Tuning Metamaterials by using Amorphous Magnetic Microwires

In this work, we demonstrate theoretically and experimentally the possibility of tuning the electromagnetic properties of metamaterials with magnetic fields by incorporating amorphous magnetic microwires. The large permeability of these wires at microwave frequencies allows tuning the resonance of the metamaterial by using magnetic fields of the order of tens of Oe. We describe here the physical basis of the interaction between a prototypical magnetic metamaterial with magnetic microwires and electromagnetic waves plus providing detailed calculations and experimental results for the case of an array of Split Ring Resonators with Co-based microwires.


S.1 Experimental set-up used for the microwave experiments
was 20 cm. The sample was placed between the antennas using a microwave transparent and diamagnetic support. Two Helmholtz coils (R= 16 cm) are used to apply a homogeneous DC magnetic field in the sample region, between -20 and 20 Oe.

S.2 Raw spectra measured by the scattering coefficient S 21
Fig. S.2 depicts the raw spectra measured for the SRR array and the magnetic microwires as a function of the applied DC magnetic fields. As the results reveal, the magnetic field changes the resonance peak of the SRR metamaterial, but these variations are difficult to observe in the raw data. For this reason in the manuscript, the spectra are referenced to the spectrum recorded at 16.6 Oe, to observe the modulation of the absorption peak due to the application of a DC magnetic field. The microwave spectrum of the SRR array plus the magnetic microwires was also measured by means of the scattering coefficient S 11 . As in the case of the S 21 coefficient, each spectrum was  Variations of the scattering coefficient S 11 at different magnetic fields. All the spectra are referenced by the spectrum recorded at 16.6 Oe.
As the results reveal the same trend is observed than in the case of the scattering coefficient S 21 showed in the main manuscript.

S.4 Induced electric current and magnetic permeability at the antenna resonance of a magnetic microwire
The electric current induced in the microwires upon illumination with 3.75 GHz microwaves is represented in Fig. S

S.5 Comparison of the microwave spectra between the microwires and the SRR array
The same microwave experiments realized to the SRRs, it described in the main manuscript, were also conducted for a set of 100 microwires, without the SRR array, of the same length used in the experiments (4 cm). As in the case of the SRR array, the microwave magnetic field, H 0 , was parallel to the axis of the microwires, so the thin antenna resonance was not excited 1 . Furthermore, different spectra were recorded applying a DC magnetic field parallel to the axis of the microwire such as in the experiments with the SRR array. For direct comparison, each spectrum was referenced by the spectrum recorded at 16.6 Oe. In Fig. S.5 is represented the spectrum at 0 Oe and the same spectrum when the microwires are inserted in the SRR array. As it is clearly seen in Fig. S.2, the spectrum of the isolated microwires is not modulated by the application of a magnetic field, whereas when they are included inside an SRR array their magnetic state modulates the response of the metamaterial at the resonance frequency of the array confirming that the observed resonance is due to the SSRs array.

S.6 Theoretical deduction of the effective permeability for an array of Split Ring Resonators within Magnetic Microwires
If we consider an array of Split Ring Resonator (SRR) of radius r [1] and we include a magnetic microwire (radius r MW <r and length l) placed at the center of each SRR, two new contributions have to be considered in the magnetic flux across the resonator: the first one associated with the magnetization of the microwire, M , and the second to the demagnetizing field of the microwire. Therefore, in presence of the microwire, the flux across the resonator is given by: where  0 is the vacuum magnetic permeability, refers to the feature that the flux of the magnetization only occurs at the cross section of the wire, α is a parameter that takes into account the fraction of the demagnetizing field of the microwire that is enclosed within the resonator, and H DC is a dc applied magnetic field along the microwire axis. The values of the parameter α ranges from zero to as described in the manuscript. The first case corresponds when the length of the microwire can be considered as infinite length in comparison with the dimension of the array. In this case, the part of the demagnetizing field of all the microwires that closes within the array can be neglected. On the other hand, the last case corresponds when the length of the microwire is much shorter than the radius of the SSRs. All the demagnetizing field is closed within the SSR and the net magnetic flux introduced by the microwire is null, α .
While in the manuscript we considered the case α=0, we describe here the more general case for any value of α. The time variation of the magnetic flux depicted by equation (1) generates an electromotrive force in the SRR of the form: Since the oscillating fields in this system are small, the time derivative of the magnetization can be expressed in terms of the differential magnetic susceptibility: The magnetic field B ave is averaged at the surfaces of the unit cell. In absence of microwires, the average value of the fields j and are zero, so that . However, the magnetization of the wire only occurs in the region enclosed by the radius of the wire. Therefore, the magnetic field of the microwire averaged outside the SRR surface is bM MW , where . The average of the demagnetizing field will be , where the coefficient α * ranges between the values 0 and , that is in the whole cell. It is important to take into account that the coefficient α * is different from the coefficient α used to define the demagnetizing field of the wires in equation (1). Thus the average field B is: Thus, we find that the effect of the magnetic microwires in the average field is equivalent to assume that the whole array is embedded in a medium of magnetic permeability μ c. As indicated above, the oscillating fields are fairly smaller than the applied fields, H 0 and H DC , then: and ( ) . Note that with this approximation and the infinite length wire, equation (11) turns into: which corresponds to the result exposed in the manuscript.
Finally, the effective permeability of the metamaterial can be computed from the average fields: Where . This effective permeability exhibit a resonance for the frequency: