Magnetoelectric effect

The first example of a magnetoelectric effect was discussed in 1888 by Wilhelm Röntgen, who showed that a dielectric material moving through an electric field would become magnetized.[3] The possibility of an intrinsic magnetoelectric effect in a (non-moving) material was conjectured by P. Curie[10] in 1894, while the term "magnetoelectric" was coined by P. Debye[11] in 1926. A mathematical formulation of the linear magnetoelectric effect was included in L. D. Landau and E. Lifshitz's famous book series on theoretical physics.[4] Only in 1959 did I. Dzyaloshinskii,[6] using an elegant symmetry argument, derive the form of a linear magnetoelectric coupling in Cr₂O₃. The experimental confirmation came just a few months later when the effect was observed for the first time by D. Astrov.[7] The general excitement which followed the measurement of the linear magnetoelectric effect lead to the organization of the series of MEIPIC (Magnetoelectric Interaction Phenomena in Crystals) conferences. Between the prediction of I. Dzialoshinskii and the MEIPIC first edition (1973), more than 80 linear magnetoelectric compounds were found. Recently, technological and theoretical progress, driven in large part by the advent of multiferroic materials,[12] triggered a renaissance of these studies[8] and magnetoelectric effect is still heavily investigated.[1]

If the coupling between magnetic and electric properties is analytic, then the magnetoelectric effect can be described by an expansion of the free energy as a power series in the electric and magnetic fields ${\displaystyle E}$ and ${\displaystyle H}$:[1]

${\displaystyle {\begin{aligned}F(E,H)&=F_{0}-P_{i}^{s}E_{i}-\mu _{0}M_{i}^{s}H_{i}-{\frac {1}{2}}\epsilon _{0}\chi _{ij}^{e}E_{i}E_{j}-{\frac {1}{2}}\mu _{0}\chi _{ij}^{v}H_{i}H_{j}\\&\qquad -\alpha _{ij}E_{i}H_{j}-{\frac {1}{2}}\beta _{ijk}E_{i}H_{j}H_{k}-{\frac {1}{2}}\gamma _{ijk}H_{i}E_{j}E_{k}+\ldots \end{aligned}}}$

Differentiating the free energy will then give the electric polarization ${\displaystyle P_{i}=-{\frac {\partial F}{\partial E_{i}}}}$ and the magnetization ${\displaystyle M_{i}=-{\frac {1}{\mu _{0}}}{\frac {\partial F}{\partial H_{i}}}}$. Here, ${\displaystyle P^{s}}$ and ${\displaystyle M^{s}}$ are the static polarization, resp. magnetization of the material, whereas ${\displaystyle \chi ^{e}}$ and ${\displaystyle \chi ^{v}}$ are the electric, resp. magnetic susceptibilities. The tensor ${\displaystyle \alpha }$ describes the linear magnetoelectric effect, which corresponds to an electric polarization induced linearly by a magnetic field, and vice versa. The higher terms with coefficients ${\displaystyle \beta }$ and ${\displaystyle \gamma }$ describe quadratic effects. For instance, the tensor ${\displaystyle \gamma }$ describes a linear magnetoelectric effect which is, in turn, induced by an electric field.[13]

The possible terms appearing in the expansion above are constrained by symmetries of the material. Most notably, the tensor ${\displaystyle \alpha }$ must be antisymmetric under time-reversal symmetry.[4] Therefore, the linear magnetoelectric effect may only occur if time-reversal symmetry is explicitly broken, for instance by the explicit motion in Röntgens' example, or by an intrinsic magnetic ordering in the material. In contrast, the tensor ${\displaystyle \beta }$ may be non-vanishing in time-reversal symmetric materials.

There are several ways in which a magnetoelectric effect can arise microscopically in a material.

• Piezoelectricity
• Multiferroics
• Exchange interaction