Frequency-thickness normalized electromagnetic parameter design for broadband electromagnetic wave absorption in FeCoCu alloy foams

Synthesis of FeCo foam and FeCoCu foam

FeCo alloy foam was fabricated through a sol-gel auto-combustion process followed by hydrogen reduction. Briefly, 3.030 g (7.5 mmol) of Fe(NO₃)₃·9H₂O, 0.728 g (2.5 mmol) of Co(NO₃)₂·6H₂O, and 1.239 g (16.5 mmol) of glycine were dissolved in 100 mL of deionized water under magnetic stirring at 600 r min^-1 and room temperature for 30 min to obtain a homogeneous precursor solution. The solution was subsequently heated to 220 °C to remove excess water, during which self-sustained combustion was triggered. The resulting oxide precursor was then reduced at 550 °C for 6 h in a 5% H₂/95% Ar atmosphere with a heating rate of 3 °C min^-1, yielding the FeCo alloy foam.

FeCoCu alloy foams with varying Cu contents were prepared using an identical synthetic protocol. Specifically, 5 at.%, 10 at.%, and 20 at.% of Fe were nominally substituted by Cu in the precursor formulation. The obtained samples were denoted as Fe₆₅Co₃₀Cu₅ foam, Fe₆₀Co₃₀Cu₁₀ foam, and Fe₅₀Co₃₀Cu₂₀ foam, respectively, according to the nominal atomic feeding ratios of Fe, Co, and Cu in the precursor solutions. It should be noted that these sample names represent nominal atomic ratios rather than the directly measured elemental compositions.

Characterization

Scanning electron microscopy (SEM) images were collected using a Phenom scanning electron microscope operated at an accelerating voltage of 15 kV, and the samples were sputter-coated with gold before observation. TEM observations and elemental mapping were performed on a Tecnai F20 transmission electron microscope operated at 300 kV. The transmission electron microscopy (TEM) samples were prepared by dispersing the powders in anhydrous ethanol under ultrasonication for 30 min. Powder X-ray diffraction (XRD) patterns were recorded on a Rigaku D/MAXRC diffractometer equipped with Cu Kα radiation (λ = 1.5406 Å), operated at 40.0 kV and 40.0 mA, with a 2θ scanning range of 10-90°. Magnetic hysteresis loops were measured using a vibrating sample magnetometer (VSM, LakeShore, Model 7,404 series) at room temperature. Inductively coupled plasma atomic emission spectroscopy (ICP-AES) was carried out using an Optima 8300 instrument (PerkinElmer, USA), and the samples were dissolved using aqua regia before measurement. Mercury intrusion porosimetry was performed using a Micromeritics AutoPore IV 9520 instrument.

For EM parameter measurements, the as-prepared FeCoCu foam powders were mixed with paraffin at a mass ratio of 60:40. Before mixing, paraffin was melted at 60 °C, and the powders were thoroughly mixed by manual grinding for 15 min to obtain a homogeneous composite. The mixture was then loaded into a standard coaxial mold and pressed into toroidal specimens under a pressure of 1.0 MPa for 1.0 min. The obtained specimens had an outer diameter of 7.00 mm, and an inner diameter of 3.04 mm. The sample surfaces were kept flat and compact to ensure good contact with the coaxial fixture. The complex permittivity and permeability were measured using an Agilent N5234A vector network analyzer in the frequency range of 2.00-18.00 GHz with a frequency step of 0.08 GHz. Samples from different preparation batches were measured three times to evaluate repeatability. Owing to the sensitivity of foam-like alloys to specimen preparation, slight variations in the EM parameters and calculated EM absorption performance can be observed, but consistent EM wave absorption behavior was obtained across different batches.

Theoretical calculation and CST simulation

To establish the quantitative relationship between EM parameters, thickness, and frequency for optimal absorption, the reflection loss (RL) was calculated based on classical transmission line theory. The RL values are expressed as^[30-34]:

where Z_in is the normalized input impedance, c is the velocity of EM wave in vacuum, and d is the coating thickness of EWAMs. The frequency was varied from 6.00 to 18.00 GHz (step: 0.01 GHz), and the thickness ranged from 1.00 to 3.00 mm (step: 0.25 mm).

For each fixed combination of f and d, μ_r was set to 1.00-0.00j to isolate the dielectric contribution and establish a reference permittivity-frequency-thickness relationship for RL minimization. This assumption is mainly applicable to nonmagnetic or weakly magnetic EWAMs and does not exclude the contribution of magnetic loss; for magnetic EWAMs, permeability provides additional degrees of freedom for impedance regulation and bandwidth broadening, as discussed below. The real (ε_r') and imaginary parts (ε_r'') of permittivity were scanned over a wide range of 0-200 with a step size of 0.10, and the corresponding values yielding the minimum RL were extracted. This range was selected to cover a broad dielectric parameter space from weakly lossy dielectric materials to highly conductive or strongly dielectric-loss-dominated EWAMs, and it is sufficiently wider than the typical permittivity range of most practical EWAMs in the GHz region. The step size of 0.10 provides a balance between numerical resolution and computational efficiency, because the calculated RL varies continuously with ε_r′ and ε_r″. During the extraction process, the optimal permittivity values were located inside the selected scanning domain rather than at the upper boundary, indicating that the range of 0-200 was adequate for identifying the RL-minimized parameters.

To evaluate the intrinsic bandwidth limitation of pure conduction loss, the free electron model was adopted. In this case, ε_r' was treated as a frequency-independent constant (ε_r' = k), which represents the real permittivity in the conduction-loss model, while ε_r'' follows^[35,36]:

where σ is the electrical conductivity and ε₀ is the vacuum permittivity. The parameter k was varied from 3.00 to 18.00 (step: 3.00), and σ ranged from 0.00 to 6.00 S m^-1 (step: 0.01 S m^-1).

For polarization-dominated loss, the Debye relaxation model was employed to determine the maximum achievable bandwidth under a single relaxation process^[37,38]:

where σ, ε_∞, ε_s and τ refer to the electrical conductivity, the dielectric permittivity at high-frequency limit, the static permittivity and the relaxation time, respectively. The parameter ε_∞ was varied from 0.00 to 10.00 (step: 0.10), ε_s-ε_∞ ranged from 0.00 to 20.00 (step: 0.10), and 1/τ was scanned within 5.0 × 10¹⁰ s^-1 -20.0 × 10¹⁰ s^-1 (step: 1.0 × 10⁹ s^-1).

All EM simulations were conducted using CST Microwave Studio 2019. For radar cross-section (RCS) simulations, a square perfect electric conductor (PEC) plate (thickness: 1.0 mm, side length: 180 mm) was coated with a 1.75 mm EM absorbing layer (Fe₇₀Co₃₀, Fe₆₅Co₃₀Cu₅, Fe₆₀Co₃₀Cu₁₀, and Fe₅₀Co₃₀Cu₂₀ foam, respectively). The simulation frequency ranged from 2.00 to 18.00 GHz (step: 0.08 GHz). Open boundary conditions were applied in all directions, and a plane wave was incident along the -z direction. For angular-dependent RCS evaluation, the frequency was fixed at 9.20 GHz, and the incident angle was scanned from -90° to 90° (step: 1°), with other parameters unchanged. The EM absorption performance of the multilayer periodic structure was simulated under similar conditions, except that a waveguide port excitation (Port 1 along -z) was used. The structure consisted of a PEC backing layer (1.0 mm) and four EM absorbing layers (each 1.5 mm thick), with lateral dimensions of 6.0, 7.0, 8.0, and 10.0 mm from top to bottom. The corresponding EWAMs were Fe₇₀Co₃₀, Fe₆₅Co₃₀Cu₅, Fe₆₀Co₃₀Cu₁₀, and Fe₅₀Co₃₀Cu₂₀ foam, respectively. Due to the presence of the PEC backing, all incident EM waves are reflected, and thus the EM absorption performance can be directly evaluated from the reflection coefficient^[2]:

Theoretical construction optimized EM parameters

Achieving broadband EM wave absorption with a single-layer homogeneous EWAM backed by a metal plate with a relatively small coating thickness (≤ 2.0 mm) remains a critical challenge. Conventional strategies primarily rely on empirical optimization of composition and microstructure, lacking direct theoretical guidance. In practical design, the EM wave absorption performance is typically evaluated by applying transmission line theory to the measured complex permittivity and permeability, which inherently leads to an empirical correlation between frequency-dependent EM parameters and EABs. However, if the functional relationship between the optimal EM parameters and the thickness-frequency combination can be established, EWAMs design can be rationally guided by tuning the EM parameters toward this target relationship, which is of great significance for the development of broadband EWAMs. The RL values of EWAMs are governed by the ε_r and μ_r [Equations (1) and (2)]. For simplification, the magnetic contribution was neglected (μ_r = 1.00-0.00j), allowing the determination of the required ε_r' and ε_r'' that minimize RL values at given coating thicknesses and frequencies. As shown in Figure 1A and B, the extracted permittivity distributions lead to RL values below -40.00 dB across multiple thicknesses (1.50, 1.75, and 2.00 mm), indicating nearly ideal EM absorption performance [Figure 1C]. To gain further insight into the governing mechanism, two characteristic parameters were introduced, defined as:

Figure 1. Theoretical determination of optimal permittivity for broadband EM wave absorption based on transmission line theory. (A) ε_r', and (B) ε_r'' as a function of frequency with different coating thicknesses (1.50, 1.75, and 2.00 mm); (C) Corresponding RL values calculated using the extracted permittivity. Frequency dependence of the parameter (D) C₁, and (E) C₂; (F) Simulated RL values using the constructed optimal permittivity; (G) Electrical thickness calculated from the constructed permittivity defined by Equations (9) and (10). (H) Electrical thickness, and (I) normalized dielectric loss factor recalculated from the RL-minimized permittivity extracted at each frequency-thickness point. EM: Electromagnetic; RL: reflection loss.

Remarkably, both C₁ and C₂ remain nearly constant with respect to frequency [Figure 1D and E], with values of approximately 76 and 95, respectively. This observation suggests that the optimal permittivity follows the relationships:

Under this condition, excellent EM absorption performance is expected over a wide frequency range.

To validate this hypothesis, the permittivity was constrained to satisfy the above relations at 1.50, 1.75, and 2.00 mm, with (μ_r' = 1.00) and (μ_r'' = 0.00). The calculated RL values remain below -20.00 dB across the entire frequency range, confirming the robustness of the proposed relationship [Figure 1F]. Furthermore, the universality of this model was examined by extending the thickness to 1.00-3.00 mm (step: 0.25 mm). The corresponding permittivity distributions [Supplementary Figure 1A and B] and RL curves [Supplementary Figure 1C] demonstrate consistently strong absorption, indicating that the proposed functional form provides a reliable guideline for broadband absorber design.

To further clarify the physical meaning of C₁ and C₂, the effective electrical thickness and normalized dielectric loss factor were analyzed. The constants C₁ and C₂ are not merely numerical fitting parameters, but reflect the phase-matching and attenuation requirements of the EWAM. For the optimized relationship ε_r′ ≈ (C₁/fd)², C₁ directly determines the electrical thickness. When f is expressed in GHz and d in mm, the value of c/4 is approximately 75, which is very close to the extracted C₁ value of approximately 76. This indicates that the ε_r′ scaling governed by C₁ maintains the EWAM close to the quarter-wavelength phase-cancellation condition over different frequency-thickness combinations. In comparison, C₂ controls the ε_r" required for energy dissipation. The ratio C₂/C₁² is approximately 0.0164, corresponding to the normalized tanδ_e, and gives tanδ_e ≈ (C₂/C₁²) fd. Therefore, the tanδ_e increases with fd, providing stronger EM attenuation at higher frequencies or larger thicknesses to suppress residual reflection. From this viewpoint, C₁ mainly reflects the phase/electrical-thickness requirement, whereas C₂ reflects the normalized attenuation requirement. Their combination enables simultaneous regulation of quarter-wavelength cancellation, impedance matching, and internal EM attenuation. The effective electrical thickness was analyzed using^[39]:

Figure 1G shows the electrical thickness calculated from the constructed permittivity defined by Equations (9) and (10), whereas Figure 1H shows the electrical thickness recalculated from the RL-minimized permittivity extracted at each frequency-thickness point. The consistency between these two results confirms that the constructed parameter relationship captures the near-quarter-wavelength condition required for optimal absorption. As shown in Figure 1G, the electrical thickness remains nearly constant and is centered around $$ \frac{\pi}{2} \\ $$, corresponding to approximately a quarter-wavelength condition. This indicates that the quarter-wavelength destructive interference condition is approximately satisfied over a wide range of frequencies and thicknesses, enabling quasi-scale-invariant impedance matching^[40]. In addition, the dielectric loss tangent (tanδ_e) follows^[4]:

which increases linearly with the product of f and d. This implies that at higher frequencies or larger thicknesses, the dielectric loss is automatically enhanced, ensuring sufficient attenuation to suppress multiple reflections from the metallic backing. Essentially, the proposed functional form realizes a normalization of frequency and thickness, allowing both impedance matching and attenuation to approach optimal conditions simultaneously over a broad parameter space. Further calculations reveal that the electrical thickness corresponding to optimal absorption remains consistently close to $$ \frac{\pi}{2} \\ $$ across different thicknesses [Figure 1H], while the normalized loss parameter $$ \tan \delta_{\mathrm{e}} /(f d) \\ $$ fluctuates narrowly around 0.0164 [Figure 1I]. These results collectively validate the physical rationality of the proposed parameter construction.

Bandwidth limitation of single loss mechanism

Although the optimal permittivity required for broadband absorption has been theoretically established, it remains highly challenging to realize such a functional relationship through practical material design. This difficulty arises from the fact that the frequency-dependent permittivity is fundamentally governed by intrinsic loss mechanisms, which can generally be classified into conduction loss and polarization loss^[41,42]. Importantly, the frequency dependence associated with these mechanisms is inherently fixed and deviates from the idealized permittivity-frequency relationship derived above. For conduction-dominated EWAMs, the dielectric response originates from leakage currents and can be equivalently described by an ideal capacitor in parallel with a resistor. In this case, the real part of permittivity remains constant, determined by the charge storage capability, while the imaginary part follows an inverse frequency dependence governed by the electrical conductivity [Equation (3)]. Here, k denotes the frequency-independent ε_r' in the pure conduction-loss model, and is used together with electrical conductivity to evaluate the bandwidth limitation of conduction-dominated EWAMs. The EABs as a function of permittivity and conductivity with coating thicknesses of 1.50, 1.75, and 2.00 mm are presented in Figure 2A-C. It is evident that, regardless of increasing the permittivity or conductivity, the effective bandwidth exhibits a clear upper limit. Although increasing coating thickness can extend this limit, the maximum EAB remains below 5.0 GHz with a coating thickness of 2.0 mm. This indicates that conduction loss alone is insufficient to achieve broadband absorption under thin-layer conditions.

Figure 2. EAB maps of conduction-loss-dominated EWAMs as functions of electrical conductivity σ and the constant real permittivity k (ε_r' = k) with coating thicknesses of (A) 1.50, (B) 1.75, and (C) 2.00 mm, respectively. Three-dimensional maps of EABs as a function of ε_∞, ε_s-ε_∞, and τ for polarization-loss-dominated materials with coating thicknesses of (D) 1.50, (E) 1.75, and (F) 2.00 mm, respectively. Corresponding EAB distributions after introducing magnetic loss (μ_r' =1.2 and μ_r" = 0.2) with coating thicknesses of (G) 1.50, (H) 1.75, and (I) 2.00 mm, respectively. EAB: Effective absorption bandwidth; EWAM: electromagnetic wave absorbing material.

When the dielectric response is dominated by polarization loss, the permittivity can be described by the Debye relaxation model [Equations (4) and (5)], where ε_∞, ε_s-ε_∞, and τ represent the high-frequency permittivity limit, relaxation strength, and relaxation time, respectively. The corresponding EABs under different parameter combinations are shown in Figure 2D-F. For thicknesses of 1.50, 1.75, and 2.00 mm, the maximum EABs are 5.25, 5.97, and 6.60 GHz, respectively, achieved at (ε_∞, ε_s-ε_∞, τ) = (5.8, 11.2, 10.8 ps), (4.2, 10.3 ,12.2 ps), and (3.1, 9.5, 12.8 ps). These results clearly demonstrate that polarization loss, owing to its additional tunable parameters, provides a higher upper limit of EABs compared to pure conduction loss.

The introduction of magnetic components can further improve impedance matching characteristics and simultaneously introduce additional magnetic loss mechanisms, such as eddy current loss and ferromagnetic resonance to further broaden the upper limit of EABs^[42]. Therefore, EWAMs with combined dielectric and magnetic losses are expected to exhibit superior broadband absorption performance compared to purely dielectric systems^[43]. To verify this hypothesis, the relative permeability was artificially set to μ_r' = 1.2 and μ_r" = 0.2, and the above calculations were repeated. As shown in Figure 2G-I, the maximum EAB is further enhanced to 6.20, 7.08, and 7.88 GHz with coating thicknesses of 1.50, 1.75, and 2.00 mm, respectively. The corresponding optimal parameters are (5.1, 9.3, 10.1 ps), (4.0, 8.8, 11.9 ps), and (3.3, 8.6 ,13.9 ps).

These results collectively indicate that neither conduction loss nor polarization loss alone can fully satisfy the requirements for thin-layer broadband absorption. Instead, the synergistic combination of conduction loss, polarization loss, and magnetic loss is essential to approach the theoretically optimal absorption condition. This also highlights the necessity of designing EWAMs with multi-mechanism loss coupling to bridge the gap between ideal permittivity construction and practical realization.

It should be emphasized that the pure conduction-loss and polarization-dominated models discussed above represent idealized limiting cases rather than completely independent descriptions of practical EWAMs. In real EWAMs, these dielectric loss mechanisms usually coexist and are strongly coupled with composition, phase structure, defects, interfaces, and conductive networks. Although the ideal permittivity-frequency-thickness relationship defines a clear target for broadband absorption, it represents a theoretical constraint rather than a directly attainable material property. Achieving such behavior in practice requires precise regulation of composition, phase structure, and microstructure^[44]. As indicated by the preceding analysis, broadband absorption under thin-layer conditions relies on tailoring the dielectric response toward the optimal parameter distribution, while introducing appropriate magnetic contributions to further expand the effective bandwidth^[45]. Based on this rationale, FeCoCu foams were selected as a model system to validate the proposed design strategy. FeCo-based alloys inherently exhibit strong dielectric and magnetic losses, providing a robust foundation for EM attenuation^[46]. The introduction of Cu is expected to regulate conduction loss by modifying charge transport and conductive pathways, while phase separation during alloy formation generates abundant heterogeneous interfaces that promote interfacial polarization and dipolar relaxation^[47]. Consequently, the FeCoCu system offers an ideal platform for the synergistic coupling of conduction, polarization, and magnetic losses.

