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# Experimental Determination of Anisotropy Fields, First and Second Anisotropy Constants, and Remanent Magnetizations in Self-Polarized Zn–Ti and Co–Ti Equally Co-Substituted BaM Ferrites

Antoine Hoz, Jean-Luc Mattei, Alexis Chevalier

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Antoine Hoz, Jean-Luc Mattei, Alexis Chevalier. Experimental Determination of Anisotropy Fields, First and Second Anisotropy Constants, and Remanent Magnetizations in Self-Polarized Zn–Ti and Co–Ti Equally Co-Substituted BaM Ferrites. INTERMAG2023, May 2023, Sendai, France. pp.1-1, 10.1109/TMAG.2023.3282604. hal-04141402

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# Experimental Determination of Anisotropy Fields, First and Second Anisotropy Constants, and Remanent Magnetizations in Self-Polarized Zn–Ti and Co–Ti Equally Co-Substituted BaM Ferrites

Antoine Hoez¹, Jean-Luc Mattei¹, Alexis Chevalier¹

¹Lab-STICC, UMR CNRS 6285, Université de Bretagne Occidentale, Brest, FRANCE

M-type hexagonal barium ferrites Ba(MₓTiₓ)Fe₁₂-₂xO₁₉ (M = Co or Zn, x = 0–0.5) were synthesized by chemical topotactical reaction. Both randomly oriented and self-polarized samples were prepared. Experimental values of the anisotropy fields, first and second anisotropy constants, and remanent magnetizations were extracted by combining hysteresis loops and microwave measurements. A salient result of this study is that, compared with K₁, K₂ cannot be disregarded for these co-doped BaM hexaferrites, even at low doping levels. Results show that the BaM ferrites of composition Ba(CoxTiₓ)Fe₁₂-₂xO₁₉ are more suitable for microwave applications that operate at frequencies between 30 GHz and 40 GHz.

***Index Terms***— Co-substituted hexaferrites, Magnetocrystalline anisotropy, Remanent magnetization, Self-polarized hexaferrites

## I. INTRODUCTION

M-TYPE BARIUM hexaferrites (BaM) have long attracted attention for microwave and millimeter wave (MMW) applications [1], [2]. The interest of self-polarized BaM material properties is evident for the design of integrated and miniature MMW devices such as circulators, in which the external permanent magnet would be eliminated [3]. The modeling of MMW devices that use self-polarized materials requires, among other inputs, the values of the saturation magnetization M<sub>S</sub>, remanent magnetization M<sub>R</sub>, and magnetocrystalline anisotropy field H<sub>K</sub>. For unsubstituted BaM, H<sub>K</sub> = 1360 kA/m [4]. It was noted that lowering this value of H<sub>K</sub> might be desirable for new MMW devices [5]. Iron substitution that could allow H<sub>K</sub> lowering leads the easy axis of magnetization to become closer to the basal plane [4], this process being driven by the values of the anisotropy constants K<sub>1</sub> and K<sub>2</sub>. The present paper reports a study of Co–Ti and Zn–Ti equiatomic co-substitution of M-type hexagonal barium ferrites Ba(M<sub>x</sub>Ti<sub>x</sub>)Fe<sub>12-2x</sub>O<sub>19</sub> (BaM-MTi, with M = Co or Zn, and x = 0–0.5) with two objectives. Firstly, to adjust anisotropy field H<sub>K</sub> to meet different MMW application requirements (reduction in the value of H<sub>K</sub> in order to lower the operating frequency, while maintaining a value of M<sub>R</sub>/M<sub>S</sub> close to 0.8), and secondly to determine the first (K<sub>1</sub>) and second order (K<sub>2</sub>) anisotropy constants for each of these compositions. To do this a law of approach to saturation (LAS), that was developed with the first and second order anisotropy constants K<sub>1</sub> and K<sub>2</sub>, was used once the anisotropy field H<sub>K</sub> had first been determined from microwave experiments. The presented method for the determination of K<sub>1</sub> and K<sub>2</sub> from magnetic experiments on polycrystalline material is new. The reduced remanent magnetization (m<sub>R</sub> = M<sub>R</sub>/M<sub>S</sub>) is also reported. To the best of our knowledge, no previous published works have reported simultaneous studies of the variations of the anisotropy constants of H<sub>K</sub> and of m<sub>R</sub> on self-polarized BaM ferrites.

## II. PREPARATION OF MATERIALS AND EXPERIMENTAL DETAILS

### A. Synthesis of self-polarized and isotropic co-doped BaM materials

Chemical syntheses were made of two co-doped barium ferrites of composition Ba(MₓTiₓ)Fe₁₂-₂xO₁₉, where M represents Co or Zn, and with x = 0–0.5. Two sets of materials were prepared because of the different requirements in the experimental processes that were followed. The first set consisted in isotropic polycrystalline materials formed by randomly dispersed particles. Samples of this set are noted MTi-x-R, where R stands for random. The samples forming the second set are highly anisotropic, of the self-polarized type. Samples of this set are noted MTi-x-SP, where SP stands for self-polarized. They were prepared by controlled topotactic reaction [6]. The difference between the manufacturing of these two sets of materials lies exclusively in the nature, and consequently the geometric form, of the iron oxide that is used as a precursor: Hematite is used for the isotropic materials, goethite for the anisotropic materials. This is summarized in Table 1. Samples of both sets were of cylindrical shape (height 1.6 mm and diameter 6.8 mm). Details on the synthesis are as follows. Anisotropic polycrystalline samples of barium ferrites of BaM-MTi were fabricated from barium carbonate BaCO₃ (Acros Organics, Germany, 99%), titanium oxide TiO₂ (99.9%, Sigma-Aldrich), zinc oxide ZnO (99.9% Sigma-Aldrich), cobalt hydroxide Co(OH)₂ (99.9%, Sigma-Aldrich) and goethite FeO(OH) synthesized in the laboratory from iron III hexahydrate (Sigma-Aldrich, USA, 98%). The main key for a successful process in manufacturing BaM self-polarized materials is to use lath-shaped goethite crystallites, with a shape factor (thickness to length ratio) close to 0.15. For reasons of confidentiality, the process of goethite synthesis is not detailed. These components were carefully mixed using a manual mortar. The resulting mixtures were pressed into cylindrical pellets (7 mm in diameter and 5 mm thick) at a pressure of 320 MPa(uniaxial pressing). The direction of the applied pressure defines the c axis, perpendicular to the upper surface of the pellets. The pellets were then sintered at 1040°C, without steps. The goethite crystallites are in the form of laths, with a
crystallographic axis [001] oriented perpendicular to the largest face of the crystallite (Fig. 1a). The pressure applied during the molding step leads to a spontaneous preferential orientation of the goethite crystallites, with the [001] axes predominantly parallel to the direction of the applied pressure (Fig. 1c). Regarding the isotropic polycrystalline samples of BaM ferrites, the manufacturing process remained the same as for the anisotropic samples, as illustrated in Fig. 1c), except that spherical hematite crystallites were used, instead of goethite crystallites.

MTi-x-R materials were used for magnetic measurements (first magnetization curve, K₁ and K₂ values) and for microwave measurements (Hₖ values). MTi-x-SP were used to determine Mₛ/Mₘ from hysteresis loops.

## B. Experimental details

X-ray diffraction analysis was carried out on a Panalytical Empyrean (Cu Kα radiation). Rietveld refinements revealed that the insertion of the dopants into the BaM crystal structure was effective to within about 10%. Magnetic measurements were performed using a Vibrating Sample Magnetometer EZ9 from MicroSense, with a maximum applied field intensity of 1500kA/m. The broadband complex permeability of each isotropic sample was measured in the frequency band 10 GHz–60 GHz by the transmission/reflection method using a Rohde and Schwarz ZVA67 vector network analyzer. The sample was placed on the top of a high-frequency microstrip line to measure the transmission (S21) and reflection (S11) parameters. For each composition, the values of Hₖ were measured using the MTi-x-R set of samples (Table 1). With this objective, the anisotropy fields Hₖ were deduced from the resonant frequency fₘ determined by the maximum of the imaginary part μ'' of the intrinsic permeability of the samples from the MTi-x-R set. Actually, the relationship between Hₖ and fₘ is written as fₘ = γHₖ, where γ is the gyromagnetic ratio (γ = 35.12 10⁻³ MHz/A·m⁻¹).

<table><caption>TABLE I. FEATURES OF THE PRECURSORS AND TEXTURING OF THE OBTAINED BAM MATERIALS</caption><thead><tr><th>Chemical precursors</th><th>Barium<br>Carbonate</th><th>Hematite</th></tr></thead><tbody><tr><td>Chemical formula</td><td>BaCO<sub>3</sub></td><td>Fe<sup>2</sup>O<sub>3</sub></td></tr><tr><td>Shape</td><td>Spherical</td><td>Spherical</td></tr><tr><td>Texturing of BaFe<sub>12</sub>O<sub>19</sub> material<br>after pressing process</td><td colspan="2">No texturing. Isotropic material.<br>Random spatial distribution of the<br>BaM particles (Figure 2c)</td></tr></tbody></table><table><thead><tr><th>Chemical precursors</th><th>Barium<br>Carbonate</th><th>Goethite</th></tr></thead><tbody><tr><td>Chemical formula</td><td>BaCO<sub>3</sub></td><td>FeO(OH)</td></tr><tr><td>Shape</td><td>Spherical</td><td>Lath</td></tr><tr><td>Texturing of BaFe<sub>12</sub>O<sub>19</sub> material<br>after pressing process</td><td colspan="2">Texturing from BaM particles<br>stacking. Anisotropic material<br>(Figure 2c)</td></tr></tbody></table>

The X-ray diffraction patterns of the goethite before and after the pressing process confirm the preferential orientation of the particles: the [101] direction, which is closest to [001] among

those contributing to the diffraction pattern, is clearly the one that remains with high intensity.

<figure><img src="image_2.png" /><figcaption>Fig. 1. TEM images of (a) lath-shaped goethite crystallites and (b) spherical hematite crystallites used for the manufacture of anisotropic and isotropic BaM ferrites, respectively; (c) Schematic representation of the powder compaction process. In this illustration, the mixture is constituted by barium carbonate spheres (red color) and, in blue color, either hematite (isotropic material, bottom) nor goethite laths (anisotropic material, top). The alignment in the basal plane of goethite crystallites is caused by the uniaxial pressure.</figcaption></figure>

All other Bragg diffraction peaks that existed before compaction are extinguished (Figs. 2b and 2d). The SEM images show that the stacking of particles in the case of a sample made from goethite is clearly evident (Fig. 2c), in contrast with the random distribution of the particles elaborated from hematite (Fig. 2a).

Our objective was to determine the anisotropy constants K₁ and K₂ of each of the Ba(MₓTiₓ)Fe₁₂-₂ₓO₁₉ compositions. For this purpose, a saturation approach law (LAS) was used, together with the anisotropy field Hₖ value that had previously been determined from microwave experiments. MTi-x-R materials were used for the magnetic measurements (first magnetization curve, K₁ and K₂ values) and for the microwave measurements (Hₖ values). MTi-x-SP were solely used to determine Mₛ/Mₘ from their hysteresis loops.

## III. MAGNETIC AND MICROWAVE MEASUREMENTS

For the study of remanent magnetization, hysteresis loops were measured on all MTi-x-R samples, in directions parallel and perpendicular to their basal plane.

