Frequency and magnetic field dependence of the skin depth in Co-rich soft magnetic microwires

Tuneable metamaterials consisting of thin ferromagnetic wires exhibiting magnetoimpedance effect present tuneability of the effective permittivity by a weak magnetic field or a mechanical stress. Magnetoimpedance effect is related to magnetic field dependence of the skin depth of magnetic conductor. We studied giant magnetoimpedance (GMI) effect in magnetically soft amorphous Co-rich microwires in the extended frequency range. From obtained experimentally dependences of GMI ratio on magnetic field and different frequencies we estimated the penetration depth and its dependence on applied magnetic field and frequency.


Introduction
Studies of tuneable metamaterials utilising thin ferromagnetic wires exhibiting magnetoimpedance effect (MI) attracted considerable attention along the last few years [1,2].These metamaterials contain arrays of metallic wires and present strong dispersion of the dielectric function in the GHz frequency range.
Magnetic wires presenting MI effect are the key factors for design of these tuneable metamaterials.Studies of magnetic wires have attracted considerable attention of researchers and engineers along many years [3,4].Perfectly cylindrical symmetry is quite favorable for achievement of high MI effect [3,4].Consequently giant magnetoimpedance (GMI) effect has been discovered [3] and later intensively studied [4][5][6][7] in different families of soft magnetic wires.
The origin of the aforementioned GMI effect is satisfactory explained considering the change in the penetration depth of the alternating current flowing through the magnetically soft conductor caused by the applied static magnetic field.Recently we modified the experimental facility that allowed us to extend the frequency range and measure GMI effect at GHZ frequencies [5].
In soft magnetic amorphous wires subjected to an external magnetic field the GMI is in the range of 100% even at frequencies of few GHz [4][5][6].Recently we reported on high MI effect in Co and Fe-rich microwires [5][6][7].As it is well-known, the penetration depth, δ, depends on the current frequency.For observation of high MI effect the penetration depth must be smaller than the magnetic wires diameter.
Consequently we present our recent studies on the penetration depth of the alternating current flowing through the magnetically soft conductor caused by the applied static magnetic field in thin amorphous wires.

Experimental details
We studied various Co-rich (Co 67 Fe 3.85 Ni 1.45 B 11.5 Si 14.5 Mo 1.7 and Co 67.71 Fe 4.28 Ni 1.57 Si 11.24 B 12.4 Mo 1.25 C 1.55 ) with different metallic nucleus diameter, d, and total microwire diameter, D, produced by the modified Taylor-Ulitovsky method described elsewhere We have measured the magnetic field dependence of impedance, Z, and GMI ratio, ΔZ/Z, for various Co-rich microwires [5,6].
We used a specially designed micro-strip sample holder described elsewhere [5].The sample holder was placed inside a sufficiently long solenoid that creates a homogeneous magnetic field, H.The sample impedance, Z, was measured using a vector network analyzer from reflection coefficient S 11 .
The magneto impedance ratio, ΔZ/Z, has been defined as: An axial DC-field with a maximum value H max up to 8 kA/m was supplied by magnetizing coils.
The frequency range for the diagonal impedance component has been measured from 1 MHz up to 7 GHz.

Experimental results and discussion
As described above the diameter reduction must be associated with the increasing of the optimal MI frequency range: a tradeoff between dimension and frequency is required in order to obtain a maximum MI effect [5,8].Consequently we measured GMI effect at different frequencies.
Co-rich microwires present linear hysteresis loops and high GMI effect in as-prepared state (Fig 1a) and a maximum on frequency dependence of ΔZ/Z max can be observed (Fig. 1b).
For Co 67 Fe From ΔZ/Z(H) dependences it is possible to estimate the penetration depth at different frequencies using the model previously described in ref. (9) considering that the changes in the real component of the impedance are related to changes in the effective area where the AC-current flows as a consequence of the skin-effect.In this model the penetration depth, δ, as a function of the ratio R DC /R AC (R DC is the DC-resistance of the wire, and R AC is the real component of the impedance ) , can be expressed as: (2) where r is the wire radius.
Consequently we measured ΔZ/Z(H) dependences for various Co-rich microwires and tried to estimate the δ (H) dependences.
Obtained δ (H) dependences demonstrate that at high frequencies the minimum penetration depth of Co 67 Fe 3.85 Ni 1.45 B 11.5 Si 14.5 Mo 1.7 microwire is about 1.5 µm (Fig. 2).
Consequently we observed non-monotonic δ (H) dependences that must be attributed to GMI effect (Figs2b and 3b) and monotonic decreasing of the δ min with increasing the frequency.The penetration depth, δ, is given by: