Linear and bilinear magnetoelectric effects in magnetically biased magnetite (Fe3O4)

It is shown that if a ferromagnetic crystal is biased by means of a static magnetic field ${\overrightarrow{\mathbf{H}}}_0$, then the electric field dependence of the macroscopic magnetic anisotropy energy gives rise to linear as well as to two kinds of bilinear magnetoelectric effects. When expressed as functions of the perturbing electric and magnetic fields $\overrightarrow{\mathbf{e}}$ and $\overrightarrow{\mathbf{h}}$, these effects appear as the terms $-{\alpha }_{\mathrm{ij}}{e}_i{h}_j$, $-\frac{{\beta }_{\mathrm{ijk}}{e}_i{h}_j{h}_k}{2}$, and $-\frac{{\gamma }_{\mathrm{ijk}}{e}_i{e}_j{h}_k}{2}$ in the appropriate free-energy density. Calculations as well as measurements are reported of the dependences on ${\overrightarrow{\mathbf{H}}}_0$ of the magnetoelectric susceptibilities ${\alpha }_{\mathrm{ij}}$, ${\beta }_{\mathrm{ijk}}$, and ${\gamma }_{\mathrm{ijk}}$ in a magnetite (${\mathrm{Fe}}_3$ ${\mathrm{O}}_4$) crystal at 4.2 °K. For ${\alpha }_{\mathrm{ij}}$ and ${\beta }_{\mathrm{ijk}}$, but not for ${\gamma }_{\mathrm{ijk}}$, the calculated and measured ${\overrightarrow{\mathbf{H}}}_0$ dependences are in good agreement. The measured value of a certain $\left|{\beta }_{\mathrm{ijk}}\right|$ at $H_0=0\mathrm{}$ is the largest $\left|{\beta }_{\mathrm{ijk}}\right|$ observed in any material.