Synthesis and characterization of FeCoCu alloy foam

FeCoCu alloy foams were synthesized via a nitrate-glycine-assisted solution combustion method followed by reduction [Figure 3A]. This process enables homogeneous precursor mixing and rapid gas evolution, resulting in a three-dimensional interconnected porous framework. The oxide precursors exhibit typical foam-like architectures with highly open and interconnected pores [Figure 3B and Supplementary Figure 2A-C], arising from intense gas release during combustion. After reduction, the overall foam morphology is preserved, while the skeleton evolves into a rough, nanoparticle-assembled structure [Figure 3C and Supplementary Figure 2D-F]. This hierarchical architecture, combining microscale channels with nanoscale building units, enhances multiple scattering and provides abundant interfacial sites for polarization^[48]. TEM and elemental mapping [Figure 3D] confirm the uniform distribution of Fe, Co, and Cu within interconnected nanoparticles, while local contrast variations suggest nanoscale compositional fluctuations. Such heterogeneity likely leads to FeCo-rich and Cu-rich domains, forming interfacial regions that facilitate Maxwell-Wagner polarization.

Figure 3. Synthesis process, structural evolution, and physicochemical characterization of FeCoCu alloy foams with different Cu contents. (A) Schematic illustration of the synthesis process; SEM images of (B) the Fe₆₅Co₃₀Cu₅ oxide precursor; and (C) corresponding reduced Fe₆₅Co₃₀Cu₅ alloy; (D) TEM image Fe₆₅Co₃₀Cu₅ alloy and corresponding elemental mappings of Fe, Co, and Cu; (E) XRD patterns of FeCoCu alloys with varying Cu contents; and (F) corresponding enlarged XRD patterns in the 40-50° range; (G) ICP-derived elemental compositions of FeCoCu alloys with varying Cu contents; (H) Mercury intrusion cumulative pore volume curves of Fe₆₅Co₃₀Cu₅ oxide precursor and reduced alloy; and (I) corresponding pore size distributions. (J) Magnetic hysteresis loops of FeCoCu alloys with varying Cu contents. SEM: TEM: XRD: X-ray diffraction; ICP: inductively coupled plasma.

XRD results [Figure 3E] reveal a transition from a single BCC phase in Fe₇₀Co₃₀ to a BCC/FCC dual-phase structure with increasing Cu content, consistent with limited Cu solubility and its tendency to induce phase separation^[49]. The emergence of dual phases introduces additional interfaces, enhancing polarization loss. Meanwhile, the systematic peak shift from 44.84° to 45.05° [Figure 3F] indicates lattice modulation and increased structural distortion, further contributing to dielectric loss.

ICP-AES analysis was further performed to determine the experimental elemental compositions of the FeCoCu alloy foams. The compositions obtained from ICP-AES are presented as weight percentages in Figure 3G. The measured results confirm the intended composition regulation, with the Cu content increasing and the Fe content correspondingly decreasing, while the Co content remains nearly constant. Although slight deviations from the nominal atomic feeding ratios are observed, the overall composition trend is consistent with the designed Fe/Co/Cu ratios. Mercury intrusion results [Figure 3H and I] show that, although partial densification occurs during reduction (intrusion volume decreasing from 3.7 to 2.3 mL g^-1), the hierarchical pore structure is largely retained. The resulting multiscale porosity simultaneously favors impedance matching and enhances multiple scattering, while the metallic framework supports conduction loss. Magnetic measurements [Figure 3J] indicate typical soft ferromagnetic behavior. The saturation magnetization decreases from 208.4 to 166.7 emu g^-1 with increasing Cu content due to magnetic dilution, while coercivity increases from 105.6 to 177.0 Oe, reflecting structural evolution and enhanced anisotropy. Despite reduced magnetization, all samples retain sufficient magnetic response to support resonance-based magnetic losses, which are critical for impedance matching and bandwidth expansion.

Overall, FeCoCu foams achieve a synergistic regulation of composition, phase structure, and hierarchical porosity. Cu incorporation not only drives the BCC-to-BCC/FCC transition and increases interfacial density, but also introduces structural heterogeneity that enhances polarization loss. Combined with the three-dimensional porous framework and retained magnetic response, these features provide an integrated platform for tuning conduction, polarization, and magnetic losses. This synergy is essential for approaching the theoretically optimal absorption condition and enables efficient broadband EM wave absorption under thin-layer constraints.

Dielectric and magnetic loss behaviors of FeCoCu foam

To further elucidate the EM loss mechanisms and their composition dependence in FeCoCu foams, the complex ε_r, μ_r, and corresponding loss factors were systematically analyzed. As shown in Figure 4A and B, all samples exhibit pronounced frequency dispersion in the range of 2.00-18.00 GHz, with clear composition-dependent evolution. Specifically, both the ε_r' and ε_r'' increase markedly with increasing Cu content. The Fe₇₀Co₃₀ sample shows relatively low ε_r', decreasing from 6.0 to 4.8, whereas Fe₆₅Co₃₀Cu₅ and Fe₆₀Co₃₀Cu₁₀ exhibit higher and more stable values (8.0-10.5). Further increasing Cu to 20 at.% results in a substantial enhancement of ε_r', indicating significantly improved charge storage. A similar trend is observed for ε_r'', where Fe₅₀Co₃₀Cu₂₀ maintains the highest values across the entire frequency range, confirming its strong energy dissipation capability. The tanδ_e values [Supplementary Figure 3] follow the same trend as ε_r'', increasing with Cu content. Fe₇₀Co₃₀ exhibits negligible loss, while Fe₆₅Co₃₀Cu₅ and Fe₆₀Co₃₀Cu₁₀ show moderate values. In contrast, Fe₅₀Co₃₀Cu₂₀ reaches significantly higher levels, indicating more efficient EM energy dissipation. However, excessive dielectric loss may deteriorate impedance matching due to overly strong dielectric responses, implying that optimal EM wave absorption requires a balance between EM attenuation capability and impedance matching rather than maximizing EM loss alone.

Figure 4. Composition-dependent EM parameters and dielectric loss mechanism decomposition of FeCoCu foams. (A) ε_r' values; (B) ε_r'' values; (C) ε_c'' values extracted from ε_r''; and (D) ε_r''-ε_c'' values of FeCoCu foams with various Cu contents; (E) Electrical conductivity values extracted from dielectric decomposition; and (F) ε_s-ε_∞ values of FeCoCu foams with various Cu contents derived from dielectric analysis. (G) μ_r' values; (H) μ_r" values; and (I) C₀ values of FeCoCu foams with various Cu contents. EM: Electromagnetic.

According to the Debye relaxation model and free electron theory, the imaginary permittivity can be expressed as the sum of conduction and polarization contributions^[42,50]:

By fitting the experimental ε_r'' data using a least-squares method, the conduction loss component ε_c'' and polarization contribution ε_p'' were separated [Figure 4C and D]. As shown in Figure 4C, ε_c'' decreases with increasing frequency and increases significantly with Cu content. The Fe₇₀Co₃₀ sample exhibits negligible conduction loss, whereas the Cu-containing samples show progressively enhanced contributions, indicating improved electron transport. This is consistent with the fitted effective conductivity values extracted from the conduction loss term in the dielectric decomposition [Figure 4E], which increase from 0.02 to 1.01 S m^-1 with increasing Cu content. Nevertheless, conduction loss alone cannot fully account for the observed dielectric behavior. As shown in Figure 4D, significant polarization loss remains after subtracting the conduction contribution, and this component also increases with Cu content. Fe₇₀Co₃₀ shows weak polarization, while Fe₆₅Co₃₀Cu₅ and Fe₆₀Co₃₀Cu₁₀ exhibit more pronounced fluctuations. The Fe₅₀Co₃₀Cu₂₀ sample shows the strongest and most complex polarization behavior, with multiple peaks indicating active and diverse relaxation processes. This enhancement of polarization loss is closely associated with crystal structure evolution. As revealed by XRD analysis, increasing Cu content induces a transition from a single BCC phase to a BCC/FCC dual-phase structure, significantly increasing the density of heterogeneous interfaces. In addition, alloying-induced compositional fluctuations, lattice distortion, defects, and particle-particle contacts further contribute to interfacial charge accumulation under alternating EM fields, leading to Maxwell-Wagner polarization. The hierarchical porous structure, combined with nanoscale heterogeneity, further promotes dipolar relaxation. Therefore, the role of Cu extends beyond improving conductivity - it fundamentally enhances polarization by introducing structural heterogeneity and interfacial complexity, enabling the synergistic coupling of conduction and polarization losses.

To further investigate relaxation behavior, Cole-Cole plots were constructed from ε_r' and ε_r'' [Supplementary Figure 4A-D]. According to classical Debye theory, a single relaxation process corresponds to a perfect semicircle, whereas multiple semicircles, distorted arcs, and tailing features indicate the coexistence of multiple relaxation processes and conduction effects. All samples exhibit multiple semicircles and distorted arcs, confirming that FeCoCu foams involve complex and multi-scale relaxation processes rather than a single Debye relaxation. This further supports the conclusion that broadband dielectric loss in this system originates from the cooperative contribution of conduction loss, interfacial polarization, and dipolar relaxation.

To more intuitively quantify the synergistic evolution of conduction and polarization losses induced by Cu incorporation, an equivalent circuit model [Supplementary Figure 5] was employed to describe EWAMs that simultaneously exhibit conduction and polarization dissipation.^[51] In this model, the parallel branch represents conduction loss dominated by free electron transport, while the series branch accounts for polarization loss associated with dipolar relaxation and interfacial polarization. By establishing a correspondence between the complex permittivity of the equivalent circuit and the modified Debye model, the frequency-dependent EM response can be transformed into physically meaningful circuit parameters, including the parallel conductivity (σ₁), parallel capacitance (ε₁), series conductivity (σ₂), and series capacitance (ε₂).

The equivalent complex permittivity of the circuit can be expressed as^{[52, 53]}:

which can be rearranged into:

According to the modified Debye relaxation model, the complex permittivity can also be written as^[54]:

By comparing Equations (16) and (17), the following relationships can be obtained:

In this equivalent circuit model, σ₁ represents the conductivity of the parallel conduction branch, which mainly reflects free-electron transport and leakage-current loss. ε₁ is the absolute permittivity term of the parallel branch and corresponds to the high-frequency dielectric storage contribution, i.e., ε₁ = ε₀ε_∞. The series branch is used to describe polarization relaxation. ε₂ represents the absolute permittivity contribution associated with the relaxation strength, i.e., ε₂ = ε₀(ε_s-ε_∞), while σ₂ is the equivalent dissipative conductivity of the polarization branch and is given by σ₂ = ε₂/τ = ε₀(ε_s-ε_∞)/τ. It should be noted that ε₁ and ε₂ are not dimensionless relative permittivity or lumped capacitance of a specific circuit geometry; rather, they are equivalent material parameters with units of pF m^-1. By contrast, σ₁ and σ₂ have units of S m^-1. Therefore, once σ, ε_∞, ε_s- ε_∞, and τ are determined, the corresponding equivalent circuit parameters can be quantitatively calculated.

Because the Cole-Cole plots of FeCoCu foams are distorted by conduction effects and multiple relaxation processes, the dielectric response cannot be described by an ideal single-Debye semicircle over the whole frequency range. Therefore, the equivalent circuit analysis in this work should be regarded as an effective and semi-quantitative description of the dominant dielectric relaxation behavior. To reduce the influence of conduction loss, the measured imaginary permittivity was first analyzed using the modified Debye model containing both polarization and conduction terms. The conduction contribution was separated according to the free-electron term, and the remaining polarization-related response was used to estimate the dominant relaxation parameters. In the Cole-Cole representation, the fitted dominant relaxation arc satisfies^[54]:

where the center and radius of the fitted arc provide ε_s + ε_∞ and ε_s - ε_∞, respectively. The relaxation time τ was determined from the characteristic relaxation frequency corresponding to the maximum polarization loss, according to ωτ = 1. It should be noted that these parameters represent effective values of the dominant relaxation process rather than unique microscopic constants. Nevertheless, the extracted parameters are reliable for comparing the composition-dependent dielectric evolution because the same fitting procedure was applied to all samples, the obtained τ values remain nearly unchanged, and the monotonic increase in ε_s - ε_∞ is consistent with the enhanced interfacial polarization induced by the BCC/FCC dual-phase structure and compositional heterogeneity [Figure 4F]. Therefore, the subsequent equivalent circuit analysis mainly reflects the relative enhancement of conduction and polarization branches with increasing Cu content.

The extracted ε_s + ε_∞ values are provided in Supplementary Figure 6A. The characteristic τ values for Fe₇₀Co₃₀, Fe₆₅Co₃₀Cu₅, Fe₆₀Co₃₀Cu₁₀, and Fe₅₀Co₃₀Cu₂₀ are 11.84, 11.73, 11.76, and 11.88 ps, respectively [Supplementary Figure 6B], showing negligible variation. The equivalent circuit parameters are summarized in Table 1. From the parallel branch perspective, σ₁ increases monotonically from 0.02 to 1.01 S m^-1 with increasing Cu content, indicating a continuous enhancement of charge transport and conduction loss. This behavior originates from both the intrinsically high conductivity of Cu and the improved connectivity of conductive pathways, reduced transport barriers, and enhanced particle contact within the porous framework. Meanwhile, ε₁ increases from 47.59 to 100.49 pF m^-1, suggesting a strengthened high-frequency dielectric response and enhanced energy storage capability. More importantly, the series branch parameters exhibit a much stronger increase. The polarization-related capacitance ε₂ rises sharply from 2.83 to 31.70 pF m^-1, while σ₂ increases from 0.24 to 2.67 S m^-1. Given that τ remains nearly constant (11.8 ps), this enhancement is primarily attributed to the increase in ε_s - ε_∞ rather than changes in relaxation dynamics. This indicates that Cu incorporation does not introduce new relaxation mechanisms but amplifies existing ones by increasing the density and strength of polarization-active sites. Notably, the increase in ε₂ and σ₂ is significantly more pronounced than that of σ₁ and ε₁, especially at high Cu content. This suggests that polarization loss is more sensitive to Cu incorporation than conduction loss. Such behavior is closely related to structural evolution: Cu induces a transition from a single BCC phase to a BCC/FCC dual-phase system, increasing interfacial density, compositional heterogeneity, and defect concentration. These features promote Maxwell-Wagner interfacial polarization and dipolar relaxation, thereby significantly enhancing dielectric loss. The nearly invariant τ further indicates that the dominant polarization mechanism remains fundamentally unchanged. Instead, Cu enhances dielectric loss primarily by amplifying polarization intensity rather than shifting relaxation frequency.

In addition to dielectric behavior, the magnetic response also shows clear composition dependence. As shown in Figure 4G, μ_r' values decrease with increasing Cu content due to magnetic dilution, while μ_r" values [Figure 4H] remain in the range of 0.1-0.6 and exhibit frequency-dependent fluctuations, indicating multiple magnetic loss mechanisms. The magnetic loss factor (tanδ_m, Supplementary Figure 7) remains within a moderate range (0.1-0.4), suggesting that although Cu reduces overall magnetization, sufficient magnetic dissipation is retained. This moderate and frequency-selective magnetic loss is beneficial for improving impedance matching. To further clarify the magnetic loss mechanism, the eddy current coefficient was calculated^[41]:

As shown in Figure 4I, C₀ is not constant with frequency, indicating that magnetic loss is not solely governed by eddy current effects but involves natural resonance and exchange resonance.

Overall, Cu incorporation induces a synergistic enhancement of conduction, polarization, and magnetic losses. While conductivity increases strengthen conduction loss, the formation of heterogeneous interfaces significantly amplifies polarization loss. Meanwhile, sufficient magnetic loss is preserved despite partial magnetic dilution. Therefore, the FeCoCu system achieves a cooperative coupling of multiple loss mechanisms, which is essential for realizing broadband electromagnetic wave absorption.

EM absorption performance and loss mechanism analysis

Further evaluation of the EM wave absorption performance of FeCoCu alloy foams with different Cu contents was conducted, as shown in Figure 5. The three-dimensional reflection loss maps reveal that the Cu content plays a decisive role in determining the absorption behavior. The Fe₇₀Co₃₀ sample exhibits a minimum reflection loss of -20.8 dB [Figure 5A], indicating a certain absorption capability, although both the absorption intensity and bandwidth remain limited. After introducing 5 at.% Cu, the RL_min of Fe₆₅Co₃₀Cu₅ decreases markedly to -52.9 dB [Figure 5B], demonstrating a substantial improvement in microwave attenuation. When the Cu content is further increased to 10 at.%, Fe₆₀Co₃₀Cu₁₀ still maintains strong absorption ability, with an RL_min of -26.7 dB [Figure 5C]. However, when the Cu content reaches 20 at.%, the RL_min of Fe₅₀Co₃₀Cu₂₀ increases to only -11.8 dB [Figure 5D], indicating a significant deterioration of absorption performance. These results suggest that Cu incorporation is not monotonically beneficial. Although excessive Cu can enhance dielectric loss, it may simultaneously weaken magnetic loss and induce impedance mismatch, thereby suppressing the penetration of incident electromagnetic waves into the absorber. The two-dimensional reflection loss maps further clarify the differences in EABs among the samples. Fe₇₀Co₃₀ achieves a maximum EAB of 5.36 GHz with coating thickness of 2.18 mm [Figure 5E], whereas Fe₆₅Co₃₀Cu₅ delivers the widest bandwidth of 7.36 GHz with coating thickness of only 1.74 mm [Figure 5F], exhibiting the best broadband absorption performance. Fe₆₀Co₃₀Cu₁₀ also maintains a considerable effective bandwidth of 6.00 GHz with coating thickness of 1.67 mm [Figure 5G]. In contrast, Fe₅₀Co₃₀Cu₂₀ shows a much narrower maximum effective bandwidth of only 1.36 GHz with a coating thickness of 1.72 mm [Figure 5H]. Combined with the preceding electromagnetic parameter analysis, these results indicate that an appropriate Cu content can simultaneously enhance conduction loss and interfacial polarization loss while preserving sufficient magnetic loss. In contrast, excessive Cu leads to overly strong dielectric response and weakened magnetic response, resulting in severe impedance mismatch and enhanced surface reflection, thereby limiting broadband absorption.

Figure 5. EM wave absorption performance and loss mechanism analysis of FeCoCu alloy foams with different Cu contents. Three-dimensional reflection loss maps of (A) Fe₇₀Co₃₀, (B) Fe₆₅Co₃₀Cu₅, (C) Fe₆₀Co₃₀Cu₁₀ and (D) Fe₅₀Co₃₀Cu₂₀, respectively. Corresponding two-dimensional reflection loss contour maps of (E) Fe₇₀Co₃₀, (F) Fe₆₅Co₃₀Cu₅, (G) Fe₆₀Co₃₀Cu₁₀ and (H) Fe₅₀Co₃₀Cu₂₀, respectively. The dashed contour lines indicate RL = −10 dB, and the maximum EABs are marked in the corresponding panels. Frequency-dependent (I) attenuation constant, (J) input impedance, (K) interference loss contribution, and (L) total energy loss contribution are presented to clarify the EM absorption mechanism. EM: Electromagnetic; RL: reflection loss; EAB: effective absorption bandwidth.

A comparative summary of representative magnetic metal-, alloy-, foam/aerogel-, and Cu-containing EWAMs is provided in Table 2 to place the EM wave absorption performance of the present FeCoCu alloy foam in context. The comparison includes coating thickness, minimum RL, EAB, and frequency range. Compared with many reported EWAMs, the present FeCoCu alloy foam exhibits a broad EAB with relatively thin coating thickness, demonstrating the effectiveness of the frequency thickness normalized EM parameter design strategy.

To further elucidate the origin of these performance differences, the attenuation capability (α), input impedance characteristics (η_in), interference loss (E_1/4λ), and total energy dissipation (E_{total loss}) can be calculated using the following equations^{[33,41,62,63]}.

where the factor T reflects the attenuation of the EM wave, Π represents reflection at the metal backplate, and ϕ is the phase difference between Π and Γ. As shown in Figure 5I, the attenuation constant increases with increasing Cu content, and Fe₅₀Co₃₀Cu₂₀ exhibits the highest attenuation capability, suggesting the strongest conductive and polarization losses. However, strong attenuation alone does not necessarily guarantee superior EM wave absorption performance, because EM attenuation can only contribute effectively after incident EM waves enter the EWAM. Excessive Cu incorporation greatly enhances dielectric loss but simultaneously weakens the magnetic response, leading to a large impedance mismatch between the EWAM and free space. As a result, strong surface reflection occurs at the air/EWAM interface, and only a limited fraction of the incident EM wave can penetrate into the EWAM for subsequent dissipation. Therefore, in the high-Cu sample, impedance matching becomes the decisive prerequisite rather than EM attenuation strength. In contrast, Fe₆₅Co₃₀Cu₅ possesses moderate dielectric loss, sufficient polarization/conduction dissipation, and retained magnetic response, which collectively lead to a more favorable input impedance. Meanwhile, its EM parameters better satisfy the quarter-wavelength cancellation condition over a broader frequency range. Thus, Fe₆₅Co₃₀Cu₅ achieves the best balance among impedance matching, attenuation capability, and destructive interference, resulting in the widest EAB and strongest overall EM wave absorption performance.

Figure 5J shows that the input impedance decreases with increasing Cu content, indicating that high-Cu samples possess a larger impedance difference from free space, which hinders the effective entry of incident EM waves into the EWAM. Therefore, although Fe₅₀Co₃₀Cu₂₀ exhibits the strongest attenuation, its poor impedance matching results in the weakest actual absorption performance.

The interference loss contribution shown in Figure 5K further confirms the importance of the quarter-wavelength cancellation condition in determining the absorption peak position and effective bandwidth. Fe₆₅Co₃₀Cu₅ and Fe₆₀Co₃₀Cu₁₀ exhibit relatively favorable interference loss contributions over a broad frequency range, indicating that their EM parameters and matching thicknesses are closer to the ideal absorption conditions proposed in the theoretical analysis^[64]. The total energy loss contribution in Figure 5L further demonstrates that Fe₆₅Co₃₀Cu₅ and Fe₆₀Co₃₀Cu₁₀ can both achieve high energy dissipation in the middle- and high-frequency regions. Among them, Fe₆₅Co₃₀Cu₅ achieves the best balance among strong attenuation, impedance matching, and destructive interference, thereby yielding the lowest RLmin and the broadest EAB.

Overall, the EM absorption performance of FeCoCu alloy foams is not governed by a single loss mechanism, but by the synergistic effects of conduction loss, polarization loss, magnetic loss, impedance matching, and interference cancellation. An appropriate Cu content enables the construction of a moderately conductive network and abundant heterogeneous interfaces while retaining the magnetic response of the FeCo matrix. Such a balance realizes the coordinated optimization of attenuation capability, impedance matching, and interference loss, which is essential for achieving thin, efficient, and broadband EM wave absorption.

