# Functional magnetoelectric composites with magnetostrictive microwires

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## Abstract

This paper discusses the electric polarization properties of ferromagnetic microwires at microwave frequencies. At the vicinity of the antenna resonance, a strong magnetic control of the wire polarization is possible owing to the magnetoimpedance (MI) effect. In order to realize efficient tunable properties, magnetic microwires of Co-rich compositions in amorphous state are considered. The absence of the crystalline structure is an important condition to establish well-defined magnetic anisotropies of small magnitudes, which results in the magnetization processes sensitive to the external parameters such as a magnetic field and a mechanical stress. Tunable soft magnetic properties of amorphous microwires are responsible for the MI effect, which can be observed at high frequencies (1–10 GHz). Here we demonstrate that the electric polarization of a microwire depends on its impedance and, hence, can be modulated changing the wire magnetization state in response to the application of a magnetic field and a mechanical stress. The polarization problem is treated theoretically by solving the scattering problem from a cylindrical ferromagnetic wire with the impedance boundary conditions. The modelling results agree well with the available experimental data.

## Keywords

Magnetoelectric effects Ferromagnetic microwire Microwave electric polarization Magnetic control of the electric polarization Magnetoimpedance## 1 Introduction

Competitive technological developments require functional materials that combine several properties which may be tuned by external stimuli. Considering materials with magnetic and (or) electric orders, it is natural to control the magnetization \(\varvec{M}\) and permeability \(\mu\) with a magnetic field \(\varvec{H}\), and the electric polarization \(\varvec{P}\) and permittivity \(\varepsilon\) with an electric field \(\varvec{E}\). The realization of cross-dependences \(\varvec{P}\left( \varvec{H} \right)\) and \(\varvec{M}\left( \varvec{E} \right)\) is of considerable practical interest which requires strong magnetoelectric coupling. Thus, in magnetically induced ferroelectrics \(\varvec{P}\) sensitively changes with \(\varvec{H}\) since the origin of electric polarization is ascribed to a complex magnetic order due to the inverse Dzyaloshinskii–Moriya mechanism [1, 2]. A large change in magnetization caused by an electric field was recently observed in hexaferrites with a conical spin structure [3, 4] and also in M-type hexaferrites with a co-linear magnetic structure [5, 6]. In the latter case the mechanism of spontaneous electric polarization can be still related to the magnetic order due to symmetrical exchange striction stimulating the crystal distortions.

The magnetoelectric effects are also realized in composites that combine coupled electric and magnetic dipoles [7]. The coupling mechanism typically involves piezoelectric and magnetostrictive interactions. In the present work we are dealing with the electrical dipoles induced in ferromagnetic wires by a high-frequency electric current generated by the electromagnetic field.

The underlying mechanism of magnetic control of \(\varvec{P}\) is the high-frequency magnetoimpedance (MI) effect which is observed in amorphous microwires of Co-based compositions [14, 15, 16]. Theoretically, this is explained by solving the scattering problem at a finite length ferromagnetic wire with the impedance boundary conditions imposed at the wire surface [8, 17]. It is demonstrated that the relaxation frequencies \(\omega_{{{\text{rel}},n}}\) depend on the wire surface impedance \(\varsigma\) and in the vicinity of resonance the wire polarization also strongly depends on \(\varsigma\). At certain conditions the variations in the wire magnetic structure caused by the external stimuli \(H_{\text{ex}}\) and \(\sigma_{\text{ex}}\) lead to large high-frequency impedance changes. This requires a well-defined magnetic anisotropy of a small magnitude which can be established by various annealing treatments [18]. In the case of amorphous alloys with a near-zero magnetostriction the induced anisotropy can be tuned through the magnetoelastic interactions [19, 20, 21, 22] to achieve large stress-MI.

Since the wire polarization depends on a magnetic field the scattering of electromagnetic waves by such wires can be modulated with a low-frequency magnetic field. Then, the scattered signal from a single microwire may be easily detected with lock-in techniques. This was proposed for embedded wireless sensing applications [9, 23, 24]. At certain conditions the amplitude of the modulated signal sensitively depends on the measured parameters such as an applied magnetic field and a mechanical stress.

The other range of applications is related to composite materials containing MI-wires. The effective permittivity of composites also depends on the wire magnetic configuration. Such composites are of interest as tunable electromagnetic materials [25, 26].

## 2 Electric polarization of a ferromagnetic wire

*z*-axis) induces the current \(\varvec{i}\) which must be zero at the wire ends:

The diagonal component \(\varsigma_{zz}\) relates the longitudinal electric field \(\bar{e}_{z}\) and circular magnetic field \(\bar{h}_{\varphi }\).

This can be compared with the static case when \(\bar{h}_{\varphi } = 2 I/{\text{ca}}\) where \(I\) is the total current. The difference is due to the retarding effects.

*z*-component of \(\varvec{A}\) describes the scattered electric field from a straight wire and is found from Eqs. (5) and (7). With the use of boundary condition (4) the generalized antenna equation is obtained for the current density in the ferromagnetic wire which has a form of an integro-differential equation:

Equation (10) involving the second derivatives with respect to \(z\) is completed with boundary condition (2) which requires zero current at the ends. The wire surface impedance \(\varsigma_{zz}\) and the real part of convolution \(\left( {G_{\varphi } *i} \right)\) determine the internal losses. The radiation losses are described by the imaginary parts of \(\left( {G*i} \right)\) and \(\left( {G_{\varphi } *i} \right)\). In the case of a moderate skin effect, the radiation losses are small. Then, the solution of Eq. (10) is represented as a series over a small parameter involving the imaginary parts of \(G\) and \(G_{\varphi }\). The zero approximation corresponds to neglecting the radiation losses (putting the imaginary parts to zero). In this case, a simplified analytical solution can be obtained which gives a reasonable approximation if the skin effect is not very strong. Moreover, a moderate skin effect (\(a\sim\delta_{m}\), \(\delta_{m}\) is the magnetic skin depth) is needed for the manipulation of electric polarization by changing the wire magnetization.

Equation (11) contains the permeability parameter \(\tilde{\mu }\) which is expressed via the diagonal and off-diagonal components of the internal permeability tensor. It has a meaning of a circular permeability in the cylindrical coordinate system with the dc magnetization \(\varvec{M}_{0}\) directed along the \(z^{\prime }\) axis; \(\sigma\) is the wire electric conductivity.

## 3 Tunable magnetic configuration in an amorphous wire

Here \(\varvec{n}_{\varvec{k}}\) and \(K\) are the direction and strength of the averaged short-range anisotropy, \(\hat{\sigma }\) is the total stress tensor including the internal macroscopic stresses \(\hat{\sigma }_{in}\) and external tensile stress \(\sigma_{ex}\), and \(\lambda_{s}\) is the linear saturation magnetostriction which is uniform in the amorphous state. Typically, the main contribution to the magnetic anisotropy comes from the magnetoelastic interactions. The short-range anisotropy may be enhanced or modified by annealing in the presence of a magnetic field or stress [21, 22].

## 4 Tuning the dynamics characteristics: impedance, current distribution and polarization

## 5 Application to tunable dielectrics and wireless sensors

Here \(\left\langle \alpha \right\rangle\) is the averaged polarizability of the magnetic wires. (The averaging could involve different spatial orientations of the wires.) Therefore, the effective permittivity will have a similar to \(\alpha\) frequency dispersion (see Fig. 5) and will show strong changes in the presence of external field \(H_{\text{ex}}\) and/or stress \(\sigma_{\text{ex}}\). The experimental results on scattering spectra (reflection and transmission) of composites with Co_{68}Fe_{4}Cr_{3}B_{14}Si_{11} microwires with induced helical anisotropy confirm this conclusion [12]. Near the dipole resonance, the transmission minimum deepens (from − 12 to − 17 dB at 2 GHz) under the application of \(\sigma_{\text{ex}}\) which strengthens the circumferential anisotropy decreasing the impedance and magnetic losses. On the contrary, the application of \(H_{\text{ex}}\) rotates the magnetization towards the axis, the impedance increases, and the dipole losses also increase. Therefore, composites with ferromagnetic wires exhibit tunable microwave spectra.

The other range of applications includes wireless sensors for measuring remotely the local stresses inside materials, which is based on the dipole scattering from a ferromagnetic wire. The microwave scattering defined by S-parameters from a single microwire is small in comparison with the incident wave. A special experimental procedure is needed to detect the scattered signal and its dependence on the environment stress. The method is based on the magnetic field dependence of the wire polarization [13, 15]. The wire is subjected to a microwave field and a low-frequency magnetic field \(H_{b}\) that modulates the amplitude of the scattered signal. The modulated signal of doubled frequency can be sensitively detected with lock-in techniques. This doubled-frequency signal reflects a symmetric shape of the magnetic hysteresis and impedance versus field plots. The magnitude of these modulations will depend on the external dc stimuli: magnetic field \(H_{\text{ex}}\) and stress \(\sigma_{\text{ex}}\) that can be originated by strains inside a material.

## 6 Conclusion

We demonstrated that magnetostrictive microwires behave as magnetically tunable electrical dipoles, which is based on microwave magnetoimpedance and resonance scattering. This dynamic magnetoelectric effect can be used for developing tunable microwave materials and wireless stress sensors with remote operation. For practical applications, it is useful to apply a low-frequency magnetic modulation that produces a signal of doubled frequency, the amplitude of which strongly depends on the external stimuli, such as a dc magnetic field or mechanical stress. The signal of a specific low frequency can be sensitively detected.

## Notes

### Acknowledgments

The authors gratefully acknowledge the financial support of Ministry of Science and Higher Education of the Russian Federation in the framework of Increase Competitiveness Program of MISiS (No. P02-2017-2-00, No 211) and RFBR (18-38-00637, 18-58-53059/18). L. V. Panina acknowledges the support for this work under the Russian Federation State Contract for organizing a scientific work (Grant No. 3.8022.2017).

### Compliance with ethical standards

### Conflict of interest

The authors declare that they have no conflict of interest.

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