The LBZFO samples with chemical formula of La_xBa_1-xZr_0.3Fe_11.7O₁₉ (x = 0-0.2) were synthesized using a sol-gel process, and the detailed flow chart of the preparation is shown in Fig. 1a. Additionally, Fig. 1b presents SEM images of the as-prepared LBZFO samples with varying proportions of La³⁺ ions. As can be observed, each sample exhibits a characteristic hexagonal plate-like microstructure, consistent with the morphology of original BaM (Fig. S1) [9,13]. The irregular accumulation of hexagonal plate-like grains inevitably leads to numerous voids formation, which could potentially enhance multi-scattering effects for microwaves in the LBZFO samples. Besides, it is noteworthy that the grain size of all LBZFO samples remains similar despite extra-doping La³⁺ ions into BaM lattice structure, thus suggesting minimal impact on its morphology by these additional dopants.

To precisely determine the phase composition, Rietveld refinement analysis was performed on XRD patterns by GSAS-II software as depicted in Fig. 1c. Clearly, successful attainment of M-type barium ferrite phase is confirmed without any obvious impurity phases related to La or Zr elements detected across all samples examined here. The implication here suggests that it is highly probable for foreign La³⁺ or Zr⁴⁺ ions integrating into the lattice structure of BaM phase. Virtually, the respective ionic radii values of La³⁺, Ba²⁺, Fe³⁺, and Zr⁴⁺ are 1.06, 1.34, 0.64, and 0.72 Å [10,12,14,15,16]. Based on the principle of radius similarity, it is initially speculated that La³⁺ might substitute for Ba²⁺; while, Zr⁴⁺ replace Fe³⁺ in BaM. Furthermore, the Rietveld refinement results pertaining to lattice constants and cell volumes of the LBZFO samples are listed in Table 1. Compared with those of original BaM summarized in Fig. S2, it is seen that the lattice constants “a”, “c” and cell volume increase from 5.893 Å, 23.207 Å and 697.941 ų of the original BaM to 5.904 Å, 23.313 Å and 703.862 ų by initially Zr⁴⁺ ions doping to BaZr_0.3Fe_11.7O₁₉. While they subsequently show a continuous reduction to 5.898 Å, 23.265 Å and 700.912 ų with an increment of La³⁺ doping content to x = 0.2 of La_0.2Ba_0.8Zr_0.3Fe_11.7O₁₉. The initially increased lattice constants and cell volume provide a powerful evidence for the fact that Zr⁴⁺ ions (0.72 Å) primarily take the place of smaller Fe³⁺ (0.64 Å) ions, whilst the following decrease in lattice constants and cell volume confirm that La³⁺ ions (1.06 Å) tend to occupy the position of larger Ba²⁺ ions (1.34 Å) in the BaM lattice.

The XPS spectra of Fe 2p and O 1s for the LBZFO samples and the pure BaM were thoroughly measured to determine the changes in concentrations of Fe²⁺ and oxygen vacancy induced by Zr⁴⁺ and La³⁺ ions doping. The Fe 2p, O 1s spectra of all specimens exhibit characteristic peaks of Fe²⁺ 2p_3/2, Fe³⁺ 2p_3/2, Fe²⁺ 2p_1/2, and Fe³⁺ 2p_1/2 at ~ 709.9, ~ 711.5, ~ 723.5, and ~ 725.0 eV, and lattice oxygen, oxygen vacancy at ~ 530.0 and ~ 531.5 eV respectively (Figs. 1d, e and S3a, b). These resolved peaks positions are consistent well with previous reports on the binding energy of Fe³⁺, Fe²⁺, lattice oxygen and oxygen vacancy in XPS [17,18]. Meanwhile, the proportions of Fe²⁺ ions and oxygen vacancy in the barium ferrite were calculated according to the corresponding peak areas (Table 2 and Fig. S3). Apparently, both the contents of Fe²⁺ ions and oxygen vacancy boost with increasing Zr⁴⁺ or La³⁺ ions doping. This observation can be attributed to a charge balance mechanism, where the lattice ions being replaced by higher-priced foreign ions would cause a part of Fe³⁺ ions to change into Fe²⁺ ions in the ferrites [19]. The results hence further confirm the aforesaid fact that Zr⁴⁺/La³⁺ ions are substituting for Fe³⁺/Ba²⁺ in BaM respectively as the following expressed defect reactions:

Meanwhile, the oxygen vacancy also could be induced in the ferrites to maintain the stability of the crystal lattice after foreign ions doping [20], which could benefit the generation of extensive electric dipoles in the LBZFO samples.

In fact, as shown in Fig. 2a, the O^2- ions adopt diverse oxygen polyhedral structures (octahedral, tetrahedral, and triangular bipyramids) within the lattice structure of BaM. As a result, Fe³⁺ ions occupy five distinct sublattice positions (12k, 2a, 2b, 4f₁ and 4f₂) due to variations in their surrounding oxygen environments. The contribution of Fe³⁺ ions at these five positions exhibits pronounced disparities in magnetic properties, including H_a, magnetic saturation strength (M_s) and etc. [21,22]. Therefore, for a comprehensive understanding of the regulation mechanism underlying millimeter-wave absorption through Zr⁴⁺-La³⁺ ions doping, it is crucial to precisely identify which Fe³⁺ sites are substituted by Zr⁴⁺ ions and how this substitution varies with changes in doping content. By analyzing the Raman spectra shown in Fig. 2b, it can be affirmed that La³⁺ ions doping significantly affects the position distribution of Zr⁴⁺ substituting for Fe³⁺. To our knowledge, a single unit cell of BaM contains a total of 64 atoms to generate 189 optical modes, and there exist a total of 42 Raman active optical modes (11A_1g + 14E_1g + 17E_2g) based on the D_6h factor group symmetry [23,24]. Among them, the characteristic peak at 682 cm^-1 belongs to the triangular-bipyramidal 2b site of Fe³⁺ ions. The scattering peak near 715 cm^-1 is ascribed to the tetrahedral 4f₁ site; while, the peaks at 411 cm^-1 and 616 cm^-1 are attributed to the octahedral 12k/2a and 4f₂ sites induced by the A_1g mode. Through a comparison with pure BaM in Fig. S4, the intensity of the scattering peaks at 715 and 411 cm^-1 changes noticeably after doping Zr⁴⁺ to composition of BaZr_0.3Fe_11.7O₁₉, suggesting that the initial addition of Zr⁴⁺ ions primarily occupy the Fe³⁺ ions at 4f₁ and (12k or 2a) sites. While a slight change for the peak at 616 cm^-1 emerge then after further doping La³⁺ ions. It is inferred that the subsequent La³⁺ ions doping has a non-negligible impact on Fe³⁺ ions at 4f₂ position.

To further investigate the specific effect of La³⁺ ions doping on the occupation position distribution of Zr⁴⁺ for Fe³⁺, Mössbauer spectra were obtained for both the original BaM and LBZFO samples. The well-fitted results, consisting of five magnetic sextets, are presented in Figs. S5 and 2c, respectively. Considering the structural position multiplicity and the resonance absorption coefficient of Fe at each position (12k, 4f₁, 4f₂, 2a, 2b), the theoretical ratio of these five positions is identified to be 50:17:17:8:8 [25,26]. According to Fig. S6a, it can be seen that the proportion distribution of Fe³⁺ sites in pure BaM sample closely matches with the theoretical value. However, the proportion of the five positions varies significantly after doping behavior as exhibited in Fig. 2d. Specifically, for the component of BaZr_0.3Fe_11.7O₁₉, it is confirmed that Zr⁴⁺ ions preferentially occupy positions at 12k and 4f₁ initially due to their location far from the barium layer which makes them less bound and hence more easily replaced by doping ions [27]. With an increase content of La³⁺ ions up to x = 0.2, there is a substantial increase in Fe occupancy at site 4f₁ even exceeding that observed for pure BaM; while, Fe at 2a and 4f₂ sites reveal an apparent decreasing trend concurrently. It indicates that the additional doping of La³⁺ ions leads to a marked rearrangement of Zr⁴⁺ ion occupation from 4f₁ site toward 2a and 4f₂ sites. Besides, the Mössbauer spectra also provide the information on hyfine interaction of Fe, including electric monopole interaction, electric quadrupole interaction and magnetic dipole interaction. In specific cases, the electric monopole interaction gives rise to the isomic shift (I.S.), which falls within the range of 0.1 ~ 0.6 mm s^-1 (Figs. 2d and S6b), indicating that Fe predominantly exists in the form of Fe³⁺ in the LBZFO structure [28]. The magnetic dipole interaction also leads to resonance spectral line splitting known as magnetic hyfine splitting (H_hf). Figures 2d and S6c demonstrate a significant reduction in H_hf upon initial Zr⁴⁺ doping into BaM due to substitution of non-magnetic Zr⁴⁺ ions for Fe³⁺ ions, weakening the superexchange interaction of Fe³⁺-O-Fe³⁺. However, subsequent replacement of rare-earth magnetic La³⁺ ions for non-magnetic Ba²⁺ ions would reinforce the superexchange effect, leading to gradual increase in H_hf values again (Fig. 2d) [29]. The electric quadrupole interaction causes quadrupole splitting of the resonance line referred to Q.S. (Figs. 2d and S6d). It can be seen that the Q.S. values remain nearly constant after La³⁺ ions incorporation into BaM lattice, suggesting minimal changes in system symmetry induced by doping behavior [30].

Moreover, first principles calculations were employed to estimate enthalpy for various sites where Zr⁴⁺ substitutes for Fe³⁺ in LBZFO sample. Figure 2e summarizes calculation results for five different structure models ranging from 0 to 3000 pm. It reveals that the crystalline structure with Zr⁴⁺ substituting for Fe³⁺ at 4f₂ position has the minimum total enthalpy beyond 3000 pm, followed by 12k, 2a and 4f₁ positions. According to the principle that lower energy levels contribute to increased stability in structures, which interprets the prominent rearrangement of Zr⁴⁺ substitution sites from 4f₁ to 2a and 4f₂ by La³⁺ doping in the perspective view of minimum energy principle. To sum up, Zr⁴⁺ ions initially tend to occupy the 12k and 4f₁ sites of Fe³⁺ ions in BaM; while, they undergo a significant rearrangement of positions from 4f₁ to 2a and 4f₂ by La³⁺ doping for structure stability.

The hysteresis loops of the original BaM and the LBZFO samples are illustrated in Figs. S7 and 3a. All the specimens are nearly saturated when the external field reaches to 20 kOe. Figure 3b depicts the magnetization curves of the LBZFO samples fitted by the law of approach to saturation (LAS) based on the following formula:

where A is the non-uniformity constant, B is a constant related to H_a and χ₀ is the high magnetic susceptibility [31]. The fitting results of M_s, H_a and the coercivity (H_c) deduced from the hysteresis loops are summarized in Fig. 3c. It is found that the initial M_s value of original BaM is 62.84 emu g^-1, which increases to a maximum value of 65.80 emu g^-1 upon Zr⁴⁺ doping to form BaZr_0.3Fe_11.7O₁₉. With further addition of La³⁺ ions, M_s firstly decreases to 58.97 emu g^-1 with x = 0.1 and then increases slightly to 60.92 emu g^-1 at x = 0.2. It has long been established that the saturation magnetization is collectively influenced by various factors, including the number, spin direction, magnetic moment of magnetic atoms/ions and the superexchange interaction, etc. [32]. In regard to BaM, the spin direction of Fe³⁺ ions at 12k, 2a, and 2b sites is upward; whereas, the spin direction of Fe³⁺ ions at 4f₁ and 4f₂ is opposite. The M_s is ultimately determined by the vector sum of spin-up magnetic ions [8,12]. Therefore, the increment of M_s in BaZr_0.3Fe_11.7O₁₉ is primarily attributed to the substitution of Fe³⁺ ions at down-spin 4f₁ site by the non-magnetic Zr⁴⁺ ions. While with the further addition of La³⁺ ions, two significant effects occur simultaneously. Firstly, the transformation of Fe³⁺ into Fe²⁺ for electric neutrality would result in a distinct magnetic dilution with magnetic moment decreasing from 5 to 4 µ_B [33]. Secondly, the rare-earth La³⁺ ions improve the number of magnetic ions and superexchange interaction in BaM, which contributes to the enhancement of M_s. Under the collaborative influence of multiple factors, the M_s consequently reveals a reduction firstly and then turns to a slight increase with La³⁺ ions content elevating (Fig. 3c). In regard to H_a, it experiences a conspicuous growth from 9.47 to 11.74 kOe for an increased La³⁺ content from x = 0 to x = 0.2. This phenomenon can be attributed to two primary causes. One is that the rare-earth La³⁺ has a larger magnetocrystalline anisotropy constant than the substituted Ba²⁺ due to the strong spin-orbit coupling effect as illustrated in Fig. 3d [11]. The other one is ascribed to the transformation of Fe³⁺ to Fe²⁺, where Fe²⁺ possesses a higher magnetocrystalline anisotropy constant for partly frozen orbital angular momentum [34,35]. As for H_c, it increases from 0.87 kOe (x = 0) to the maximum value of 1.96 kOe (x = 0.2). Considering the fact that H_c is jointly determined by H_a and grain size, improving with increasing H_a or decreasing grain size [36]. By observation of Fig. 1b, the grain size hardly changes by La³⁺ ions doping. Therefore, the improvement of H_c with La³⁺ content increasing is dominantly contributed by the correspondingly enhanced H_a.

Figures S8 and 3e display the real and imaginary parts of the complex permeability for the original BaM and LBZFO samples across a frequency spectrum of 26.5-40 GHz. Apparently, no magnetic resonance behavior occurs in the original BaM; whereas, all LBZFO samples display an asymmetric magnetic resonance phenomenon. Previous literature has reported that the natural resonance frequency of pure BaM lies approximately at 45 GHz [37], which exceeds the upper limit of our measuring capabilities. The substitution of non-magnetic Zr⁴⁺ ions for Fe³⁺ ions in BaM would lead to a decrease in H_a and thus reduces the natural resonance frequency to ~ 27 GHz with the constituent of BaZr_0.3Fe_11.7O₁₉. After the additional La³⁺ ions doping, it can be observed that the resonance peak frequency gradually shifts toward the high-frequency direction to atmospheric window of 35 GHz. According to the expression of the natural resonance frequency:

where $\gamma$ is gyromagnetic ratio, g is Landé factor. It is known that the resonance frequency is directly proportional to H_a. The reinforced H_a by La³⁺ ions doping is hence the root cause for the increase in natural resonance frequency to approach the atmospheric window of 35 GHz.

The Cole-Cole plots of permeability are drawn in Fig. 3f to elucidate the multifaceted magnetic resonance relaxation occurring beneath the asymmetric resonance peaks. As a matter of fact, the permeability induced by resonance behavior should obey the equation as follows [38]:

where µ₀ is permeability in free space, H_m and B_m are the amplitude of the magnetic field and the magnetic induction intensity, respectively. That is to say, each semicircle in µ"-µ' curves correspond to a magnetic resonance behavior. The presence of three distinct semicircles in the LBZFO samples strongly suggests the existence of three predominant resonance loss mechanisms within the specified frequency range. To gain a deeper understanding of the multiple magnetic resonance behavior, the asymmetric resonance peaks are further analyzed using the Peakfit software as depicted in Fig. 3g. Intriguingly, they all could be resolved into three distinct peaks at the frequency range of 26.5-40 GHz. The frequencies of the three resolved peaks are (27.10, 32.00, and 37.16 GHz), (30.20, 33.50, and 36.45 GHz) and (33.40, 37.20, and 39.80 GHz) when x = 0, 0.1, and 0.2, respectively. Actually, the g values of Fe³⁺ and Fe²⁺ are 2.0 and 3.54, separately. The exchange coupling between these two ion species, as illustrated in Fig. 3h, can lead to the formation of new g factors that fall within the range of 2.0 to 3.54 [12,39]. Based on the formula of natural resonance frequency, the g values for the resolved peaks are determined to be (2.04, 2.41, 2.80), (1.90, 2.11, 2.29), and (2.03, 2.26, 2.42), respectively. This analysis confirms that, apart from the natural resonance originating from the intrinsic Fe³⁺ ions with g values around 2.0 (1.90 ~ 2.04), the exchange coupling between Fe³⁺ and Fe²⁺ ions with g values ranging from 2.11 to 2.80 enhances the formation of impressive magnetic loss peaks within the higher frequency range. Furthermore, it is worth emphasizing that the exchange coupling effect by the dual high-valance ions doping is more pronounced around the atmospheric window of 35 GHz when compared to the series doped solely with Zr⁴⁺ ions in Fig. S9, Ti⁴⁺ ions in Fig. S10 or Nb⁵⁺ ions in Fig. S11. These results unambiguously demonstrate the beneficial impact of the dual high-valance ion doping strategy in enhancing the magnetic loss capacity within the target frequency range. In conclusion, the contrasting impact of La³⁺ and Zr⁴⁺ ions on H_a allows for precise tuning of inherent resonance to the desired atmospheric window of 35 GHz. Meanwhile, the co-substitution of the two high-valence ions can effectively intensify the exchange coupling effect between Fe³⁺ and Fe²⁺ to gain wide and strong magnetic loss.

The complex permittivity values of the original BaM and LBZFO samples plotted against frequency are clearly visible in Figs. S8b and 4a. Upon initial doping with Zr⁴⁺ ions, both real part (ε’) and imaginary part (ε") of the dielectric constant reveal an evident enhancement. Subsequently, with the introduction of additional La³⁺ ions, these values exhibit a notable decrease. While with La³⁺ content further rising from x = 0.1 to x = 0.2, the permittivity values remain relatively stable with a slight decrease observed only in the high-frequency range of ε". It is universally acknowledged that the permittivity is comprised by two distinct behaviors: conduction and polarization [40]. In order to gain a comprehensive understanding for the alteration mechanisms of permittivity, the Cole-Cole curves are plotted with ε’ and ε" as the horizontal and longitudinal coordinates as shown in Fig. 4b firstly. According to the Debye equation expression as follows [41,42,43]:

where ${\varepsilon}_{\infty}$ and ε_s represent the relative permittivity under infinite frequency and static permittivity. The Debye relaxing process is consequently represented as a circle in the Cole–Cole diagram. From Fig. 4b, there are three obvious semicircles in the LBZFO samples, which indicates the existence of abundant polarization relaxations. Besides, the ε"-ε' curves also display sections of partial straight lines, demonstrating that conduction loss also contributes to dielectric loss to some extent [44]. To quantitatively assess the contributions of conductive and polarization loss, the least square method was employed based on variations in the Debye equation [45]. As depicted in Fig. 4c, the conduction loss of the LBZFO samples gradually decreases with increasing La³⁺ doping proportion; while, the polarization loss exhibits an opposite increasing trend. In fact, the diminished conduction loss probably originates from the rearrangement of Zr⁴⁺ ions from tetrahedral 4f₁ site to octahedral 2a/4f₂ sites after La³⁺ doping, which is probably owing to the lower impurity energy level provided by Zr⁴⁺ substitution in the higher symmetry octahedron (Fig. 4d) [12]. On the other side, as the portions of Fe²⁺ ions and oxygen vacancy increase gradually with the La³⁺ doping ratio increasing based on the aforementioned XPS results in Fig. 1d, e, numerous Fe³⁺/Fe²⁺, Zr⁴⁺/VӦ, La³⁺/VӦ dipoles are thus generated in the LBZFO samples as illustrated in Fig. 4e for efficiently reinforcing the polarization loss [8,12]. By reason of the foregoing, the additional incorporation of La³⁺ ions is expected to meticulously regulate the weight coefficient of conduction loss and polarization loss for achieving a suitable dielectric loss performance.

The microwave absorption performance is usually evaluated by the reflection loss (RL) values calculated by transmission line theory based on the following equations [46,47,48,49,50,51,52,53]:

where Z_in is the input impedance of absorber, Z₀ is the characteristic impedance of free space, ε_r is complex permittivity (ε_r = ε'-jε"), µ_r is the permeability (µ_r = µ'-jµ") of the ferrites, f is wave frequency, d is the thickness of the absorption layer and c is the velocity of light. Figure S12 displays that the original BaM is not equipped with absorption capacity in the range of 26.5 ~ 40 GHz owing to the deficiency in magnetic and dielectric loss. However, as shown in Fig. 5a, the effective absorption band can be observed in the matching thickness range of n = 1, n = 3 and n = 5 by Zr⁴⁺ and La³⁺ ions co-doping, and the RL has a smaller value in the matching thickness range of n = 3 with the increase in La³⁺ doping ratio. Specifically, Fig. 5b shows the two-dimensional RL curves of the LBZFO samples at various thicknesses. Among the rest, the corresponding EAB and impedance matching values are also merged, and the pink areas highlight the effective absorption range where the RL values are below - 10 dB, meaning the microwave absorption exceeds 90%. It demonstrates that the LBZFO sample with x = 0.1 exhibits the optimal absorption performance around the atmospheric window of 35 GHz. The effective absorption range (RL < - 10 dB) covers 29.4 ~ 37.4 GHz (8.0 GHz) under the first order of matching thickness (n = 1) at 1.2 mm. Intriguingly, it significantly extends to the widest EAB range of 27.5 ~ 40.0 + GHz (12.5 + GHz) under the second order of matching thickness (n = 3) at 3.2 mm, accompanied by an impressively low RL_min value of -19.8 dB. The millimeter-wave absorption properties around the atmospheric window of 35 GHz are evidently superior than the Zr⁴⁺, Ti⁴⁺ or Nb⁵⁺ ions solely doped series (Figs. S13-S15), which is attributed to the impressively modified input impedance matching on the grounds of Z_in/Z₀ values in Fig. 5b. This is dominantly contributed by the precise regulation of natural resonance peak for approaching 35 GHz via the opposite effect of La³⁺ and Zr⁴⁺ ions on H_a, strengthening the magnetic loss capacity in the corresponding frequency band as presented in Fig. 5c. Besides, the enhancive portion of polarization loss to conduction loss revealed in Fig. 4c is also believed to exert a positive effect on ameliorating the impedance matching to broaden the absorption bandwidth [54,55,56,57]. As a consequence, the co-incorporation of La³⁺ and Zr⁴⁺ ions into BaM serves as a highly competitive approach for the development of exceptional millimeter-wave absorbers, specifically designed for the 35 GHz atmospheric window.

CST Studio Suite has been utilized to meticulously simulate the propagation characteristics of millimeter-waves through LBZFO samples, both in the near-field and far-field environments. The near-field simulation diagram is illustrated vividly in Fig. 6a, providing a clear visualization of the wave propagation. The results obtained from these simulations are presented in Fig. 6b, clearly showing the significant reduction in the intensities of millimeter-waves after passing through the LBZFO samples. This observation verifies the remarkable millimeter-wave absorption capacity of the elaborately designed LBZFO samples. In addition, as shown in Fig. 6c, a simplified Predator II UAV model with a length of 0.82 m and a width of 1.34 m is applied for the far-field simulation, where the atmospheric window center of 35 GHz is chosen as the field monitoring frequency and the open boundary is roundly set in all directions. Figure 6d shows the RSC values of the UAV model from 0° to 360° before/after coating the LBZFO samples with thickness of 1 mm. Evidently, the RCS values diminish effectively at omnidirectional angles with the usage of the LBZFO coatings, which can be more clearly observed through the scattered signals of three-dimensional radar waves as shown in Fig. 6e. Besides, the near-field and far-field simulations have also been conducted for the Zr⁴⁺, Ti⁴⁺ and Nb⁵⁺ ions solely doped series as seen in Figs. S16-S19. A comparative analysis of the results reveals that the LBZFO coatings contribute significantly to reduce the radar detection probability for military equipment across a wide range of angles. This finding holds tremendous promise of the LBZFO coatings for applications in the field of millimeter-wave absorption.