Decomposing the permeability spectra of nanocrystalline finemet core

The initial permeability, μ_i, and its variation with frequency have been studied extensively in nanocrystalline soft magnetic materials by many workers.^1–6 In metallic samples the contribution to the frequency dependence of μ_i arises both from the eddy current losses and from the magnetization mechanisms described by domain rotation (DR) and domain wall (DW) bulging and displacement. Only a few attempts^7–10 have been made so far to estimate these contributions from the experimental data on μ_i. In the case of ferrites, it is obvious that eddy current can be neglected and only the DR and DW contributions are discussed.¹¹ Nevertheless, many publications neglect the eddy current contribution in metallic ribbon or powder cores and consider only the magnetization relaxation spectra of Debye type and its versions considering the distribution of the relaxation time.¹⁰ In the present work we simulate the experimental data considering both the eddy current and relevant magnetization mechanisms as a function of the amplitude of exciting field for a nanocrystalline Finemet core with Round type hysteresis loop.

The ribbon is assimilated to an infinite plane where the internal magnetic field can be calculated using the Maxwell equations.¹² The complex permeability is given by:

where μ_s is the static relative permeability, σ is the electric conductivity and t the thickness.

Decomposing in real and imaginary part, we get the formulas used for simulation of eddy current contribution:

and

is the skin depth.

The maximum of the imaginary part, μ”, correspond to the frequency where the skin depth is half of the ribbon thickness. This frequency limit for ferromagnetic metals with relatively high permeabilities (between 1000 and 100 000) and low resistivity (between 10-150 microohm.cm) correspond to audio frequency range (between 1 kHz and several 100 kHz). Whereas for the soft Mn-Zn ferrites the resistivity is 7-8 orders of magnitude larger and correspondingly the frequency limit is 4 orders of magnitude larger (above 100 MHz) than the limit for ferromagnetic metals of similar sizes. This is why the eddy current contribution can be neglected for ferrites but it is essential for ferromagnetic metals even for soft magnetic composites made of ferromagnetic metallic powders, where the local resistivity in powders is low although the overall resistivity of the composite can be very large. It is worth mentioning that in the case of the eddy current modelled complex permeability for metals the imaginary part never becomes higher than the real part. In practice, however the high frequency branch of the imaginary part, in general intersect and surpass the real part. Therefore, the contributions from the magnetization mechanisms should also be considered.

Considering the domain wall (DW) having an effective mass (Döring¹³) due to the angular momenta of spins forming the wall, the small-amplitude periodic motion of the wall can be taken as a damped harmonic oscillator. Equation of motion for a unit area of 180^o wall for small displacements x from the equilibrium will be:

where m is the domain wall effective mass, x is the displacement of the wall, β is the damping coefficient and α is the stiffness parameter. The stiffness is a structure sensitive parameter; whereas the mass and the damping are structure insensitive, depend on the magnetic characteristics of the material only. J_s is the saturation polarization and determines the pressure on the wall. Looking for the solution of differential equation as x = x_o·exp(ωt), the permeability will be proportional to 2J_sx/H:

where 4J_s²/α is the static permeability, ω²_DW = α/m and ω_c = α/β.

This is a Lorentz type equation and correspond to the resonant behavior of permeability characterized by a resonance frequency, f_DW, and a damping factor dependent frequency, f_c.

Decomposing in real and imaginary part we get the formulas used for simulation of DW contributions:

Due to the extreme small effective mass of DW in nanocrystalline alloys the frequency f_DW is well above the experimental frequency range (f/f_DW ∼ 0), consequently the above formulas simplify to the usual Debye type equations valid in low frequency region.

The spin rotation within the domains evolves exponentially in time with a relaxation time ∼1/f_DR after applying a step signal. The damping is described by a Debye type equation:

where f_DR is usually higher than the f_c describing the DW relaxation.

Decomposing in real and imaginary part we get the formulas used for simulation of DR contributions:

The ingot of Finemet type alloy with a composition of Fe₇₃Si₁₆B₇Nb₃Cu₁ was prepared from pure elements by induction melting under Argon using water cooled copper mold. A homemade equipment was used to prepare amorphous ribbons by planar flow casting in air on Cu(Zr) disc (diameter 30 cm) applying a peripheral speed of 40 m/s. The ribbon width was 4mm and the thickness 20 μm. The amorphous nature of the precursor material was checked by x-ray diffraction. Cores with outer diameter of 25 mm and inner diameter of 17 mm were prepared by tight winding of the amorphous ribbon. A normal heat treatment without magnetic field was applied under Argon for 1h at 823 K in a vertical tubular furnace. To avoid the induced anisotropy in self-magnetic field of the magnetic domains, the samples were withdrawn after 1 h annealing into the cold zone of the furnace.

The permeability spectra were measured using the Agilent RLC meter type 4294 A measuring the series equivalent inductance (Ls) and resistance (Rs) data for different exciting field amplitudes in the frequency range of 40-110 MHz. We tried to keep constant the amplitude of the exciting AC current during the frequency sweep by applying a one turn for the inductance and by selecting the latest available Agilent instrument (type 4294A) with superior current supply properties compared to the earlier editions (type 4274 A and 4285 A). It is worth mentioning that the importance of domain wall contribution is decreasing rapidly towards high frequencies. Its contribution is present at low frequencies (below about 10 kHz) where the constant amplitude can be provided by the instrument. The permeability components were determined by the usual relations:

where L and R are the measured inductance and resistance of the sample, L_o and R_o are those of empty solenoid.

The quasi-static hysteresis loop was measured at f = 0.05 Hz and is represented in Fig. 1. The coercive field was found to depend on frequency even in this low frequency range where the drift of the integrator can hardly be avoided. As a reference value we take the Hc = 1.2 A/m obtained at 0.05 Hz.

The amplitude of the exciting field, H_AC, should be compared to this value where the DW movements (Barkhausen jumps) dominate the magnetization process. Below the coercive field there is a critical value Hp that delimitates two different magnetization mechanisms: below Hp the DW bulging and above it the DW jumping dominate.¹³ In accordance with these processes,¹⁴ the real part of permeability as a function of frequency has a constant section for H_AC<H_p and start decreasing immediately when H_AC > H_p. In Fig. 2 the permeability spectra are presented for some selected values of H_AC.

Unfortunately, due to the limitation of the equipment working in current generation mode, the constant excitation between 40Hz -110 MHz could not be maintained above H_AC ∼ 1 A/m. Therefore the maximum exciting field amplitude was 0.9 A/m. Practically, all the spectra coincide below 0.18 A/m, this is the initial permeability range with a constant section of the real part of the permeability as a function of frequency. The maximum of the imaginary part of the permeability occurs at 32.5 kHz, which gives a Snoek product (μs·f) ∼ 3 GHz. Interestingly, this value is very similar to that obtainable from the frequency limit of eddy current skin depth. From formula (5) we obtain:

for ρ_o ∼ 120 μΩ.cm and t ∼ 20 μm, one can obtain μ_s·f ∼ 3 GHz. This numerical result emphasize that the eddy current is a non – negligible component of the permeability spectra.

Simulating the permeability spectra taking H_AC as a parameter have been carried out using the formulas (2-3), (8) and (10). The result is shown in Fig. 3:

The spectra were decomposed using the Levenberg–Marquardt algorithm running under Origin 9 software in four contributions: i) eddy current; ii) Debye relaxation of magnetization rotation (noted as reversible Debye), iii) Debye relaxation of damped domain wall bulging (noted as irreversible Debye) and iv) resonant type DW motion (noted as DW resonant). Practically, the last free components are all of Debye type with different characteristic frequencies, because the resonance frequency of domain walls is very high, well above the working frequency range. For f/f_DW ∼ 0 the equation (8) take the form of equations (10).

The DW displacement start to have contribution to the permeability when the exciting field exceeds 0.18 A/m – 0.4 A/m (see Fig. 2.a), so the initial permeability range ends at about 0.3 A/m. In this initial permeability range (for H_AC = 0.011, 0.022 and 0.045 A/m) the permeability components are from the eddy current and magnetization rotation (reversible Debye type). At higher H_AC the reversible and irreversible parts of the DW start to dominate the small frequency region (smaller than 10 kHz). Two components appear: a resonant type at very low frequencies (DW resonant) and a Debye type which correspond to the damped DW contribution (noted as irrev. Debye).

In this way, we have obtained by simulation those two Debye like permeability’s mentioned in the literature¹ and in addition a resonant DW contribution which determines the losses (the imaginary component of permeability) at very small frequencies and high excitation field amplitudes. The eddy current plays a significant role in determining the permeability spectra at all the excitation amplitudes.

Simulating the permeability spectra of the Round type nanocrystalline Finemet core for different exciting field amplitudes, four contributions have been identified: i) the eddy current plays sizable role at all excitations, the domain wall movement give rise to ii) an irreversible Debye type and a iii) resonant type contribution at small frequencies and high excitations and the magnetization rotation gives a iv) reversible Debye type contribution at all excitations and high frequencies. The role of the eddy current is shown also by the coincidence of the experimental Snoek product with the theoretical one calculated from the skin depth formula.

This work was realized within the framework of the project KFI 16-1-2016-0079 supported by Hungarian National Research, Development and Innovation Office. This work was also supported by VEGA grants of the Scientific Grant Agency of the Ministry for Education of the Slovak Republic No. 1/0413/15, 2/0173/16, 1/0377/16 and 1/0164/16, by the Slovak Research and Development Agency under the contract APVV-15-0621 and by the Ministry of Education Agency for Structural Funds of EU in frame of projects Nos. ITMS 26220120055 and ITMS 26220220061.