XRD patterns of oriented and non-oriented BaM (x = 0.3) are compared in Figure 3A. Both sets of diffraction peaks match well with the standard BaM-type hexaferrites pattern (PDF#27-1029), confirming the formation of a pure M-type hexaferrite phase with space group P6₃/mmc. After normalization to the strongest peak, the oriented sample shows increased relative intensities of the (00l) reflections. Combined with the Lotgering factor f_L = 0.57 calculated from the XRD patterns, these results confirm a well-developed c-axis texture. Figure 3B shows XRD patterns of oriented Ba(ZnTi)_xFe_12-2xO₁₉ with different x. All samples exhibit single-phase M-type structure, but diffraction intensities vary irregularly, reaching a maximum at x = 0.3. Figure 3C displays a magnified view of the (203) peak. The ionic radius of Zn²⁺ (0.74 Å) is larger than that of Fe³⁺ (0.64 Å), while that of Ti⁴⁺ (0.60 Å) is smaller^[14]. If ionic size alone governed the structural response, the interplanar spacing of the (203) plane would be expected to shift monotonically as the substitution level x increases. However, as shown in Figure 3C, the (203) peak exhibits a non-monotonic angular variation: it shifts progressively to higher angles for x = 0.1 - 0.3. Then it shifts back to lower angles for x = 0.4 and 0.5. The underlying mechanism responsible for this behavior involves composition-dependent site occupancy by Zn²⁺ and Ti⁴⁺ ions and is discussed in detail in Section 3.2, supported by Raman spectroscopy analysis. Nevertheless, the observed irregular shift of the (203) peak already confirms the successful doping of Zn²⁺ and Ti⁴⁺ ions into the BaM lattice.

Rietveld refinement was carried out using the General Structure Analysis System (GSAS) program. Figure 4A-F presents the Rietveld refinement patterns for all compositions, where the observed data (Obs) are shown as × symbols, the calculated profile (Cal) as a red curve, the difference curve (Obs-Cal) as a blue curve, and the Bragg peak positions (Bragg) as vertical purple tick marks. Refined lattice parameters (a, c, c/a, V_cell) and the corresponding goodness of fit (χ²) are listed in Table 1. The relatively low χ² values confirm the reliability of the refined lattice parameters. The c/a ratios are all below 3.98, confirming the hexagonal structure^[15]. Notably, the lattice parameters exhibit a non-monotonic variation with increasing substitution level x. This behavior can be rationalized by the combined effects of site-occupation preference and doping level, a point that will be further clarified in Section 3.2 using Raman spectroscopy.

The Lotgering factor (f_L), defined by Equation (1) ^[16], provides a quantitative measure of the degree of c-axis texture in BaM as follows.

(1)

Here, ∑I (00l)/∑I (hkl) represents the ratio of the sum of XRD intensities from (00l) planes to that from all (hkl) planes for the oriented sample. Moreover, ∑I₀ (00l)/∑I₀ (hkl) is the corresponding ratio for randomly oriented BaM without preferred orientation. Based on the equation and XRD patterns, the calculated f_L values are summarized in Table 1. For instance, at x = 0.1, f_L reaches 0.57. Generally, f_L increases with a higher c-axis orientation^[17]. While a high degree of c-axis orientation plays an important role in achieving a high remanence ratio, the extent of grain growth during sintering also has a strong influence on the final remanence ratio of the material. Bulk density (ρ) was determined using the drainage method of Archimedes [Table 1]. Moreover, the overall porosity (p) is defined by Equation (2) as follows^[18].

(2)

(3)

In these expressions, ρ_x denotes the X-ray density as given in Equation (3)^[18]. At the same time, the symbols Z, M, N_A, and V_cell correspond to the number of formula units per unit cell, the molecular weight, Avogadro’s constant, and the unit cell volume, respectively. Previous studies have consistently shown that porosity significantly influences the FMR behavior of hexaferrites, with lower-porosity samples exhibiting narrower FMR linewidths^[19]. A quantitative analysis of the FMR linewidth and its dependence on porosity is presented below.

SEM images of oriented Ba(ZnTi)_xFe_12-2xO₁₉ are presented in Figure 5A-F. The microstructural morphology shows a clear dependence on the amount of Zn²⁺ and Ti⁴⁺ incorporated into the BaM lattice. A typical hexagonal platelet morphology is observed across the entire sample surface, consistent with the c/a ratio obtained from XRD refinement. Moreover, no obvious agglomeration is observed in the SEM images, resulting in a compact microstructure. The sample with x = 0.1 shows the most uniform distribution of hexagonal grains with minimal inter-grain voids. As shown in the cross-sectional images in Figure 5G and H, the grains tend to develop mainly along the direction normal to the sample surface. As shown in Figure 5G, the x = 0.1 sample displays noticeably more compact grain packing with fewer visible voids than the x = 0.5 sample shown in Figure 5H, which is in good agreement with its higher bulk density and lower porosity values listed in Table 1. As shown in Figure 6, the grain size distributions for both compositions exhibit a pronounced concentration, confirming their uniform grain size. Statistical analysis in Figure 6A-F indicates an overall trend of an initial increase and subsequent decrease in the average grain size, despite some irregular fluctuations.

Raman spectroscopy arises from the inelastic scattering of incident photons by material constituents, a process that involves energy transfer. This method is capable of probing local structural features, such as vibrational modes and bond strengths^[20]. Group theory analysis based on D_6h symmetry indicates that M-type hexaferrites possess 42 Raman-active modes (11A_1g + 14E_1g + 17E_2g), 30 infrared-active modes (13A_2u + 17E_1u), and 54 silent modes (3A_1u + 4A_2g + 13B_1g + 4B_1u + 3B_2g + 12B_2u + 15E_2u). The letters A, B, and E label different optical modes, while the subscripts g (gerade) and u (ungerade) indicate whether a given mode is symmetric or antisymmetric with respect to the inversion center^[21,22].

As shown in Figure 7A, the BaM crystal structure consists of alternating rhombohedral (R) and spinel (S) blocks stacked along the c-axis. This arrangement is conventionally denoted as RSR^*S^*, where the asterisk indicates a 180° rotation of the corresponding block about the c-axis^[13]. Within the oxygen interstices of the R and S blocks, Fe³⁺ ions occupy five distinct crystallographic sites: three octahedral sites (2a, 12k, 4f₂), one bipyramidal site (2b), and one tetrahedral site (4f₁), among which 2a, 12k, and 2b spin up, while 4f₁ and 4f₂ spin down^[23].

For M-type hexaferrites, the octahedra (FeO₆), bipyramids (FeO₅), and tetrahedra (FeO₄) each give rise to characteristic Raman vibration modes. The Raman spectra acquired from our samples in the range of 100-800 cm^-1 are presented in Figure 7B. Their peak positions and intensities agree well with the literature values for typical BaM^[18]. Furthermore, Figure 7C depicts the result of applying a Gaussian fit to the Raman spectrum. The major Raman peak parameters, such as frequencies (Freq) and intensities (cts), are summarized in Table 2 based on the fitting results. The Raman peaks corresponding to the tetrahedral (4f₁) and octahedral (4f₂) sites show the most pronounced trend: as x increases, the peak intensity (cts) generally decreases, indicating that Zn²⁺-Ti⁴⁺ preferentially substitute Fe³⁺ at these lattice positions. This substitution disrupts crystal symmetry, increases disorder, and induces local strain^[24]. No additional peaks are observed in the Raman spectra, confirming the phase purity of the samples and consistent with the XRD results.

Figure 7D presents a more detailed view of the Raman shifts of the five crystallographic sites as a function of Zn²⁺-Ti⁴⁺ substitution. As x increases, the Raman vibrational modes associated with the 4f₁ and 4f₂ sites show a noticeable shift. However, the mode positions corresponding to the remaining sites, namely 12k, 2a, and 2b, remain essentially unchanged until x reaches 0.1. It suggests that Zn²⁺-Ti⁴⁺ initially occupy the 4f₁ and 4f₂ sites in BaM. Although Raman spectroscopy probes local bonding environments and does not directly measure cation site occupancy, the clear shifts observed in the vibrational modes of the 4f₁ and 4f₂ polyhedra strongly suggest that these two sites are the most affected by the introduction of Zn²⁺ and Ti⁴⁺ ions into the BaM lattice. As discussed in the Introduction, the site preference of Ti⁴⁺ in M-type hexaferrites remains unsettled. The site-occupation tendencies for Zn²⁺ and Ti⁴⁺ inferred from these Raman shifts are qualitatively consistent with the model proposed by González-Angeles et al.^[10]. According to this model, Zn²⁺ and Ti⁴⁺ tend to occupy the 4f₁ and 4f₂ sites, respectively, thereby reducing the spin-down contribution and leading to an initial enhancement of the saturation magnetization. Upon further increase in x, Raman shifts are observed for all five sites, indicating that Zn²⁺-Ti⁴⁺ progressively occupy all five positions.

Based on the site occupation evolution revealed by Raman spectroscopy, the non-monotonic variations of both the (203) diffraction peak angle in Figure 3C and the lattice parameters in Table 1 can be understood in terms of site occupation preference and doping level. For x = 0.1, where Zn²⁺ and Ti⁴⁺ preferentially occupy the 4f₁ and 4f₂ sites, the smaller Ti⁴⁺ substituting Fe³⁺ produces a dominant local contraction, leading to a slight right-shift of the (203) peak and relatively stable lattice parameters. For x = 0.2 and 0.3, the (203) peak continues to shift rightwards, and the lattice parameters show a slight further contraction, indicating that the contraction effect from Ti⁴⁺ remains dominant during this stage. For x = 0.4 and 0.5, the dopant ions gradually occupy all five Fe³⁺ sites. The larger Zn²⁺ (0.74 Å) now exerts a net expansion effect that overcomes the contraction from Ti⁴⁺, causing the (203) peak to shift leftwards and the lattice parameters to expand, reaching their maximum at x = 0.5. The non-monotonic trends observed in both the lattice parameters and the (203) peak angle, therefore, directly reflect a transition from preferential site occupation at low doping levels to broad multi-site substitution at higher doping levels, driven by the competing size effects of the larger Zn²⁺ and the smaller Ti⁴⁺ ions^[25].

Room-temperature magnetic characterization of the oriented samples was performed using a VSM. Figure 8A shows the hysteresis loops recorded for x = 0.0 under two field orientations: perpendicular and parallel to the hexagonal plane. When the magnetic field was applied perpendicular to the natural surface plane, which corresponds to the crystallographic easy axis, the sample reached magnetic saturation more readily. It showed a remanence ratio (M_r/M_s) of 0.91. This value is substantially higher than the theoretical upper limit of 0.5 expected for randomly oriented, non-interacting grains^[13]. In contrast, when the field was applied parallel to the hexagonal plane along the hard axis, the remanence ratio fell sharply to just 0.20, and a much stronger applied field was needed before the sample reached saturation. These observations indicate that after magnetic-field alignment, the material displays strong direction-dependent magnetic anisotropy. Specifically, the anisotropy field is lower along the c-axis but higher within the basal plane. The hysteresis loops measured with the VSM field parallel to the c-axis for oriented samples at various x values are shown in Figure 8B. From these curves, we extracted the coercivity (H_c) and remanent magnetization (M_r) for each composition, and the results are compiled in Table 3.

When the applied field is sufficiently strong, the magnetization curves of polycrystalline magnetic materials gradually converge toward a common shape as the material approaches saturation. This behavior can be well described by the Law of Approach to Saturation (LAST) as follows^[26].

(4)

In Equation (4), the parameters A and B denote material-dependent coefficients with dimensions of Oe and Oe², respectively. They are associated with the resistance to magnetization during the technical magnetization process. Where χ_p denotes the paramagnetic susceptibility, and H is the applied magnetic field. For M-type hexaferrites, A is close to zero, and B is given as follows^[27].

(5)

Here, μ₀ is the permeability of free space. In strongly magnetic materials like BaM, the paramagnetic susceptibility is sufficiently weak that the term χ_pH can be omitted^[28]. Thus, the LAST expression can be simplified to the form given in Equation (6) as follows.

(6)

Here, M varies linearly with 1/H². Under high-field conditions (19-20 kOe), the values of M_s, a material constant for a given composition, and B are obtained by linearly fitting M against 1/H², as illustrated in Figure 8C. The anisotropy constant K₁ and the anisotropy field H_a are then determined using Equation (5)^[29,30]. The corresponding magnetic parameters derived from the grain structure are summarized in Table 3, and the compositional variation trends of these parameters are shown in Figure 8D.

As seen in the data, the trends of H_a and K₁ are essentially identical, consistent with the theoretical expectation that a larger H_a corresponds to a larger K₁. The saturation magnetization (M_s) of the samples initially rises to a peak of 72.84 emu/g at x = 0.4, then declines. It should be noted that although the highest M_s (72.84 emu/g) is attained at x = 0.4, its remanence ratio drops sharply to 0.47 as shown in Table 3, which is unfavorable for self-biased operation. Therefore, M_s must be evaluated together with M_r/M_s when assessing composition suitability for self-biased circulators. All Zn-Ti substituted compositions show higher M_s values than their undoped counterparts. It suggests that the ionic composition plays a key role in determining the hysteresis loop shape and associated magnetic parameters^[31]. This overall enhancement in M_s is primarily due to the preferential occupation of the tetrahedral 4f₁ and octahedral 4f₂ sites by non-magnetic Zn²⁺ and Ti⁴⁺ ions, which replace the original spin-down Fe³⁺ moments at these positions, reducing the total spin-down contribution and increasing the net magnetization. However, with increasing Zn²⁺-Ti⁴⁺ substitution, both M_r and M_s show non-monotonic, irregular trends, with M_s peaking at x = 0.4, and then decreasing. These trends can be attributed to the site preference of the dopant ions and their different ionic radii relative to those of the host ions, which alter interatomic distances and consequently modify the superexchange interactions of Fe-O-Fe^[32]. Such modifications lead to either an enhancement or a reduction in magnetic properties, thereby explaining the irregular changes in both M_r and M_s^[30]. The remanence ratio (M_r/M_s) reaches its maximum value of 0.87 after doping at x = 0.1 and x = 0.5. It is also reflected in the SEM images shown in Figure 5G and H, which reveal the most distinct and uniform layered morphology for Ba(ZnTi)_xFe_12-2xO₁₉ at these compositions, features generally associated with improved magnetic performance. Differences in cross-sectional morphology correlate with different M_r/M_s values. Specifically, a more pronounced layered structure is associated with a higher remanence ratio. The increase in M_r/M_s from the theoretical maximum of 0.5 for randomly oriented non-interacting grains to 0.87 is mainly due to the effective c-axis grain alignment achieved by magnetic-field orientation during pressing. The high Lotgering factors and the cross-sectional SEM images in Figure 5G and H confirm that the platelet-like grains are aligned perpendicular to the sample plane.

Coercivity (H_c) is a key parameter distinguishing hard from soft magnetic behavior. Its variation in polycrystalline ferrites can be understood using Equation (7)^[27]:

(7)

An increase in grain size D reduces the total grain boundary area and weakens the overall pinning effect, thus lowering H_c. However, a larger K₁/M_s ratio strengthens pinning and acts in the opposite direction^[33]. In the samples studied here, the observed non-monotonic behavior of H_c, which first decreases and then increases, is governed by the competition between these two factors. The minimum H_c does not strictly coincide with the maximum grain size D because the K₁/M_s ratio varies considerably with composition. When this ratio decreases sufficiently, it becomes the dominant factor controlling pinning behavior, driving H_c to lower values even as the grain size decreases. The subsequent rise in H_c at higher substitution levels is then driven by the combined effect of increasing K₁/M_s and decreasing D.

In polycrystalline hexaferrites, the total FMR linewidth ΔH can be expressed as the sum of intrinsic linewidth ΔH_i, magnetocrystalline anisotropy contribution ΔH_a, porosity-related contribution ΔH_p, and surface roughness contribution ΔH_oth^[34,35], as follows.

(8)

ΔH_i is typically negligible for M-type hexaferrites, while ΔH_oth can be minimized by careful surface polishing^[36]. Therefore, the following analysis focuses mainly on ΔH_a and ΔH_p. Importantly, ΔH_a and ΔH_p are positively related to the crystalline anisotropy and porosity-induced linewidth broadening contributions^[37]. Because the samples consist of polycrystalline assemblies of oriented grains, the experimentally measured ΔH reflects an average of the collective gyromagnetic response arising from a large number of individual crystallites. Figure 9A-D displays the FMR absorption spectra of the BaM sample (x = 0.1) obtained by sweeping the external field (H₀) at different frequencies (f). H₀ was applied perpendicular to the sample plane, parallel to the c-axis. The experimental data measured in the 64-67 GHz frequency range are plotted as circular markers. The orange curve represents a fit using a Gaussian (GaussAmp) lineshape, while the green curve corresponds to a Lorentzian fit. Quantitatively, the Gaussian fit yields a higher coefficient of determination (R² = 0.975) than the Lorentzian fit (R² = 0.968) for the 64 GHz spectrum, indicating a better agreement with the experimental data. Therefore, the FMR linewidth (ΔH) was extracted from the Gaussian fit. The FMR peak position shifts with the external field across different frequencies.

To comprehensively evaluate the effect of Zn²⁺-Ti⁴⁺ co-substitution on high-frequency magnetic losses, the FMR linewidths of all compositions (x = 0.0 - 0.5) were measured across the 40-67 GHz range. Figure 10 summarizes the dependence of ΔH on substitution level x, where ΔH was measured at a representative frequency for each sample. The ΔH exhibits a clear non-monotonic trend where it initially decreases with increasing x, reaches a minimum at x = 0.2, and then increases significantly at higher substitution levels. Specifically, the narrowest linewidth of 289 Oe was achieved at 59 GHz for x = 0.2, while the broadest linewidth of 1,322 Oe was observed at 48 GHz for x = 0.5.

To quantitatively evaluate the role of porosity, Table 1 lists the porosity p of each oriented BaM sample. A direct comparison of the porosity data with the ΔH values shown in Figure 10 reveals a clear correlation between the two, as both quantities follow the same trend with increasing x and reach their lowest values at x = 0.2. As x increases beyond 0.3, porosity rises sharply to 18.70% at x = 0.5, accompanied by a sharp increase of ΔH to 1,322 Oe (at 48 GHz). In addition to porosity, the magnetocrystalline anisotropy field H_a also plays a role. As listed in Table 3, the H_a values for x = 0.1 (11,930 Oe) and x = 0.2 (14,533 Oe) are among the lower levels across all substituted samples, especially when compared with the much higher value at x = 0.4 (22,428 Oe). A lower anisotropy field generally reduces the anisotropy-related linewidth broadening contribution. Combining the two factors above, the narrow FMR linewidths observed at x = 0.1 (355 Oe at 67 GHz) and x = 0.2 (289 Oe at 59 GHz) can be attributed to the synergistic effect of minimal porosity and relatively low magnetocrystalline anisotropy field.

For a finite rectangular plate sample, the internal demagnetizing field H_d = -N_z·M_s is spatially nonuniform, where N_z is the position-dependent demagnetizing factor and M_s the saturation magnetization, due to magnetic surface charges at the sample boundaries^[38]. For the 4.0 × 4.0 × 0.8 mm³ specimens measured here (M_s ≈ 344 emu/cm³, x = 0.2), a surface-integral calculation yields N_z = 0.825 at the center, decreasing to approximately 0.47 at the edge midpoint and approximately 0.41 near the corners. The standard deviation of the internal field across the midplane is approximately 40 Oe, contributing approximately 14% to the narrowest FMR linewidth (ΔH_min = 289 Oe). Within the central 70% of the sample area, the field variation is only about 12 Oe, which is approximately 4% of ΔH_min. The edge demagnetizing effect is therefore a measurable but non-dominant source of linewidth broadening in our samples.

Compared with typical values for commercially available polycrystalline hexaferrites (~2,000 Oe)^[39], all substituted samples in this study, especially those with x = 0.1 - 0.3, demonstrate markedly lower linewidths in the millimeter-wave band. Notably, the Ba(ZnTi)_xFe_12-2xO₁₉ hexaferrite with x = 0.2 exhibits the narrowest FMR linewidth of 289 Oe at 59 GHz, compared with previously reported values as shown in Figure 11. Among the investigated compositions, the x = 0.2 sample achieves the narrowest FMR linewidth in this study (ΔH = 289 Oe at 59 GHz), underscoring the effectiveness of Zn-Ti co-substitution in suppressing magnetic losses. For practical self-biased circulator applications, the x = 0.1 composition offers the most well-rounded set of properties, as it simultaneously delivers the highest remanence ratio among all substituted samples at M_r/M_s = 0.87, a sufficiently large anisotropy field H_a = 11,930 Oe for millimeter-wave self-biasing, and a narrow FMR linewidth ΔH = 355 Oe at 67 GHz that compares favorably with most reported values for polycrystalline M-type hexaferrites.