These cycles are completely similar, with mₛ = 0.5, demonstrating the isotropy of the materials. Whereas, on the whole MTi-x-SP samples, these hysteresis loops are significantly different, characteristic of anisotropic material (Fig. 3).
<figure><img src="image_3.png" /></figure>

<figure><img src="image_4.png" /></figure>

<figure><img src="image_5.png" /></figure>

<figure><img src="image_6.png" /><figcaption>Fig. 2. SEM images of samples constituted either by randomly dispersed BaM particles (a) or oriented BaM particles (c). Rietveld analysis of goethite before (b) and after (d) the compaction process, respectively.</figcaption></figure>

As an illustration, Fig. 4 shows the hysteresis cycles (in reduced magnetization) obtained from measurements in the $\vec{n}$ direction on the doped materials with $x = 0.1$, for the samples whose particles are either oriented or randomly dispersed. The fits of $M(1/H_i^2)$ in a strong field ($H_i = 1340-1500$ kA/m) are presented in Fig. 5, for $x = 0, 0.1, 0.3$, and $0.5$, for $Ba(Co_xTi_x)Fe_{12-2x}O_{19}$. They concern the first magnetization curve measured on materials with a random distribution of particles. A linear variation of $M(1/H_i^2)$ is observed in this field range.

<figure><img src="image_7.png" /><figcaption>Fig. 3. Hysteresis cycles of the material $BaFe_{12}O_{19}$, prepared either from hematite or goethite as iron oxide precursor. The former (circles) is typical of isotropic material, with similar measured magnetizations along or perpendicular to $\vec{n}$, while the latter (squares) shows a behaviour characteristic of anisotropic material.</figcaption></figure>

An illustration of the results obtained from measurements in the microwave range, allowing us to obtain the anisotropy field $H_K$ values, is given in Fig. 6, where the frequency evolution of the isotropic material $BaFe_{12}O_{19}$ is shown. The anisotropy field $H_K$ was deduced from the resonant frequency determined by the maximum of the imaginary part $\mu''$ of the intrinsic permeability, from the relationship $f_R = \gamma H_K$.

<figure><img src="image_8.png" /><figcaption>Fig. 4. Hysteresis cycles of the material $Ba(Co_0.1Ti_{0.1})Fe_{11.8}O_{19}$, depending on whether it consists of randomly oriented particles (black dots) or stacked particles (solid line); the red dotted curve is the first magnetization curve.</figcaption></figure>

The maximum of the imaginary $\mu''(f)$ occurs for $f_R = 46.6$ GHz, which corresponds to $H_K = 1330$ kA/m (i.e., 16.6 kOe). This result agrees with the value of the anisotropy field $H_K$ of BaM ferrites, which is close to 1360 kA/m (i.e., 17 kOe). The values of $H_K$ thus obtained for the doped hexaferrites are reported in Table II.

## IV. $K_1$ AND $K_2$ DETERMINATION. DISCUSSION.

The law of approach to saturation (LAS), which is widely used for the analysis of the magnetization curves of polycrystalline magnetic materials, can be used to estimate the magnetic anisotropy field $H_K$. For a polycrystalline magnetic material consisting of randomly oriented crystallites, the magnetization $M(H)$ tends towards its saturation value MS according to the empirical expression [7]–[9]:

$$M(H) = M_s(1-a/Hi - b/Hi^2) + \chi_p Hi \quad (1)$$
where $H_i = H_0-H_d$, $H_0$ is the applied field and $H_d$ is the demagnetizing field.

<figure><img src="image_9.png" /><figcaption>Fig. 5. $M(1/H_i^2)$ linear behavior, for some of the $Ba(Co_xTi_x)Fe_{12-2x}O_{19}$ compositions.</figcaption></figure>

<figure><img src="image_10.png" /><figcaption>Fig. 6. Complex permeability spectra for an isotropic $BaFe_{12}O_{19}$ sample.</figcaption></figure>

The term a/H, which results from the presence of inclusions and defects, must vanish at sufficiently high magnetic fields. The last term, $\chi_p$, is a small high-field susceptibility, called the paraprocess. It is due to high-field band splitting and can be ignored below moderately applied field intensities [8]. The b/H² term arises from the reorientation of magnetic moments when the anisotropy axis is misaligned with the applied field. It is directly related to the magnetocrystalline anisotropy (even to $H_K$ or to the first anisotropy constant $K_1$). Then, within a range of high enough internal fields, the following relation (2) is commonly used to derive $M_S$ and the anisotropy field for an isotropic distribution of magnetic moments which can change direction only by rotating against the magnetic anisotropy [10]:

$$M(H) = M_S(1-b/H_i^2) \quad (2)$$

For uniaxial systems such as BaM particles, the expression for b is directly related to the magnetocrystalline anisotropy by $b = -H_K^2/15$ [10]. The calculation of the first order anisotropy constant $K_1$ is often done from here, then using the relation [11]:

$$H_K = (2K_1 + 4K_2)/(\mu_0 M_S) \quad (3)$$

In situations where $K_1 \gg K_2$, this leads to $b = -4K_1^2/15$. However, this common assumption cannot be fulfilled for doped materials with $x>0$ because the condition $K_1 \gg K_2$ might not be satisfied even at low doping values (although this is often assumed to be the case). On the one hand, both constants can be

deduced from the Sucksmith–Thompson plot of H/M versus M², but this method was developed for single crystals and is not suitable for polycrystalline materials [12] as it leads to underestimated $K_1$ and overestimated $K_2$. It is, therefore, difficult to obtain the two anisotropy constants $K_1$ and $K_2$ of polycrystalline samples clearly. However, on the other hand, it can be shown that the LAS for randomly oriented crystallites with uniaxial anisotropy, and now considering the first and second anisotropy constants, can be written (where $M(H_i)$ is the component of the magnetization along the field $H_i$):

$$M(H_i) = M_S \left( 1 - \frac{4 \frac{21K_1^2 + 48K_1 K_2 + 32K_2^2}{(\mu_0^2 M_S^2)}}{315 H_1^2} \right) \quad (4)$$

Details of this calculation are given in Appendix A. From the measured first magnetization curves, the linear variation of M vs. $1/H_i^2$ was observed for all samples in the appropriate field range, after correction for the demagnetizing field. The determination coefficients $R^2$ in M vs. $1/H_i^2$ linear relations are in any case better than 0.999 (Fig. 5).

The variations of $K_1$ and $K_2$ as functions of x were determined from the measurements of $H_K$ using microwave experiments and its relationship with $K_1$ and $K_2$ (1), on one hand, and from the slope of straight line $M(H_i)$ against $1/H_i^2$ (2), on the other. $K_1(x)$ and $K_2(x)$ are shown fig. 7.

A previous study reported on comparable, although lower, $K_1$ and $K_2$ values for a co-doped $BaFe_{11.2}Co_{0.4}Ti_{0.4}O_{19}$ single crystal [13]: $1.033 \times 10^5$ J/m³ and $K_2 = 0.08 \times 10^5$ J/m³. The values of the constants $K_1$ and $K_2$ presented in [13] were obtained using the Sucksmith and Tompson method, dedicated to single crystals. The method we use is suitable for polycrystalline materials, and does not have the limitations of Sucksmith and Tompson's method [14]. We therefore believe that the values we present are reliable. It is clear that, compared with $K_1$, $K_2$ cannot be disregarded for co-doped (Co–Ti) and (Zn–Ti) BaM hexaferrites, even at low doping levels. The data in Table 1 show that both $M_R/M_S$ and $H_K$ decrease with increasing doping for the $Ba(M_xTi_x)Fe_{12-2x}O_{19}$ self-polarized samples. However, the decrease of $H_K$ is more rapid, and that of $M_R/M_S$ slower, for $Ba(Co_xTi_x)Fe_{12-2x}O_{19}$ than for $Ba(Zn_xTi_x)Fe_{12-2x}O_{19}$. In particular, $Ba(Co_xTi_x)Fe_{12-2x}O_{19}$ with $x = 0.3$ shows quite a high $M_R/M_S = 0.77$ value, together with $H_K = 11.6$ kOe, which corresponds to the resonance frequency $f_R = 32.5$ GHz. These features make barium hexaferrites of composition $Ba(Co_xTi_x)Fe_{12-2x}O_{19}$ more suitable for MMW applications.

