Multiferroic Coupling of Ferromagnetic and Ferroelectric Particles through Elastic Polymers

Abstract

Multiferroics are materials that electrically polarize when subjected to a magnetic field and magnetize under the action of an electric field. In composites, the multiferroic effect is achieved by mixing of ferromagnetic (FM) and ferroelectric (FE) particles. The FM particles are prone to magnetostriction (field-induced deformation), whereas the FE particles display piezoelectricity (electrically polarize under mechanical stress). In solid composites, where the FM and FE grains are in tight contact, the combination of these effects directly leads to multiferroic behavior. In the present work, we considered the FM/FE composites with soft polymer bases, where the particles of alternative kinds are remote from one another. In these systems, the multiferroic coupling is different and more complicated in comparison with the solid ones as it is essentially mediated by an electromagnetically neutral matrix. When either of the fields, magnetic or electric, acts on the 'akin' particles (FM or FE) it causes their displacement and by that perturbs the particle elastic environments. The induced mechanical stresses spread over the matrix and inevitably affect the particles of an alternative kind. Therefore, magnetization causes an electric response (due to the piezoeffect in FE) whereas electric polarization might entail a magnetic response (due to the magnetostriction effect in FM). A numerical model accounting for the multiferroic behavior of a polymer composite of the above-described type is proposed and confirmed experimentally on a polymer-based dispersion of iron and lead zirconate micron-size particles.

Keywords: elastic properties; ferroelectric particles; ferromagnetic particles; magnetoelectric composite; multiferroics.

Figure 1

Diagram of the coupling hierarchy in a three-component composite. Particles 1 and 2 interact as magnetic or electrical dipoles and exert forces $F_d^{12}$ and $F_d^{21}$ on each other. This interaction causes elastic stresses which are transferred by spring links (solid lines) to the polymer particle 3 that, together with the springs attached to it, resembles the matrix. Due to the spring strains, particle 3 experiences elastic forces $F_E^{13}$ and $F_E^{23}$, the vector sum of which is transferred as a single force $F_E$ via the appropriate spring to particle 4 (the force projection is $F_E^{\text{′}}$ ) that transfers it outside the presented fragment of the system. We point out that $F_E^{\prime }$ is drawn twice to show its translation from particle 3 to particle 4.

Figure 2

Snapshot of the test box; blue spheres are FM particles, orange ones are FE particles, white arrows show their magnetic/electric polarizations, green spheres are polymeric beads, green lines denote elastic links; the snapshot region is 10 × 50 × 50 μm along Ox, Oy, and Oz axes, respectively.

Figure 3

Simulated polarization (a) and magnetization (b) curves for a soft elastic matrix filled with a mixture of FM and FE particles. (c) Polarization curves under an external (bias) magnetic field: H = 0 and 400 kA/m; (d) magnetization curves under an external (bias) electric field: E = 0 and 90 MV/m.

Figure 4

Simulated (a) direct $\mathsf{\Delta }P\left(E\right)=P{\left(E\right)}_H-P{\left(E\right)}_0$ and (b) inverse $\mathsf{\Delta }M\left(H\right)=M{\left(H\right)}_E-M{\left(H\right)}_0$ magnetoelectric effects; the vertical axis is scaled in arbitrary units.

Figure 5

(a) Dependence of polarization increment $\mathsf{\Delta }P$ on the strength of bias magnetic field strength; (b) dependence of magnetization increment $\mathsf{\Delta }M$ on the strength of bias electric field.

Figure 6

Magnetization increment $\mathsf{\Delta }M\left(H\right)=M{\left(H\right)}_E-M{\left(H\right)}_0,$ normalized units under an electric bias of 5 MV/m; experimental data (black) and simulation data (red) with error bars (grey).