RCS reduction simulation and multilayer structural design

To further assess the application potential of FeCoCu foams in practical metal-backed stealth scenarios, CST simulations were conducted to evaluate their radar cross-section reduction capability. As illustrated in Figure 6A, FeCoCu foams with a thickness of 1.75 mm were coated onto a 1.0 mm-thick PEC layer to construct a metal-backed EM absorbing structure, and the RCS response was analyzed in the frequency range of 2.00-18.00 GHz. Figure 6B shows that the RCS of the bare PEC layer increases with frequency, confirming its strong reflection toward incident EM waves. In contrast, after coating with FeCoCu EM absorbing layers, the RCS values decrease to different extents. Among all samples, Fe₆₅Co₃₀Cu₅ exhibits the most pronounced RCS reduction, with values significantly lower than those of the bare PEC layer and other compositions. This result is consistent with the reflection loss analysis, in which Fe₆₅Co₃₀Cu₅ exhibits both the lowest RL_min and the widest EAB, indicating that its optimized impedance matching and synergistic loss mechanisms can effectively suppress the backscattering of metallic targets. Figure 6C further compares the RCS reduction of the Fe₆₅Co₃0Cu₅-coated PEC layer with respect to the bare PEC layer, together with the absolute value of the reflection loss at the same thickness. The RCS reduction curve agrees well with the -RL curve in terms of frequency dependence, and both reach their maximum near 13.00-14.00 GHz. This consistency suggests that the absorption peak revealed by the transmission-line model can be effectively translated into practical scattering suppression. In other words, Fe₆₅Co₃₀Cu₅ not only exhibits excellent EM absorption performance in the theoretical reflection loss calculation, but also achieves significant RCS reduction in a more realistic electromagnetic scattering model. For comparison, the RCS reduction behaviors of the other FeCoCu-coated PEC structures are shown in Supplementary Figure 8A-C. Their RCS reduction curves also exhibit frequency-dependent trends that generally follow the corresponding -RL curves, confirming that the reflection loss characteristics of these EWAMs are closely related to their scattering suppression capability. However, their RCS reduction intensities are weaker than that of Fe₆₅Co₃₀Cu₅. Fe₇₀Co₃₀ shows limited RCS reduction because of insufficient dielectric attenuation, whereas Fe₅₀Co₃₀Cu₂₀ exhibits reduced scattering suppression despite its high attenuation capability, which can be attributed to severe impedance mismatch caused by excessive dielectric loss. Fe₆₀Co₃₀Cu₁₀ presents an intermediate behavior, consistent with its moderate absorption performance. These results further demonstrate that effective RCS reduction requires a balanced combination of impedance matching and energy dissipation rather than simply maximizing the loss capability.

Figure 6. RCS simulation and structural design of FeCoCu-based EWAMs. (A) CST simulation model of FeCoCu foam coated on a PEC layer; (B) RCS values of PEC and FeCoCu-coated PEC layers; (C) RCS reduction of Fe₆₅Co₃₀Cu₅-coated PEC compared with bare PEC, together with the corresponding -RL curve. Three-dimensional RCS patterns of (D) bare PEC; and (E) Fe₆₅Co₃₀Cu₅-coated PEC, respectively; (F) Angular RCS distributions of PEC and FeCoCu-coated PEC layers. (G and H) Schematic model of the multilayer frequency-selective structure based on FeCoCu alloys with different Cu contents; (I) Reflection loss curve of the multilayer frequency-selective EWAMs. RCS: Radar cross-section; PEC: EWAM: electromagnetic wave absorbing material.

The angular RCS distributions further confirm the scattering regulation capability of the EM absorbing coating. Figure 6D shows that the bare PEC layer exhibits strong scattering over different azimuth and elevation angles, especially with a pronounced main scattering lobe along the specular reflection direction. In comparison, after coating with 1.75 mm Fe₆₅Co₃₀Cu₅, the overall RCS intensity is greatly reduced, the main scattering lobe is significantly compressed, and the side scattering is also effectively suppressed, as shown in Figure 6E. The three-dimensional RCS patterns of Fe₇₀Co₃₀, Fe₆₀Co₃₀Cu₁₀, and Fe₅₀Co₃₀Cu₂₀ are provided in Supplementary Figure 8D-F. Although these coatings also suppress the scattering intensity relative to the bare PEC layer, their reduction effects are less pronounced than that of Fe₆₅Co₃₀Cu₅, further verifying the superior angular scattering suppression capability of the optimally Cu-modulated sample. Figure 6F presents the angular RCS curves at different incident angles. The bare PEC layer shows the strongest scattering near 0°, whereas all FeCoCu-coated structures exhibit different degrees of RCS reduction over a broad angular range. Among them, Fe₆₅Co₃₀Cu₅ shows the most obvious scattering suppression under normal and near-normal incidence, indicating that its absorption capability is not limited to frequency-domain optimization but also possesses favorable angular adaptability. Based on the preceding dielectric, magnetic, and impedance analyses, this performance can be attributed to the optimized balance among conduction loss, interfacial polarization loss, and magnetic loss induced by moderate Cu incorporation. Such a balance allows EM waves to enter the EWAMs efficiently while being sufficiently dissipated. In contrast, excessive Cu enhances attenuation capability but tends to cause impedance mismatch, thereby limiting RCS reduction.

On this basis, a multilayer frequency-selective surface structure was further designed to broaden the EM absorption bandwidth, as shown in Figure 6G and Figure 6H. The multilayer unit consisted of a 1.0 mm-thick PEC backing layer and four EM absorbing layers, each with a coating thickness of 1.5 mm. The lateral dimensions of the four EM absorbing layers were 6.0, 7.0, 8.0, and 10.0 mm from top to bottom, respectively. In this structure, FeCoCu foams with different Cu contents were introduced into different layers of the frequency-selective units, enabling each layer to contribute to impedance regulation and energy dissipation in different frequency regions. Because FeCoCu foams with different Cu contents possess distinct dielectric losses, magnetic responses, and impedance matching characteristics, the multilayer configuration can construct a gradient distribution of EM parameters along the propagation direction. This gradient structure reduces the impedance discontinuity at the air/EWAM interface and promotes broadband synergistic EM absorption through multiple reflections, destructive interference, and multi-mechanism energy dissipation. As shown in Figure 6I, the multilayer frequency-selective structure exhibits an ultrabroad absorption bandwidth, with the RL below -10 dB reaching 11.8 GHz and that below -15 dB still reaching 11.3 GHz, significantly outperforming the single-layer homogeneous EWAMs. This result demonstrates that integrating the composition-dependent EM responses of FeCoCu foams with structural design can overcome the trade-off between impedance matching and loss intensity in a single material system, thereby enabling efficient EM wave absorption over a much wider frequency range^[65]. It also further validates the design principle proposed in this work: broadband absorption should not rely solely on high loss, but requires the coordinated optimization of conduction loss, polarization loss, magnetic loss, impedance matching, and interference cancellation.