Frequency dispersion model of the complex permeability of soft ferrites in the microwave frequency range

The complex permeability of Cu-doped nickel-zinc polycrystalline ferrites is strongly dependent on microstructure, particularly, on relative density ( 𝜙 ) and average grain size ( 𝐺 ). In this study, a mathematical model, able to fit the measured magnetic permeability spectra from 10 6 to 10 9 Hz, is proposed and validated for a width range of average grain sizes (3.40–23.15 µ m) and relative densities (0.83–0.96). To the authors’ knowledge, domain-wall motion and spin rotation contributions to magnetic permeability have been integrated jointly with the microstructure for the first time in the proposed model, highlighting the relative influence of each magnetizing mechanism and microstructure on the magnetic permeability at different angular frequencies.


Dependence of complex magnetic permeability on angular frequency
When an alternating current (AC) magnetic field (H) of angular frequency () is applied to the material, the magnetic permeability becomes a complex property with a real ( ′ ) and an imaginary part ( ′′ )  () =  ′ () −  ⋅  ′′ () (1)   where real part represents the material storage capacity of magnetic field whereas imaginary part represents losses and power dissipation.Magnetic susceptibility must also be defined as a complex property  () =  ′ () −  ⋅  ′′ () Permeability and susceptibility are related as follows ′′ () =  ′′ () 18 Accordingly, a magnetic susceptibility for domain-wall motion (  ) and a magnetic susceptibility for a gyromagnetic spin rotation (  ) can be defined, and Equations ( 3) and ( 4) can be rewritten as follows: ′′ () =  ′′  () +  ′′ DW () The frequency dispersion of Cu-doped Ni-Zn ferrites, in the angular frequency range studied, exhibits curves where real part is described as a relaxation curve while the imaginary part describes a curve passing through a maximum.Literature [4][5][6][7][12][13][14][16][17][18][19][20][21][22][23][24] reports several physicomathematical models defining angular frequency functions for  ′  (),  ′  (),  ′′  (), and  ′′  () in order to calculate and accurately reproduce the frequency dispersion from Equations ( 5) and (6). One of the ost extended model in polycrystalline ferrites was proposed by Nakamura, 13,18 who kept in mind the influence of spin rotation and domain-wall motion mechanisms on magnetic complex susceptibility, considering that the spin rotation component is of relaxation-type and that the domainwall motion component is of resonance type, both depending on the angular frequency, leading to the following equations: where   is the static susceptibility of the spin rotation,   is the static susceptibility of the domain-wall motion,   is the resonance frequency of the spin rotation,   is the resonance frequency of the domain-wall motion, and  is the damping factor of the domain-wall motion.Equations ( 7) and (8) allow determining the complex magnetic permeability at any angular frequency but for a sample with a specific microstructure.These equations do not take into account the influence of the microstructure on frequency dispersion.

Dependence of complex magnetic permeability on microstructure
Literature reports the strong influence of microstructure on complex magnetic permeability.5][26][27][28][29][30][31][32][33][34][35][36][37]39 However, very few models relating magnetic permeability to microstructure have been developed in the past, and all of them have been uniquely tested at a specific constant angular frequency.In this regard, it is worth mentioning the models proposed by Globus, 36 Rikukawa, 38 Johnson and Visser, 25 and Pankert. 40Authors have tested these models in polycrystalline ferrites and proven that they are not able to predict the influence of average grain size and porosity on magnetic permeability.Hence, models were modified, and the following relationship, derived from Pankert model, was validated with a good accuracy with the experimental imaginary part of complex permeability 2, 11 : where  ′′ is defined as follows: and  is defined from relative density (), which, in turn, is the ratio of the density to the theoretical density, as: In Equations ( 9) and (10),  ′′  is the imaginary partcomplex magnetic permeability spin rotation contribution,  ′′  is the imaginary part-complex magnetic permeability domain-wall motion contribution,   is the domainwall width,  is the average grain size, and  is the densification defined by Equation (11).Densification  is an effective concept when comparing systems of different initial porosities, 41 and  lim stands for the relative density below which the value of the magnetic permeability is virtually zero.This experimental value can be estimated by plotting relative density versus magnetic permeability and extrapolating magnetic permeability to zero. 2,11For the studied polycrystalline ferrite, a  lim value of 0.65 has been reported. 11uthors have satisfactorily tested Equation ( 9) at four specific angular frequencies (10 6 , 10 7 , 10 8 , and 10 9 Hz), determining for each of the angular frequencies studied the experimental values of the parameters  ′′  ,  ′′ DW ,  ′′ , and   . 11

Joint dependence of complex magnetic permeability on angular frequency and microstructure
Equations ( 7) and ( 8) show the influence of angular frequency on magnetic permeability whereas Equation ( 9) reveals the influence of microstructure at a specific angular frequency.Equations ( 8) and ( 9) can be combined to take into consideration simultaneously both angular frequency and microstructure, and since microstructure has the same influence on the real and imaginary parts of complex magnetic permeability, 25,38,40 the following generalized equations are proposed: where  ′ is constant, 40 and  ′′ follows the Equation (10).
Summarizing, the present paper aims to: • Test Equations ( 12) and ( 13) to verify their ability to predict complex magnetic permeability (both real and imaginary part) in a wide range of sintered microstructures and angular frequencies.• Quantify the contribution of each magnetism mechanism on the complex magnetic permeability: Spin rotation and domain-wall motion (terms of the Equations ( 12) and ( 13) containing   and   parameters, respectively).• Determine the spin rotation static susceptibility (  ), the domain-wall motion static susceptibility (  ), the resonance frequency of the spin rotation (  ), and the resonance frequency of the domain-wall motion ( DW ).All these parameters are intrinsic characteristics of the material, microstructure-independent, and constants at any angular frequency, allowing different materials to be directly compared.

EXPERIMENTAL PROCEDURE
Cu-doped Ni-Zn-polycrystalline-sintered ferrites are prepared as described in previous papers. 42,43Hereafter, a brief description of experimental procedure is given.
A polycrystalline ferrite, of chemical composition  0.12  0.23  0.65 ( 2  4 ), supplied by Fair-rite Products Corp. was used as starting powder.It consists of spray-dried granules, with an average size of 175 µm, made up of particles with an average size of 1−2 µm and narrow particlesize distribution.Real density value of 5380 kg⋅m −3 was determined by helium pycnometer.
Cylindrical and toroidal test specimens (3-mm thick and 19-mm external diameter; 6-mm internal diameter for the toroidal test specimens) were shaped by uniaxial dry pressing and sintered by the conventional air solid-state method in an electric laboratory furnace.
Six compaction pressures, (50, 75, 200, 150, 200, and 300 MPa) and 10 dwell sintering times (5, 10, 20, 30, 45, 60, 120, 300, 900, and 1800 min) were used to sinter 60 samples at 1100 • C covering a wide range of relative densities (0.83-0.96) and average grain sizes (3.40-23.15µm).Bulk densities of samples were measured according to Archimedes principle.Relative density () of each sample was calculated as the ratio between bulk density and real density of the starting powder (5380 kg⋅m −3 ), upon which porosity could be calculated as (1 − ).Scanning electron microscope (FEG-ESEM Quanta 200F) was used to observe the microstructure of ferrite specimens.Through an image analysis software, average grain size () was determined from grain size distribution of the thermal-etched surface cross-sectional area for each specimen.Experimental F I G U R E 1 Frequency dispersion spectra of complex magnetic permeability (real and imaginary part) of two specimens with different microstructure values of  and  for every specimen are shown in Table 1, together with its absolute error ε() and ε().Relative error for  is always lower than 1% whereas the maximum relative error for  is around 10%.Table 1 also shows the amplitude of grain size distribution  as the difference between  90 and  10 .Grain size distribution is, in all cases, monomodal and relatively narrow, although it becomes wider with increasing both sintering time and temperature.
The complex magnetic permeability (real  ′  and imaginary  ′′  part) was obtained in an Agilent E4991A RF impedance/material analyser in the 10 6 −10 9 Hz frequency range using an Agilent 16454A magnetic material test fixture.Each curve is made up of 800 experimental values in the aforementioned frequency range.
Experimental curves showing the frequency dispersion for  ′  (red solid line) and for  ′′  (blue solid line), for every specimen, can be found in Figures S1-S60.and etched cross-sectional surfaces of both specimens of Figure 1, where, particularly, different average grain size can be observed.5][46] Same evidence can be extended to all sixty specimens studied.The x-ray diffraction (XRD) pattern has also been verified to be the same for all 60 specimens. 47quations ( 12) and ( 13) allow to determine the values of the magnetic permeability as a function of the microstructure ( and ) and angular frequency.In order to use both equations to fit experimental frequency dispersion and to determine the characteristic material parameters, two additional considerations must be taken into account:

RESULTS AND DISCUSSION
(i)  ′ shall be considered constant 40 : (ii)  ′′ values, which were previously obtained by the authors for the four angular frequencies studied (10 6 , 10 7 , 10 8 , and 10 9 Hz), 11 have now been empirically fitted to an exponential curve of the following equation: Taking into account the last two conditions, Equations (12) and ( 13) become: A nonlinear least-squares method has been used to fit experimental data (see Figures S1-S60) to Equations ( 16) and ( 17), allowing the estimation of constants   ,  DW ,   ,  DW , , and  by minimizing the sum of squared residuals.Table 2 depicts these estimated values, which can be introduced in Equations ( 16) and ( 17) yielding to the following: As an illustrative example, Figures 3 and 4 show the calculated complex permeability using Equations ( 18) and ( 19) for a specimen with an average grain size of 15.50 µm and a relative density of 0.92 (the same information can be seen together in Figure S7), where the solid black line stands for the estimated values, and red and blue circles stand for the experimental values of real and imaginary part, respectively.Figures S1-S60 show the same representation for all 60 specimens studied in the range of angular frequencies tested.As can be observed, the calculated values are in good agreement with experimental data, being slightly better for those specimens with higher relative density and average grain size, which, indeed, are F I G U R E 3 Experimental and calculated values using Equations ( 18) and ( 19) of complex magnetic permeability -real part and calculated values of spin rotation and domain-wall motion contributions F I G U R E 4 Experimental and calculated values using Equations ( 18) and ( 19) of complex magnetic permeabilityimaginary part and calculated values of spin rotation and domain-wall motion contributions the more interesting specimens from a technological point of view.
To quantify the good agreement, the complex magnetic permeability-real and imaginary part-can be calculated from Equations ( 18) and ( 19) for the 800 points that make up each experimental curve.The representation of the values of the real and imaginary parts of the complex magnetic permeability calculated versus the experimental ones gives straight lines with a slope of 1 that pass through the origin of coordinate.The squared correlation coefficients ( 2 ) are shown in Table 1.The  2 values are very close to 1, except for specimens with relative density around 0.85.In any case,  2 is greater than 0.97, which confirms the good agreement between experimental and calculated values in the entire range of tested frequencies.
The accurately fitting of both real and imaginary part of the complex magnetic permeability strengthens the concept of a relaxation curve for the real part and reinforces the idea of a curve that passes through a maximum for the imaginary part.
Furthermore, to consider the contribution of spin rotation and domain-wall motion mechanisms to complex magnetic permeability, Equations ( 18) and (19) By way of example once again, Figures 3 and 4 (and Supplementary Fig. 1 to 60), show the contribution of every magnetizing mechanism (dot lines) to the complex magnetic permeability in the range of angular frequencies tested.Spin rotation contribution is higher than domain-wall contribution at low angular frequencies, as general rule.However, spin rotation contribution

Figure 1
Figure 1 depicts the frequency dispersion (real part -solid lines and imaginary part -dotted lines) for two specimens with the same chemical composition (that of the studied ferrite) but with different sintered microstructure.As shown in this figure, the real part of the magnetic complex permeability decreases with frequency (relaxation curve) while peaks in the imaginary part, highlighting the strong influence of this second with frequency dispersion.Figure 2 shows the scanning electron micrographs of polished Processing conditions and sintered microstructural properties of the Cu-doped Ni-Zn-polycrystalline ferrite specimens TA B L E 1 can be separated in two terms as follows: