Correlation between Low Field Microwave Power Absorption and Soft Magnetic Properties of Ferrites

Abstract

We describe studies on ferrites that use the Low Field Absorption (LFA) technique. These are made in soft magnetic polycrystalline ferrites magnetite, garnet iron, and nickel-zinc ferrite, where we show LFA spectra, highlighting their main characteristics and illustrating how magnetic phase transitions and magnetic anisotropy are detected employing these measurements. In this chapter, we also present a review of ferromagnetic resonance (FMR) studies in ferrites. Ferromagnetic materials exhibit various behaviors when subjected to ac magnetic fields, such as domain wall relaxation (DWR) and FMR. For instance, most ferrites generally find DWR at relatively low frequencies, below 10 MHz, while FMR must satisfy the Larmor equation. Microwave power absorption can be associated with magnetization processes as a function of dc fields, such as LFA and FMR. Microwave absorption centered at zero magnetic fields has been observed in various materials: superconductors, ferrites, and semiconductors. For ferrites, the LFA signal with the same phase of the ferromagnetic resonance follows the thermal behavior of the magnetization. For magnetically ordered materials, the LFA signal is evidence of the existence of a ferromagnetic phase, this signal is not present in the paramagnetic state, and it appears as the temperature is decreased below Curie temperature.

1. Introduction

Ferrites are a group of magnetic ceramic oxides containing the ferric ion Fe³⁺ in their structure and a divalent (M²⁺) or trivalent rare earth (R³⁺) cation. Accordingly, to their crystal structure, ferrites can be classified as spinels (MFe₂O₄), garnets (R₃Fe₅O₁₂), perovskites (RFeO₃), or as hexagonal structures related to the magnetoplumbite mineral (MO·6Fe₂O₃) [1, 2]. This diversity in composition and crystal structure provides a broad spectrum of magnetic and electrical properties which can be tuned and engineered for electronic and biomedical applications.

This chapter will focus on soft magnetic ferrites such as the spinel [3] and the garnet crystal structure [4, 5]. These ferrites are characterized by their high magnetic permeability, high electrical resistivity, and moderate permittivity from dc to GHz frequencies [6]. Soft ferrites are applied in electronic devices of high frequency [7] due to their ability to interact with dc and ac magnetic fields for transporting information or energy [6, 7] and by shielding electronic components from undesirable electromagnetic interference (EMI) [8].

The applicability of the soft ferrites is associated with the magnetic susceptibility ($\chi$) or its magnetic permeability (${\mu }_r=\mu /{\mu }_0=1+\chi$). As well as soft ferrite is anisotropic, losses, and dispersive media; their magnetic permeability is a complex quantity and depends on the temperature$\left(T\right)$, magnetic field frequency ($\omega$), strength and direction (H), and stress, where the characterization of $\mu \left(\omega \right)$ as a function of the frequency is fundamental for the development of applications based on soft magnetic materials. Also, the measurement techniques of $\mu \left(\omega \right)$ concerning magnetic fields, frequency, and temperature are valuable tools that provide a vision into the magnetic ordering, magnetic couplings, anisotropies, energy losses, and phase transitions in magnetic materials.

This chapter is dedicated to the microwave characterization of the soft magnetic ferrites magnetite (Fe₃O₄), nickel-zinc ferrite (NZFO: (Ni_x, Zn_1-x)Fe₂O₄), and yttrium iron garnet (YIG: Y₃Fe₅O₁₂). The focus of this chapter will be on the characterization of these systems and their magnetic anisotropies and phase transitions using the microwave power absorption techniques that historically are known as Ferromagnetic resonance (FMR) [9, 10], Low field absorption (LFA) [11] and magnetically modulated microwave absorption spectroscopy (MAMMAS) [12].

1.1 Soft magnetic materials under alternating magnetic fields

Ferromagnetic materials exhibit various behaviors when subjected to ac magnetic fields, ranging from domain wall relaxation (DWR) to ferromagnetic resonance (FMR). At radio frequency, in the 10³–10⁶ Hz range, the changes in magnetic permeability are mainly associated with the movement of domain walls (DW), where this process can be modeled to assume that DW moves like an elastic membrane driven by an externally applied ac field. In this model, it is assumed that the DW can be elastically deformed by the amount $x\left(t\right)$ in response to an ac field $H_0{e}^{-\mathit{i\omega t}}$; this process is described through the differential equation [1, 13]:

$$m\ddot{x}+\beta \dot{x}+\mathit{\alpha x}=2M_s{H}_0{e}^{-\mathit{i\omega t}}$$E1

Where $m$ is a domain wall effective mass per wall surface area, $\beta$ is a magnetic viscous damping factor, and $\alpha$ is a restoring constant, similar to a deadened harmonic oscillator. Eq. (1) can exhibit relaxation or resonance of domain walls depending on the ratios of constants $m,\alpha$, and $\beta .$ By defining the resonance ${\omega }_0=\sqrt{\alpha /m}$, and relaxation ${\omega }_r=\alpha /\beta$ frequencies, it is possible to explore the DW dynamics as the damping and restoring parameters vary.

Figure 1 shows the calculated complex inductance ($L^{\ast }\left(\omega \right)$) spectra for ferrite samples characterized by different ratios of relaxation to resonance frequencies. Inductance is directly proportional to magnetic permeability $L^{\ast }\left(\omega \right)=G_0{\mu }^{\ast }\left(\omega \right)$, where $G_0$ is a shape-dependent geometrical factor. This behavior goes from a relaxation-driven process toward a resonance-dominated one as the relaxation frequency becomes larger than the resonance one.

Figure 1.

Calculated complex inductance L^*(ω) spectra for ferrites with different ratios of relaxation to resonance frequencies.

1.2 Microwave power absorption techniques

The volumetric density of the microwave power absorbed by a magnetic sample is directly related to the electrodynamic properties of the material and is given as [14, 15]:

$$\frac{\mathit{dP}}{\mathit{dV}}=\frac{1}{2}\left({\sigma }^{\prime }{\left|E_{\mathit{ac}}\right|}^2+{\mathit{\omega \epsilon }}^{\prime \prime }{\left|E_{\mathit{ac}}\right|}^2+{\mathit{\omega \mu }}^{\prime \prime }{\left|H_{\mathit{ac}}\right|}^2\right)$$E2

Where $\omega$ is the angular frequency of the electromagnetic radiation, ${\epsilon }^{″}$ is the imaginary part of the electric permittivity, and ${\mu }^{″}$ is the imaginary part of the magnetic permeability. Each term at the right side of the equation corresponds to losses due to the Joule effect, dielectric, and magnetic mechanisms, respectively.

Microwave power absorption can be measured as a function of the radiation frequency ($\omega$) with the DC magnetic field ($H_{\mathit{DC}}$) constant; or as a function of $H_{\mathit{DC}}$ under a fixed microwave frequency. The first kind of measurement, broadband FMR [16, 17], requires a microstrip line or coplanar waveguide setting for transporting the microwave field. In this setup, the sample is mounted onto the waveguide, and the reflected and absorbed waves are analyzed while the system is kept under a fixed DC magnetic field $H_{\mathit{DC}}$.

In the second type of FMR setup [17, 18], the sample is mounted inside and in the center of a resonant cavity coupled to a microwave source through a magic-T bridge. The Microwave absorption is obtained by analyzing the reflected wave from the sample-loaded cavity due to a change in the quality factor (Q), and later it is guided to a crystal detector using a microwave circulator. These microwave measurements are performed under a fixed frequency while varying the DC magnetic field ($H_{\mathit{DC}}$). It is necessary to mention that as the resonant cavity and the waveguide are designed for specific microwave bands, e.g., X-band (8.8–9.8 GHz), they only allow for a small frequency variation, meaning that they need to be changed each time a different measuring frequency is required.

The kind of microwave absorption techniques employed in this chapter (FMR, LFA, and MAMMAS) were implemented with resonant cavities in the TE₀₁₁ mode [12, 17, 19], where the sample is placed at a maximum magnetic field ($H_{\mathit{ac}}$) and a minimum electric field ($E_{\mathit{ac}}$), see Figure 2. This setup holds that $\left|E_{\mathit{ac}}\right|<<\left|"H_{\mathit{ac}}\right|$, so the magnetic losses dominate over the dielectric losses. Also, for most non-metallic magnetic samples, the electrical conductivity is minimal; therefore, the Joule effect losses are minimal. So, the main contribution comes from magnetic losses, reflecting the high-frequency collective response of the ordered electron spins on the material.

Figure 2.

Schematic distribution of (a) magnetic and (b) electric field lines inside a cylindrical TE₀₁₁ microwave cavity. Adapted from [20].

Most of the results discussed in this chapter were obtained in an electron paramagnetic resonance (EPR) spectrometer JEOL JES-RES3X (see Figure 3). It is also necessary to mention that microwave absorption is modulated by a magnetic field ($H_{\mathit{mod}}$) of 100 kHz and is generated by Helmholtz coils on each side of the cavity. Therefore, the spectrometer signal is detected through the method of detection-homodyne with lock-in amplification, i.e., the output signal of the spectrometer ($S_{\mathit{FMR}}$) is proportional to the magnetic field derived from the microwave power absorption of the sample at a given microwave frequency ($\omega$), modulation field ($H_{\mathit{mod}}$) and temperature ($T$) conditions [21]:

Figure 3.

Block diagram for implementing FMR, LFA, and MAMMAS techniques in a JEOL JES-RES3X spectrometer at X-band. Adapted from [12, 20].

$$S_{\mathit{FMR}}={\left(\frac{\mathit{dP}}{\mathit{dH}}\right)}_{\omega ,H_{\mathit{mod}},T}$$E3

1.2.1 Ferromagnetic resonance (FMR)

The ferromagnetic resonance (FMR) phenomenon is the resonant absorption of microwave energy due to the Larmor precession of the magnetic moment in a material (Figure 4). The Larmor frequency gives FMR condition [9, 10, 22]:

Figure 4.

Schematic representation of the Larmor precession. A magnetic moment $\widehat{\mathit{m}}$ describes a precession cone around the magnetic field H₀ and absorbs the transverse microwave field h_x. Adapted from [9].

$${\omega }_0=\gamma {\mu }_0{H}_0$$E4

Here, $\gamma =\frac{g{\mu }_B}{\hslash }=1.76086\times {10}^{11}\frac{\mathrm{rad}}{\mathrm{s}\bullet \mathrm{T}}$ ($\gamma /2\pi =28.25$ GHz/T) is the gyromagnetic ratio of the electrons, $g$ is the Landé-factor of the electrons, and $H_0$ is the resonance field of the system.

The magnetization process at microwave frequencies can be modeled via the Landau- Lifshitz-Gilbert (LLG) equation [23]:

$$\frac{d\overrightarrow{M}}{\mathit{dt}}=-\gamma {\mu }_0\overrightarrow{M}\times {\overrightarrow{H}}_0+\frac{{\alpha }_G}{M}\left(\overrightarrow{M}\times \frac{d\overrightarrow{M}}{\mathit{dt}}\right)$$E5

Here, ${\alpha }_G$ is the dimensionless Gilbert damping parameter, which describes the relaxation of magnetization during Larmor precession.

LLG equation describes the dynamical response and relaxation of magnetization under given conditions of dc ($H_{\mathit{DC}}$) and ac ($H_{\mathit{ac}}$) fields. Because of the magnetic ordering, the effective field experienced by the magnetization is given by the superposition of the applied DC field and the internal field of the material:

$$H_0=H_{\mathit{DC}}+H_{\mathit{int}}$$E6

The internal field ($H_{\mathit{int}}$) represents the restoring torque effects experienced by the magnetic moment $\widehat{\mathit{m}}$ due to crystal anisotropy$\left(H_K\right)$, demagnetizing fields ($H_D$), exchange coupling interaction ($H_J$), and magnetoelastic effects ($H_{\lambda }$). Therefore, the changes in the resonance field are due to sample orientation, chemical composition, microstructure, and temperature, providing rich information about the internal field of magnetic materials.

Also, FMR experiments allow for obtaining dynamical information regarding the magnetic relaxation of materials via the analysis of the FMR linewidth.

It can be shown that, near the resonance field, the real (${\chi }^{\prime }$) and imaginary (${\chi }^{\prime \prime }$) components of the magnetic susceptibility can be approximately described by the Lorentzian resonance curves [22]:

$${\chi }^{\prime }\approx \left(\frac{\gamma M_s}{2{\alpha }_G\omega }\right)\frac{\beta }{1+{\beta }^2}$$E7

$${\chi }^{\prime \prime }\approx \left(\frac{\gamma M_s}{2{\alpha }_G\omega }\right)\frac{1}{1+{\beta }^2}$$E8

With the parameter $\beta =\frac{H-\left(\omega /\gamma \right)}{\mathit{\alpha \omega }}$ for FMR measurement at a constant frequency. ${\mathrm{\Delta }H}_{\mathit{\text{FWHM}}}$ of the imaginary part (${\chi }^{\prime \prime }$) is related to the Gilbert damping parameter by the relation [22]:

$${\mathrm{\Delta }H}_{\mathit{\text{FWHM}}}=\frac{2{\alpha }_G\omega }{\gamma }$$E9

Eqs. (7)–(9) indicate that a smaller Gilbert damping factor makes the microwave absorption peak value more intense and concentrated in a small field interval. Else, as ${\alpha }_G$ increases, the absorption takes place into a wider field interval with a lower peak value.

1.2.2 Low field absorption (LFA)

LFA technique refers to the microwave absorption observed around zero DC magnetic field within a typical field width of ${\mu }_0{H}_{\mathit{DC}}=\pm 100$mT. LFA measurements are performed in the typical microwave cavity setup for electron paramagnetic resonance (EPR) spectroscopy (Figure 3). EPR spectrometer is equipped with a zero-field cross-unit which compensates for the remanence of the electromagnet, then allows precise measurements at small field values near zero [11, 20].

The typical LFA signal may exhibit hysteresis, observed in superconducting and magnetically ordered materials [11, 12]. So, the LFA signal is associated with the absorption dynamics of microwaves for fluxoid tubes in High-T_c superconductors and the interaction of microwaves with magnetic moments in the magnetic domains for ferromagnetic materials [12, 24].

LFA signal only appears below the corresponding Curie temperature T_C, a signature of the collective spin ordering in magnetic materials [25, 26, 27, 28]. In magnetic materials, the hysteresis in the LFA signal has been associated with magnetization processes occurring in the magnetic domains of the non-saturated state [29]. Experiments on nickel samples [29] and amorphous cobalt-rich ribbons [30] also suggest that the LFA signal is heavily correlated with materials anisotropy and demagnetizing fields.

1.2.3 Magnetically modulated microwave absorption spectroscopy (MAMMAS)

Magnetically modulated microwave absorption spectroscopy (MAMMAS) [11, 12], or MFMMS (Magnetic field modulated microwave spectroscopy) as it is called in other works [15], is the study of the losses of the microwave power absorption as temperature function at a fixed magnetic field value ($H_{\mathit{DC}}$). So, the MAMMAS signal is the magnetic field derived from the microwave power absorption, measured at a fixed magnetic field ($H_{\mathit{DC}}$), as a temperature function:

$$\text{MAMMAS}={\left(\frac{\mathit{dP}}{\mathit{dH}}\right)}_{\omega ,H_{\mathit{DC}}}\left(T\right)$$E10

It is necessary to mention that the FMR and LFA spectra are measured at a fixed temperature. However, the MAMMAS signal can be measured under a continuous temperature scan. Therefore, the MAMMAS technique is ideal for studying phase transitions in magnetic materials [15, 20]. The high sensitivity of this technique to detect the magnetic phase transitions is associated with the changes in electromagnetic properties of the materials with long-range electronic ordering. As this ordering is destroyed by thermal energy, marked changes exist in the electrical conductivity, permittivity, and permeability of materials during a phase transition. Finally, according to Eq. (2), changes in these properties affect the microwave power absorption of the magnetic medium.

2. Microwave power absorption studies in soft magnetic ferrites

Microwave power absorption techniques like FMR, LFA, and MAMMAS have been applied for various studies about magnetic ordering, phase transitions, and anisotropy characterization on soft magnetic ferrites. Here, we present a survey on the studies carried out in magnetite, nickel-zinc ferrite (NZFO), and yttrium iron garnet (YIG) employing these techniques.

2.1 Magnetite (Fe₃O₄)

Magnetite (Fe₃O₄) is the natural form of iron oxide and is considered a typical ferrite material. It crystallizes in the inverse spinel structure, whose general formula is represented by AB₂O₄. This structure belongs to the high symmetry space group $\mathit{Fd}\overline{3}m$ (No. 227), and the unit cell has a lattice parameter of $a_0=8.3967\mathring{A}$ [31]. The spinel structure is formed by a closed cubic packing of oxygen anions with cations distributed between tetrahedral or A-sites and octahedral or B-sites. In a normal spinel, the divalent cation occupies the A-sites, while trivalent cations occupy octahedral B-sites as represented by: (A²⁺) [B³⁺₂]O₄. However, the magnetite is an inverse spinel, where the A-sites are occupied by half of the ferric ions while the B-sites contain the other half of Fe³⁺ ions and the ferrous Fe²⁺ cations. Then, cationic distribution in magnetite corresponds to (Fe³⁺) [Fe³⁺Fe²⁺]O₄.

Magnetite has a low-temperature phase transition known as the Verwey transition [32, 33, 34, 35], where the magnetite has a low crystal symmetry, i.e., goes changing from the cubic $\mathit{Fd}\overline{3}m$ space group to a monoclinic structure (space-group $\mathit{Cc}$) when cooling. This phase transition is also accompanied by an ordering change phenomenon of the ferric and ferrous ions in octahedral sites, and it is manifested as a sharp increase in the electrical resistivity of the magnetite at $T=T_V\cong 122$ K [33, 36]. In this transition, by diminishing the temperature, the metallic conductive cubic phase changes toward a monoclinic insulator phase. This low symmetry phase (Figure 5) is described by a superstructure where the 8 tetrahedra and 16 octahedral sites become non-equivalent among them [36].

Figure 5.

Crystal structures of magnetite in the a) high symmetry phase ($\mathrm{Fd}\overline{3}\mathrm{m}$) for $\mathrm{T}>{\mathrm{T}}_{\mathrm{V}}$, and b) in the low symmetry monoclinic phase $\left(\mathrm{Cc}\right)$ for $\mathrm{T}<{\mathrm{T}}_{\mathrm{V}}$.

The Verwey transition has been studied by different techniques [37], such as electrical resistivity, magnetoresistance [38], infrared and Raman spectroscopes [39], X-Ray diffraction (XRD), magnetometry, susceptibility, Mössbauer spectroscopy, calorimetric, nuclear magnetic resonance (NMR), and X-ray resonant absorption [32, 33, 37]. Also, FMR, LFA, and MAMMAS techniques have been applied to study this phase transition [24, 40].

In an earlier FMR study [41] on magnetite single crystals at X-band and K-band (18–26.5 GHz), the disappearance of the FMR signal has been observed at $T=T_V$ for X-band measurements. Furthermore, it could be interpreted as an increased Larmor frequency far above the X-band frequency range. This increase in FMR frequency is attributed to the larger anisotropy change of the monoclinic phase, which is added to the internal field in the magnetite (Eq. (6)).

More recently, the broadband FMR technique was used for studying the Verwey transition in an epitaxial magnetite film grown on MgGa₂O₄ (001) substrates [42]. FMR linewidth and the Gilbert damping showed an abrupt increase below the T_V = 110 K, which is consistent with the expected crystal anisotropy change in the [010] direction.

FMR experiments were carried out in magnetite nanopowders, see Figure 6, following the Verwey transition through the changes in the resonance field ($H_r$) and peak-to-peak linewidth ($∆H_{\mathit{pp}}$) in these spectra [40], where $∆H_{\mathit{pp}}$and $H_r$ have a minimum and an inflection point near the T_V (Figure 7), respectively.

Figure 6.

FMR spectra in magnetite nanopowders for selected temperatures. Adapted from [40].

Figure 7.

Temperature dependence of the resonance field (H_r) and peak-to-peak linewidth ($∆{\mathrm{H}}_{\mathrm{pp}}$) in magnetite nanopowders. Adapted from [40].

As is known, in the limit of independent, non-interacting particles, the FMR linewidth is proportional to the anisotropy field, i.e., $∆H_{\mathit{pp}}\propto H_K$ [43]. So, the minimum in the FMR linewidth occurring before the Verwey transition can be associated with the isotropic point ($T_{\mathit{iso}}\approx 130$ K [33]) in the magnetocrystalline anisotropy field for the cubic phase of magnetite, where $K_1$ passes through zero and changes to a positive sign for $T_V<T<T_{\mathit{iso}}$(Figure 8). In this way, $\mathit{\text{the}}∆H_{\mathit{pp}}$ minimum occurs due to the nearly zero crystalline anisotropy in magnetite before the Verwey transition, and the residual linewidth reflects the contribution of crystalline defects, electric losses, and dipolar coupling in the nanopowders.

Figure 8.

Temperature dependence of (a) the saturation magnetization (filled circles) and (b) the lead magneto term of the magnetocrystalline anisotropy constant (K₁-filled squares). Data were interpolated from reference [33].

LFA and MAMMAS techniques have been applied for studying the Verwey transition in bulk and nanopowders magnetite samples. Figure 9a shows LFA spectra in bulk magnetite samples at several temperatures. The linewidth ($∆H_{\mathit{LFA}}$) and the enclosed area ($A_{\mathit{LFA}})$ of the LFA signal showed a very similar temperature dependence. $∆H_{\mathit{LFA}}$ showed a minimum of 126 K, which was associated with the minimum in anisotropy. On further cooling, $∆H_{\mathit{LFA}}$ begins to grow again due to the increase in anisotropy of the monoclinic phase.

Figure 9.

LFA spectra in (a) bulk and (b) nanopowders magnetite at different temperatures. Adapted from [24, 40], respectively.

In nanoparticles [40], LFA spectra can be described by a superposition of two components, one with hysteresis convoluted with a linear one. As is shown in Figure 9b, the slope of the linear component increases with temperature, showing a significant step at T > 137 K, which has been associated with the onset of the change in electron dynamics in B-sites.

MAMMAS spectra for bulk [24] and nanopowders [40] for magnetite samples are shown in Figure 10a and b, respectively. In both MAMMAS spectra, an increment in the microwave absorption is observed, due to an increment in the absorbing centers’ quantity and which produces an increase in the magnetoconductance of the samples, until reaching a maximum value. Later, the MAMMAS spectra diminish, and this behavior is associated with a decrease in the quantity of absorbing centers by the process of antiparallel spin alignment. It is necessary to mention that in the bulk sample, MAMMAS response increases to low temperatures due to an increase of the magnetocrystalline anisotropy in the magnetite monoclinic phase.

Figure 10.

MAMMAS response for (a) bulk and (b) nanopowders magnetite samples. Adapted from [24, 40], respectively.

Wampler et al. [15] used the MAMMAS technique for studying the Verwey transition in magnetite epitaxial film growth onto MgO (001) substrates. They also provided electromagnetic modeling to explain the changes in microwave absorption for magnetite epitaxial films as a temperature function. In their model, the changes in MAMMAS spectra (renamed as magnetic field modulated microwave spectroscopy, MFMMS) are associated with changes in the magnetic moment and the magnetoresistive behavior of magnetite. MAMMAS measurements were performed under different DC magnetic field (H_DC) values in their work, producing a family of MAMMAS curves. These responses were mainly associated with changes in the magnetic moment field derivative and then with the differential susceptibility ($\chi =\frac{\partial M}{\partial H}$), where the peak feature was mainly due to the magnetoresistive effect in magnetite.

2.2 Nickel-zinc ferrites (Ni_1-x,Zn_x)Fe₂O₄

Nickel-zinc ferrite (NZFO: (Ni_1-x,Zn_x)Fe₂O₄) is a solid-state solution whose final members are the nickel ferrite (NFO), a ferrimagnetic inverse spinel (↓Fe³⁺) [↑Fe³⁺↑Ni²⁺]O²⁻₄; and the zinc ferrite (ZFO), which is a direct spinel with an antiferromagnetic order at low temperatures (Zn²⁺) [↑Fe³⁺↓Fe³⁺]O²⁻₄ [1, 44, 45]. The resulting mixed spinel ferrites have a complex cation distribution represented by (Fe³⁺_1-x Zn²⁺_x) [Fe³⁺_1 + xNi²⁺_1-x]O²⁻₄. Having a higher electrical resistivity (10⁶Ω^.m), NZFO is applied in electronic devices at frequencies between 1 and 500 MHz [2].

The magnetic properties of NZFO can be finely tuned by changing the Zn²⁺ content, as shown in Figure 11. Being diamagnetic, when Zn²⁺content increases, the magnetic moment in the A-sites is diluted. Also, the A-B superexchange coupling interaction between magnetic ions in A- and B-sites is weakened with increasing x, and it causes a non-collinear magnetic ordering. This magnetic order type is known as Yaffet-Kittel (YK), which, as a first approximation, can be regarded as a triangular ordering of magnetic moments between one A-site and two adjacent B-sites [46]. The YK angle increases with x until it reaches 90° for ZFO. So, in magnetization experiments for NZFO, an initial increase up x = 0.4–0.5, was observed while M_s decreases for larger Zn content. On the other hand, the absolute value of the first anisotropy constant decays continuously with x because Ni²⁺ is the main source of crystalline anisotropy in this system. Finally, NZFO with x > 0.8 are paramagnetic at room temperature.

Figure 11.

The variation of the saturated magnetization (M_s– Red circles) and the absolute value of the magnetocrystalline anisotropy constant (K₁.- blue squares) for NZFO as a function of the zinc content (x). With data from [1].

Figure 12 shows the extended field microwave absorption (EFMA) measurement. LFA signals more FMR spectrum for bulk powders (Figure 12a) and nanocrystalline thin film (Figure 12b) in NZFO samples with x = 0.65 [47]. The presence of the FMR signal at high fields and hysteresis around zero applied field due to LFMA are observed. For the bulk powders and thin film of NZFO, the resonance fields are $H_r=2320\mathrm{Oe}$ and $H_r=2480\mathrm{Oe}$, respectively, where this difference may be attributed to the nanocrystalline microstructure of the film. Also, the larger FMR linewidth can be due to the stress induced by the interphase between the ferrite and substrate in the thin film.

Figure 12.

The extended field microwave absorption in NZFO with x = 0.65 for (a) bulk powders and (b) thin film samples, where LFA signal is evident around H = 0 Oe. Adapted from [47].

FMR, LFMA, and MAMMAS studies have also been reported for sintered NZFO samples with x = 0.65 [27, 28]. The microwave absorption dynamics in these samples were interpreted in terms of the temperature dependence of the YK angle at low temperatures [27], while on heating, the changes were due to the Curie transition at T_C = 430 K [28].

In a polycrystalline NZFO sample, the FMR spectrum showed the ferrimagnetic to paramagnetic transition at T_C = 430 K, where the shift in resonance field toward the free electron value is because the internal field vanishes, see Figure 13a. Also, $∆H_{\mathit{pp}}$ decreased with the temperature, Figure 13b, where a kink in the linewidth versus temperature curve was observed at T_C and an inflection point at T_Y = 240 K.

Figure 13.

Temperature dependence of (a) H_res and (b) ΔH_pp in the 200–440 K temperature range for the polycrystalline NZFO sample. Adapted from [20].

On cooling the polycrystalline NZFO sample, MAMMAS response decreased, with a minimum at T_m = 240 K, followed by an increase to low temperature. By correlating the microwave absorption on NZFO with the energy changes due to YK ordering, it was possible to reproduce the main features of the MAMMAS spectrum [48]. The changes in magnetic permeability produce the MAMMAS signal due to the thermal evolution of exchange coupling in the NZFO sample, see Figure 14.

Figure 14.

MAMMAS response for the polycrystalline NZFO sample. Adapted from [20].

On the other hand, the LFA spectrum showed the development of a second component with an opposed phase at T < 239 K, see Figure 15 (up). The onset of this second mode in the LFA spectrum coincides quite well with the T_m = 240 K value observed in MAMMAS measurements. Finally, on heating, the LFMA spectrum vanishes for T > T_C = 430 K, see Figure 15 (down).

Figure 15.

LFA spectra of the polycrystalline NZFO sample in the 362–408 K (down) and 208–261 K (up) temperature ranges. Adapted from [20].

The study of microwave absorption processes at low magnetic fields is also crucial for developing some applications of soft ferrites for those applications that are expected to operate magnetic devices under small or zero applied fields. Lutsev and Shutkevich [49] reported broadband FMR measurements at 50 MHz-4.0 GHz frequencies in Mn-Zn and Ni-Zn ferrite nanocomposite films. Under these conditions, FMR absorption appears at H_DC < 500 G, and a significant superposition of LFA and FMR signals is expected. By manipulating the initial demagnetized state with a triangular ac field of low frequency (1 Hz), it was possible to change the absorption levels of the ferrite nanocomposites. This effect was due to the narrowing of the distribution of nanoparticle orientation due to the action of the triangular ac field.

2.3 Yttrium iron garnet (YIG)

YIG (Y₃Fe₅O₁₂) is a synthetic magnetic oxide that was independently discovered by Bertaut and Forrat [50], Pauthenet [51], and Geller and Gilleo [52] between 1956 and 1957. YIG adopts the garnet crystal structure, which belongs to the high symmetry space group $\mathit{Ia}\overline{3}d$ (No. 230). Its large unit cell ($a_0=12.376\mathrm{\AA }$) is described as a bcc arrangement of three kinds of polyhedral units, named octahedral (16a), tetrahedral (24d), and dodecahedral (24c), as represented in Figure 16. While the large dodecahedral site accommodates the Y³⁺ ion, the ferric ions distribute between octahedral and tetrahedral sites in a 2:3 ratio.

Figure 16.

Schematic representation of the crystal structure for yttrium iron garnet (YIG), where it is shown the coordination polyhedral for octahedral (green), tetrahedral (magenta), and dodecahedral (yellow) sites.

YIG has remarkable magnetic, electric, and magnetic optical properties, which are the basis for its applications in microwave devices for radar and telecommunications [5]. Also, YIG is a soft magnetic material with a low magnetocrystalline anisotropy with cubic anisotropy constants $K_1=-610$ J/m³ and $K_2=-$ 26 J/m³ at 300 K [53]. This behavior is due to the ferric ions, which have a spectroscopic term ⁶S meaning that they have a minimum spin-lattice coupling. YIG is also a good insulator with $\rho \approx {10}^{10}\mathrm{\Omega }\bullet \mathrm{m}$ at 300 K [54] because it only contains trivalent cations, avoiding electrical conduction via an electron hopping mechanism. Its small anisotropy, along with the very high electrical resistivity, gives single crystal YIG the lowest known Gilbert damping factor, being as low as ${\alpha }_G=1.3\times {10}^{-4}$; this manifest with the smallest know FMR linewidth (FWHM) of $∆H=0.52\mathrm{Oe}$ (0.052 mT) reported by LeCraw (Figure 17) [55].

Figure 17.

FMR spectrum for a single crystal YIG sphere measured at 9.3 GHz with H_DC applied parallel to the [100] axis. (a) Imaginary permeability representation with data interpolated from LeCraw [55]. (b) Numerically calculated field derivative. The linewidth in (a) is $∆\mathrm{H}=520\mathrm{mOe}$, equivalent to $∆{\mathrm{H}}_{\mathrm{pp}}=130\mathrm{mOe}$ in the dP/dH representation.

Since their discovery, YIG has been one of the most widely studied materials under the FMR technique. Experiments for measuring its linewidth and damping [56, 57], the power saturation effects, the excitation of spin waves [5, 58, 59, 60], and spin pumping [61], and spin-torque transference [62] have been reported.

Although the single crystals and the micrometer-thick epitaxial films have smaller values of linewidth, this parameter is extremely sensitive to microstructure, pores, and magnetic impurities. YIG has been used as a model for studying the FMR relaxation mechanisms in magnetic materials. The difference in FMR properties between single crystals and nanocrystalline YIG materials can be huge, while epitaxial thin films growth onto GGG (100) substrates can exhibit linewidth and damping with low values as $∆H=1.2-2.0\mathrm{Oe}$ and ${\alpha }_G=2.2\times {10}^{-4}$ [63]; the nanocrystalline films growth onto Si (100) can have linewidths as large as $∆H=100-400$ Oe, corresponding to Gilbert damping factors of about${\alpha }_G\sim 2\times {10}^{-2}-1\times {10}^{-1}$ [64].

LFA signal in YIG samples has a much lower amplitude, and its linewidth is smaller than in the FMR spectra ($∆H_{\mathit{pp}}^{\mathit{LFA}}<∆H_{\mathit{pp}}^{\mathit{FMR}}$). The recording of LFA signals in YIG samples usually needs higher powers and gain levels. That is the consequence of the weaker magnetocrystalline anisotropy of YIG as compared with spinel ferrites, so the initial magnetization processes involved in LFA hysteresis play a minor role in the softer garnet ferrite.

A comparative study of the microwave absorption between micropowders and nanopowders in the YIG samples was presented by Sanchez MH et al. [65], where the representative FMR spectra are shown in Figure 18. The nanopowders in YIG samples show evidence of superparamagnetic (SPM) behavior, which is evidenced by vibrant sample magnetometry (VSM); also, by their nearly constant resonance field because it does not change with the temperature, and it is centered at H_r = 3330 Gauss.

Figure 18.

FMR spectra for (a) micro powders and (b) nanopowder of the YIG samples at 300 K, and their respective LFA signals are shown in the insets. Adapted from [65].

In contrast, the micropowders in the YIG particles exhibit magnetization curves with hysteresis, and a much more complex thermal dependence of their FMR parameters, as expected for magnetic ordering. H_r in nanopowders increases with temperature due to thermal energy weakening of the internal field in the YIG sample. Also, the linewidth decreases with temperature because of magnetocrystalline anisotropy and long-range dipolar interactions, which are proportional to M_s.

As stated before, the LFA signal is a signature of magnetic order in the materials, as we show by the contrast between the FMR and LFA absorption spectra in YIG particles of micrometric and nanometric dimensions [65]. When both LFMA signals are compared, see the insets in Figure 18a (up) and 18b (down) found that the magnetically ordered micrometer-sized YIG particles have an LFA signal with evident hysteresis around zero magnetic fields. Conversely, YIG nanoparticles only exhibited a linear LFMA response with no hysteresis and just a constant slope, which slowly decreases with temperature in coherence with their SPM nature.

3. Conclusions

Microwave power absorption techniques like FMR, LFA, and MAMMAS are useful for characterizing magnetic ordering, couplings, and anisotropies in soft magnetic ferrites.

In magnetically ordered materials, the presence of the LFA signal is considered a signature of the onset of magnetic ordering. The magnetization processes in unsaturated magnets produce hysteresis on the LFA spectrum. So LFA measurements are useful for detecting magnetic phase transitions and following magnetic anisotropies.

MAMMAS signal is originated from the magnetic field dependence of conduction and magnetization process, while the electric polarization process plays a minor role whenever a cylindrical microwave cavity is implemented.

Acknowledgments

DGAPA-UNAM financially supported this work through the grant PAPIIT No. IN106621, and J.F.Barrón thanks DGAPA-UNAM for the postdoc grant.

Conflict of interest

The authors declare no conflict of interest.