<table><caption>TABLE II. REMANENT MAGNETIZATIONS AND ANISOTROPY FIELDS</caption><thead><tr><th></th><th colspan="2">M<sub>R</sub>/M<sub>S</sub></th><th colspan="2">H<sub>K</sub> (kA/m)</th></tr><tr><th>x</th><th>(ZnTi)<sub>x</sub></th><th>(CoTi)<sub>x</sub></th><th>(ZnTi)<sub>x</sub></th><th>(CoTi)<sub>x</sub></th></tr></thead><tbody><tr><td>0</td><td>0.92</td><td>0.92</td><td>1336</td><td>1336</td></tr><tr><td>0.1</td><td>0.84</td><td>0.86</td><td>1224</td><td>1288</td></tr><tr><td>0.2</td><td>0.77</td><td>0.82</td><td>1112</td><td>1208</td></tr><tr><td>0.3</td><td>0.73</td><td>0.80</td><td>1048</td><td>1088</td></tr><tr><td>0.4</td><td>0.72</td><td>0.77</td><td>968</td><td>928</td></tr><tr><td>0.5</td><td>0.70</td><td>0.75</td><td>888</td><td>824</td></tr></tbody></table>
<figure><img src="image_11.png" /><figcaption>Fig. 7. First and second order anisotropy constants plotted against the doping level x. Symbols in grey are data are extracted from [13].</figcaption></figure>

## APPENDIX A

We consider the magnetization of a ferromagnet in a narrow region around ferromagnetic saturation approaches to saturation via rotation process with a small angle between the magnetization and the applied field (Fig. A1).

<figure><img src="image_12.png" /><figcaption>Fig. A1. Spherical particle with uniaxial magnetocrystalline anisotropy.</figcaption></figure>

For a single-crystal particle of BaM, with uniaxial anisotropy, the magnetocrystalline energy variation for a saturation approach is written, at order 2: $W_K=K_1\sin^2\Theta+K_2\sin^4\Theta-\mu_0M_s\cos(\phi-\Theta)+1/2\mu_0N_dM_s^2$, where $\phi$ and $\Theta$ are the angles that the internal field $\vec{H}_i$ and the magnetization $\vec{M}_s$ make with the easy axis, respectively, $N_d$ being the shape anisotropy factor. The equilibrium position of the magnetization is deduced from the condition $dW/d\Theta=0$. Assuming moreover that in strong fields the condition $\phi-\Theta \to 0$ is verified, one easily obtain: $M_{//} = M_s(1 - 1/(\mu_0^2 M_s^2)) \cdot \frac{1}{H_i^2} \cdot (2K_1^2 \sin^2\phi \cos^2\phi + 4K_1 K_2 \sin^4\phi \cos^2\phi + 8K_2^2 \sin^6\phi \cos^2\phi)$. Then, in the case of a BaM polycrystal, where the axes of easy magnetization are randomly distributed, the component $\langle M_{//} \rangle$ of the magnetization along the field $H_i$ results from an average taken over the different orientations of the c axis writes: $\langle M_{//} \rangle = M_s(1 - 1/(\mu_0^2 M_s^2)) \cdot \frac{1}{H_i^2} \cdot (2K_1^2 \sin^2\phi \cos^2\phi + 4K_1 K_2 \sin^4\phi \cos^2\phi + 8K_2^2 \sin^6\phi \cos^2\phi)$, where the brackets indicate an average value made on the sphere of unit radius. We thus obtain: $\langle M_{//} \rangle = M_s(1 - 1/(\mu_0^2 M_s^2)) \cdot \frac{1}{H_i^2} \cdot \frac{4}{315} \cdot (21K_1^2 + 48K_1 K_2 + 32K_2^2)$.

## ACKNOWLEDGMENT

This work is partially supported by a public grant overseen by the French National Research Agency (ANR-20-ASTR-0010, CONTACT project).

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