of FeCoCr@CN, followed by subsequent ultrasonic treatment to achieve homogeneous mixing. The mixture obtained through blending was then placed in a Teflon mold and subjected to normal agitation during the drying process to achieve a uniformly distributed dispersion of the nanoparticles within the polymer. The composite material, after drying, was then sliced into small pieces and rolled into flexible thin sheets (approximately 1 mm thick) through hot pressing using compression molding. These polymeric flexible composite samples were then evaluated for electromagnetic interference shielding performance at the 8.2 to 12.4 GHz frequency range using a rectangular waveguide with an E5071C network analyzer (ENA series, 300 Hz–20 GHz, Keysight, Santa Rosa, CA, USA). The details of the calculations of S-parameters are provided in the Supporting Information (ESI).

3. Results and Discussion

The detailed analysis of the characterization of this material can be found in our earlier work on nonlinear optical absorption [30]. We briefly describe it here. Detailed morphological analysis through scanning electron microscopy (SEM) revealed tubular structures in knotted form composed of coiled and helically twisted nanotubes (Figure 1a). Transmission electron microscopy (TEM) confirmed the presence of multi-walled nanotubes with a compartmentalized “bamboo-like” structure (Figure 1b). Metal nanoparticles, identified as FeCoCr alloy, were found prominently at the tips of the nanotubes (Figure 1c), suggesting a tip-growth mechanism for nanotube synthesis. The HRTEM image (Figure 1c) clearly shows that the nanoparticles are quasi-spherical, with an average diameter of ~25–40 nm. High-resolution TEM (HRTEM) images identified crystalline alloy nanoparticles within graphitized carbon shells. The calculated d-spacing of the metallic nanoparticles was 0.409 nm. Furthermore, HRTEM images showed a gradient of graphitization; well-ordered graphitic walls at the nanoparticle core and inner walls of the nanotubes were replaced by progressively disordered (amorphous) carbon on exterior surfaces.

A peak characteristic at 2θ ≈ 26° was attributed to the (002) plane of graphitic carbon placed atop a broader signal resulting from amorphous carbon. Peaks of a bcc metallic phase were observed, indicating the formation of a FeCoCr alloy (Figure 1d) [30]. The reflections were somewhat displaced from those for pure Fe or CoFe and suggest lattice distortion with the addition of Cr. An indication of a small peak due to magnetite (Fe₃O₄) was also observed. Raman spectroscopy was used to further investigate the structure of the carbon and defect concentration (Figure 1e).

The spectrum had a number of features in the 1000–4000 cm⁻¹ range, consisting of a sharp peak for sp² carbon (graphitic G-band) and diffuse bands for structural defects (D-bands) [30]. Other low-intensity peaks below 1000 cm⁻¹ were used to mark vibrational modes of oxide phases, possibly consisting of Cr₂O₃, CoO, or Fe₃O₄. The intensity ratio of the G band to the combined D bands (I_G/I_D) was determined to be approximately 0.406, indicating a moderate level of disorder, likely arising from nitrogen’s incorporation into the carbon framework. The TGA plot suggests 73% carbon content in the material. The soft ferromagnetic character of the sample with medium saturation magnetization (M_s ≈ 28 emu/g) and low coercivity (~561 Oe) was evidenced by VSM measurements [32,33,34], which are optimum for FMR-based microwave absorption. The magnetic nature arises primarily due to the FeCoCr alloy, as carbonaceous phases are non-magnetic.

4. EMI Shielding Properties of the Composite

The polymer composites of PVDF:FeCoCr@CN were used to measure the EMI shielding property in the X-band frequency region using an ENA. The shielding effectiveness (SE) values of the composite samples were calculated from the S-parameters extracted from the ENA. The variation of SE versus frequency is shown in Figure 2a. The contributions of reflection and absorption in total shielding (SE_T) are depicted as SE_R and SE_A, respectively, in Figure 2. The prepared composite shows a maximum in the SE_A plot at 9.5 GHz frequency. This peak in absorption leads to an increase in the total shielding (SE_T) values at 9.5 GHz. The reflection (R), absorption (A), and transmission (T) coefficients in % were also calculated from the S-parameters and are shown in Figure 2b. We observed that at 9.5 GHz, there is maximum in the absorption coefficient, and therefore the transmission coefficient also decreased. To investigate the origin of the sharp peak at 9.5 GHz, we have calculated the relative permittivity and permeability values of the composite in the X-band using the Nicholson–Ross–Weir algorithm [35]. The complex permittivity (ε* = ε′ − jε″) values in the 8–12 GHz region are shown in Figure 2c. The real part of permittivity (ε′), which describes the charge storage capacity of the material, is almost constant in the entire frequency range. In contrast, the dielectric loss (ε″) shows a distinct variation. In the ε′′ plot (Figure 2c), a pronounced minimum is observed near 9.6 GHz, which lies close to the ferromagnetic resonance (FMR) frequency. This feature likely arises from the interplay between dielectric relaxation and magnetic loss. While a small contribution from dielectric polarization cannot be ruled out, the dominant absorption mechanism in this region is magnetic in nature, as confirmed by the strong resonance at ~9.5 GHz observed in the EPR spectrum (discussed later). This correlation establishes FMR as the primary contributor to energy dissipation in the composite at these frequencies. There are two additional minor peaks in the ε″ plot located at around 10.8 GHz and around the 10.1 GHz region.

The complex permeability plot is shown in Figure 2d. From the complex permeability plots, a resonance-type phenomenon is observed, which is located at 9.2 GHz frequency. The sharp peak at 9.5 GHz in the μ″ value and the fluctuation of the μ′ value at 9.5 GHz is attributed to the presence of ferromagnetic resonance phenomena. This FMR could be due to presence of FeCoCr particles, which are ferromagnetic in nature, as shown in the VSM plot. The magnetic loss due to other factors, such as eddy current, hysteresis loss, and domain wall motion, are negligible and can be neglected. To investigate the effect of FMR on the reflection loss (RL) on the composite, the RL is calculated using the following equation [12,36,37,38]:

$$RL(dB)=20\mathit{log}{\displaystyle \frac{|Z_{in}-Z_o|}{|Z_{in}+Z_0|}}$$

(1)

where, Z₀ is the impedance of free space (377 Ω) and Z_in is the input characteristic impedance of the absorbing layer, which can be calculated using ε_r and µ_r at a given frequency using the following equation:

$$Z_{in}=\sqrt{{\displaystyle \frac{{\mu }_r}{{\epsilon }_r}}}\mathit{tanh}\left(j{\displaystyle \frac{2\pi ft}{c}}\sqrt{{\epsilon }_r{\mu }_r}\right)$$

(2)

As shown in Equation (1), RL is a function of the complex permittivity, permeability, incident frequency, and thickness of the sample. Therefore, it is very difficult to understand the individual effect of each of these on the RL. On several occasions, the RL peak in misunderstood or misrepresented as an attribution of the FMR, as it could also be due to the thickness or from the matching frequency. To understand it better, we have evaluated the RL values of our specimen at five different thickness values of 1, 2, 2.5, 3, 4, and 5 mm. The corresponding RL plots are shown in Figure 2e. Here, we can observe that the RL peak at 9.5 GHz frequency does not change with the change in thickness of the sample. Because we know that FMR is an intensive property, i.e., it does not depend on the amount of the sample, by changing the thickness of the sample, the FMR peak should not change. In the RL plot, the peak at 9.6 GHz is independent of the thickness, which validates the claim that the RL is indeed mainly due to the FMR loss. Furthermore, to disentangle the effects of thickness and FMR loss, we have simulated (using MATLAB software 2018a version) the RL peak values from 8 to 12 GHz and from 0.01 to 50 mm thickness. For the simulation, in the first case, we have taken the values of complex permeability and permeability values similar to the experimental values obtained from the ENA but without the FMR peak, while for the second case we have included the FMR peak values (Figure 3 and Figure 4). The simulated results clearly match our hypothesis that FMR is the dominant contributor in RL in our composite sample.

Compared to relatively newer FeCoC and CoNi/CNT architectures with high resonant attenuation and wide effective absorption bandwidths due to complementary magnetic–dielectric losses and hierarchical carbon structures [39], the new FeCoCr@CN/PVDF composites here show comparable shielding/absorption performance at tolerable loadings with retained mechanical flexibility and easy processing, setting them squarely among state-of-the-art systems. Characteristically representative FeCoC hybrids, such as rGO/MXene/FeCoC aerogels derived from MOFs [40], and CoNi/CNT heterostructures, such as CoNi nanochains grafted with N-doped CNTs [41] and yolk–shell CoNi@NC–CoNi@CNT hybrids [42], demonstrate how magnetic anisotropy, conductive networks, and interfacial polarization merge to create reflection loss minima of great depth and wide bandwidths, trends emulated in our FeCoCr@CN design. In the future, our PVDF matrix’s light weight and solution processability suggest compatibility with scalable, large-area roll-to-roll coating for conformal, large-area “EMI-skin” layers in future electronics; recent advances in flexible PVDF-based EMI films and scalable CoNi/carbon composites enable this manufacturing path [43,44,45]. Conformable absorbers of this type are especially of interest for 5G/6G devices, where thin, impedance-matched layers can cut crosstalk and stray radiation across X–K_u bands and higher [46]. Key next step challenges are lowering material and processing costs (e.g., MOF/precursor use, filler fraction optimization), sustaining environmental stability with humidity/thermal/mechanical cycling, and demonstrating long-term performance in real device stacks, issues emphasized across recent absorber studies and reviews [47].

5. EPR Studies for Validation of Ferromagnetic Resonance (FMR)

Electron Paramagnetic Resonance (EPR) plays a significant role in understanding the magnetic properties of ferrites and magnetic samples, particularly at high frequencies. The significance of EPR studies lies in the fact that the resonance in magnetic samples stems from the interaction between electron spins and electromagnetic radiation (in the microwave region), positioning it as an essential method for examining spin dynamics and magnetic interactions in ferro/ferrimagnetic materials. EPR spectra of the FeCoCr@CN sample are shown in Figure 5.

