Estimation of permeability tensor and dielectric permittivity of ferrites using a wave guide method under a dc magnetic field

Using a vector network analyzer equipped with a calibrated rectangular wave guide the electric permittivity and the element of the magnetic permeability tensor for Y3Fe5O12, ZnFe2O4 and NiFe2O4 are measured. The electric permittivity can be estimated from the body resonances ( = /2 ) if a sufficient long sample is used. The estimation of the magnetic permeability tensors’ parameters can be estimated comparing the experimental results with computer simulations using the magnetic properties of the materials as derived from the magnetic measurements.


Introduction
The role of magnetically soft ferrites with Neel temperature larger than 300 K is very important in microwave devices.Typical examples are circulators, phase shifters and isolators [1,2].Moreover it has been demonstrated that magnetized ferrites in patch antennas induce novel properties [3].The presence of the ferrites in the substrate of the antenna can tune the polarization of the radiated field as well as other characteristics of the antenna in a controllable way through an external magnetic field.To facilitate the design of novel devices knowledge of electrical permittivity and magnetic permeability in the microwave regime of the electromagnetic spectrum is necessary.For the measurement of , H P several techniques have been proposed [4].The majority of these concern the estimation of the electrical permittivity.Only a few concern the magnetic permeability.In a paramagnetic sample the magnetic permeability is considered as a scalar quantity and it can be estimated with the Nicolson Rosss method [4].The permeability of a magnetized ferromagnetic/ferrimagnetic sample, in the microwave regime becomes an asymmetric tensor.The reason for this behavior is resonance precession of the permanent magnetic moment due to the microwave rffield.The aim of this article is to present a new methodology for the estimation of the electrical permittivity and the parameters of the permeability tensor, using rectangular magnetized samples loaded inside a rectangular waveguide.

Theory
The electromagnetic behavior of a filled waveguide with a magnetized ferrite is generally very complex [6,7].A magnetically polarized material with a static magnetic M the saturation magnetization, J is the gyromagnetic ratio of the electron, 0 P is the permeability of the free space, and D a phenomenological parameter describing the losses.In an magnetized sample with infinity dimensions, depending on the propagation configuration, one can define three types of effective magnetic permeability: 1) propagation parallel to the external magnetic field, 0 P P P N r r , for right and left circularly polarized waves.2) Propagation perpendicular to the applied dc magnetic field (βA AH0), and rf-electric field parallel to the H0, 0 P P . 3)Propagation perpendicular to the applied dc magnetic field (βAH0), and rf-electric field perpendicular to the H0, 2) In a rectangular wave guide partially loaded with ferrite magnetically polarized along the y-axis in TE10 approximation the propagation constant E is given by the roots of the transcendental equation [8]:

Results
The refined with the so-called reverse spinel structure.All the Ni 2+ ions occupy the octahedral sites while the Fe ions are distributed both on octahedral and tetrahedral sites.On the other hand, the X-ray diffraction pattern of ZnFe2O4 can be refined with the normal spinel structure.The magnetic properties of the samples are estimated using a SQUID magnetometer.The standard TRL calibration procedure is used to the zero reference planes.The external magnetic field is applied using two rectangular Nd2Fe14B permanent magnets with energy product 100 MJ.The magnetic circuit is closed with a steel yoke.Fig. 3 shows the frequency variation of the real part of the effective permeability (Eq.1.2) for the three studied samples, in several dc magnetic fields, using the values for Ms estimated from magnetization measurements.The Lande factor is taken 2 g while the loss parameter   / 2 d nO ).In addition, narrow negative peaks about a-10dB are also present.These peaks most probably arise from partially filling of the wave guide cross section with the ferrite or from magnetostatic modes.When the width of the dielectric is less than the wave guide's width the body resonances are shifted in respect to the ones occurring for perfectly loaded case    Fig. 5 shows the variation with frequency for the amplitude of the scattering parameters |S11|, and |S21| for NiFe2O4 spinel measured for Hy=0, 1.2, 3 and 6 kOe.Simulations of the zero field data demand that for this material the relative permittivity must be / 12. .Finally, Fig. 6 shows the frequency dependence for |S11|, and |S21| scattering parameters, measured for Hy=0 and 6 kOe for ZnFe2O4 spinel.In order the reflectionless peaks to be reproduced the permittivity must be / 13. .The fact that not significant changes are observed between the measurements in zero and 6 kOe magnetic field is directly related with the small frequency variation of the effective permeability for the particular sample (see Fig. 2c).For this sample the variation of e P displays the expected resonance behaviour but due to the small saturation magnetization the deviation from the vacuum permeability is small.

Conclusions
The electric permittivity and the parameters of the magnetic permeability tensor are estimated for the ferrimagnetic samples Y3Fe5O12, ZnFe2O4 and NiFe2O4 with parallelepiped shape and using a vector network analyzer equipped with a calibrated rectangular wave guide.The electric permittivity can be estimated from the body resonances (݀ = ݊ߣ/2) if a sufficient long sample is used.The estimation of the magnetic permeability tensors' parameters can be estimated comparing the experimental results with computer simulations using the magnetic properties of the materials from the magnetic measurements.

Fig. 1 .
Fig. 1.Magnetic moment per g, versus external magnetic field, at T=300 K for Y3Fe5O12, NiFe2O4 and ZnFe2O4.Note the small saturation magnetization of ZnFe2O4.

Fig. 2
Fig.2 Rectangular wave guide used in the measurements.

Fig. 1
shows the magnetization curves of the Y3Fe5O12, NiFe2O4 and ZnFe2O4.All the samples are soft ferrimagnets.The coercive fields for the Y3Fe5O12 and NiFe2O4 spinel are less than 10 Oe, while for ZnFe2O4 approximately 100 Oe.Interestingly, the saturation magnetization is about 4 Obviously these values influence the frequency variation of the effective permeability.Theoretical simulations are performed using the microwave studio of CST package[5].The ferrite samples are inserted at the center of a rectangular precision wave guide 3/4×3/8 in 2 with length 128 mm (seeFig 2).The scattering parameters ij S are measured using a vector network analyser (Anritzu 37269B).

'
is selected to have appreciable coincidence among experimental and simulated scattering parameters (see below).Fig.4shows the frequency variation of the experimental amplitude of the scattering parameters in several dc- magnetic fields for the Y3Fe5O12.The magnetic field is applied parallel to the short dimension of the wave guide (y-axis).

Fig. 5 .
Fig. 5. Amplitude of the scattering coefficients |S11| and |S21| versus frequency for NiFe2O4 in several magnetic fields (H||yaxis).The sample's dimensions are 9.2×19.4×18.8mm 3 .This arises from the fact that the dispersion relation ( ) E Z is shifted.In the other case where f b b except the shifting of the body resonances, new modes are predicted [8].For nominal magnetic field Hy=2.5 kOe, a large part of 11 | ( )| S f has a value near zero and the corresponding 21 | ( )| S f takes values less than -40dB.This behaviour originates from the negative values of the effective permeability e P (see Fig.3).Above a certain frequency

G.
). Above a certain frequency 11 | ( )| S f displays the reflectionless negative peaks a fact which implies 0 e P ! .For higher nominal fields the frequency range, where 0 e P is located on the right side, outside the studied frequency range.Clearly, the measurements with nominal field Hy=6.7 kOe, the frequency range where 0 e P is above 15 GHz.In order Joint European Magnetic Symposia 2013 06005-p.3toestimate the electric permittivity and the parameters of e P we carry out theoretical simulations using the microwave studio of the CST package.Representative examples of the theoretical curves are shown in Fig.4with solid lines.Good agreement among theoretical and experimental curves, for zero field has been achieved when the relative electric permittivity for the Y3Fe5O12 is ' It should be noted that the matching of the reflectionless modes permit a very accurate estimation of / r H .For non-zero magnetic fields, in addition with ' 16.5 correction of the external magnetic field.

Fig. 6 .
Fig. 6.Scattering parameters |S11| and |S21| versus frequency for Hy=0 and 6 kOe of ZnFe2O4.The sample's dimensions are 8.05×17.67×21.33mm 3 .The dashed line represents the theoretical calculated values of |S11| and |S21| with 13.3 r H samples were prepared with standard solid state reaction.Stoichiometric amounts of raw materials Y2O3, Fe2O3, NiO and ZnO are thoroughly mixed and heated at 1425 o C for the Y3Fe5O12 and at 1200 o C, for NiFe2O4 and ZnFe2O4 with intermediate grindings.In the final step powders are pressed in rectangular bars of appropriate dimensions to adjust the rectangular waveguide and sintered for 24h.Rietveld refinements of the x-ray diffraction patterns have shown that all the samples are single phase materials with good crystallinity.The x-ray diffraction pattern of Y3Fe5O12 can be refined with garnet crystal structure.Similarly, NiFe2O4 has been successfully 7