From the EPR spectra, several important parameters were determined, including the resonance line width (ΔH), the resonance field position corresponding to zero signal (H₀), and the effective g-factor (g). These parameters provide insights into the magnetic environment and spin interactions in the samples [48,49,50,51,52,53].

The g-factor (g) was calculated using the following equation:

$$g={\displaystyle \frac{hf}{\beta H}}$$

(3)

where h is Planck’s constant, f is the frequency of the microwave (9.4 GHz in this case), H is the magnetic field strength, and β is the Bohr magneton. The calculated g-factor is 2.00474.

A slight fluctuation in μ′ around 9.5 GHz can be attributed to the influence of magnetic anisotropy, which modifies the resonance condition by shifting the effective internal fields. This effect is commonly reflected in the EPR linewidth (ΔH), where broader lines correspond to stronger anisotropy and inhomogeneous local fields. Thus, the anomaly in μ′ may arise from small variations in magnetic anisotropy, leading to resonance broadening near this frequency.

The g-factor of 2.00474 in the FeCoCr alloy offers key insights into its magnetic behavior and the electronic structure of its constituent ions. Iron (Fe) and cobalt (Co), both ferromagnetic with partially filled 3d shells, act as the primary sources of magnetic moments, while chromium (Cr)—antiferromagnetic in its pure form—modifies the alloy’s magnetic properties. The slightly elevated g-value indicates that unpaired 3d electrons dominate the magnetism, with a small but non-negligible orbital contribution arising from spin–orbit coupling, particularly from Co and Cr. The moments are relatively localized and well-aligned, consistent with ferromagnetism, though Cr introduces disorder, diluting magnetic interactions and subtly shifting the g-factor compared to a pure FeCo alloy due to local crystal field or exchange variations. Additionally, paramagnetic centers may exist, likely due to Cr sites or defects with incomplete magnetic ordering. Despite these influences, the system remains predominantly spin-driven, with the local electronic environment playing a secondary role.

The g-factor of 2.00474 in the FeCoCr alloy highlights significant features of its magnetic properties and electronic configuration. The slightly elevated value, in comparison to the free electron g-factor, suggests that unpaired 3d electrons, mainly from iron (Fe) and cobalt (Co), are the key contributors to the alloy’s magnetism. Both Fe and Co, as ferromagnetic elements with partially filled 3d shells, produce significant localized magnetic moments and result in robust ferromagnetic alignment within the alloy. Chromium (Cr), while exhibiting antiferromagnetic properties in its pure form, contributes to a certain degree of disorder and dilution of magnetic interactions within the FeCoCr system. Chromium can disrupt local ferromagnetic order through clustering and by serving as a site for paramagnetic or frustrated spins, as demonstrated by research on Fe-Cr alloys, which indicate a decrease in the Curie temperature and slight alterations in magnetic properties due to the presence of chromium [54,55,56,57].

The g-factor value slightly above 2, further implies a modest but significant orbital contribution to the total magnetic moment, attributable to spin–orbit coupling, most notably from Co and, to a lesser extent, Cr. This orbital contribution modifies the magnetic response beyond a pure spin-only system and is supported by recent analyses detailing how spin–orbit interaction and local crystal fields shift the g-factor in Fe-based alloys [57,58]. The majority of the system’s magnetism remains spin-driven, ensuring overall ferromagnetic behavior, but the nuanced effects of local environments and disorder, especially from Cr or alloying-related defects, yield paramagnetic sites or incomplete ordering at the microscopic scale. Thus, the observed magnetic characteristics and g-factor reflect a complex interplay of spin, orbital effects, and crystal structure, with Fe and Co providing robust ferromagnetism and Cr introducing a controlled degree of magnetic dilution and orbital complexity [59].

The 0.1 GHz offset observed between the EPR frequency (9.4 GHz) and the FMR peak (9.5 GHz) falls well within typical experimental error margins encountered in high-frequency magnetic resonance measurements. Such small frequency differences can arise due to instrumental calibration uncertainties, sample inhomogeneity, or slight differences in resonance conditions associated with magnetic anisotropy and demagnetizing effects. Therefore, this offset does not indicate a significant physical discrepancy and is considered experimentally insignificant.

6. Conclusions

Here, we present a detailed investigation of electromagnetic wave attenuation phenomena in polyvinylidene fluoride (PVDF)-based polymer composites filled with carbonaceous nanostructures and ternary FeCoCr alloy nanoparticles. We provide experimental verification to differentiate ferromagnetic resonance (FMR) loss from quarter-wave interference in microwave absorption. The polymeric flexible composite sample, synthesized through solution casting, evidenced high reflection loss (>−20 dB) at 9.5 GHz with minimum thickness (2.5 mm). We distinguished between FMR and quarter-wave cancellation clearly and systematically through the integration of experimental and simulation techniques. A flat RL peak at 9.5 GHz across broad thickness ranges substantiates the dominance of FMR. Our results offer valuable information on the design of high-performance electromagnetic interference shielding materials based on tunable ferromagnetic and dielectric properties, high-performance, broadband, light-weight EMI absorbers for use in cutting-edge electronics and military systems.