Microwave Magnetochiral Effect in ${\mathrm{Cu}}_2{\mathrm{OSeO}}_3$

We theoretically find that in the multiferroic chiral magnet ${\mathrm{Cu}}_2{\mathrm{OSeO}}_3$ resonant magnetic excitations are coupled to the collective oscillation of the electric polarization, and thereby attain simultaneous activity to the ac magnetic field and ac electric field. Because of the interference between these magnetic and electric activation processes, this material hosts a gigantic magnetochiral dichroism for microwaves, that is, a directional dichroism at gigahertz frequencies in the Faraday geometry. The absorption intensity of a microwave differs by as much as $\sim 30\%$ depending on whether its propagation direction is parallel or antiparallel to the external magnetic field.

Figure 1

(a) Crystal structure of ${\mathrm{Cu}}_2{\mathrm{OSeO}}_3$. (b) Magnetic structure of ${\mathrm{Cu}}_2{\mathrm{OSeO}}_3$. (c) Phase diagram of the spin model given by Eq. (1) with $J=1\mathrm{meV}$ and $D/J=0.09$. (d) Schematic figure of the conical spin structure. (e) Symmetry axes in the chiral cubic $P{2}_{1}3$ crystal. (f) Ferroelectric polarization $\mathit{P}\parallel [001]$ under $\mathit{H}\parallel [110]$. (g) Absence of $\mathit{P}$ under $\mathit{H}\parallel [010]$. (h) Calculated net polarization $P_c$ in the field-polarized ferromagnetic state under $\mathit{H}\perp \mathit{c}$ as a function of the angle $\theta$ between $\mathit{H}$ and the $b$ axis.

Figure 2

(a)–(c) In the presence of a net magnetization $\mathit{M}\parallel \mathit{H}$ under $\mathit{H}\parallel [010]$, the oscillating magnetization component $\mathrm{\Delta }{\mathit{M}}^{\omega }(\parallel [100])$ is accompanied by the oscillating polarization component $\mathrm{\Delta }{\mathit{P}}^{\omega }(\parallel [001])$. (d) Configuration of microwave ${\mathit{H}}^{\omega }$ and ${\mathit{E}}^{\omega }$ components, for which magnetochiral dichroism occurs under $\mathit{H}\parallel [010]$: ${\mathit{k}}^{\omega }\parallel \pm \mathit{H}$, ${\mathit{H}}^{\omega }\parallel [100]$ and ${\mathit{E}}^{\omega }\parallel [001]$.

Figure 3

Imaginary parts of the calculated dynamical magnetic, dielectric, and magnetoelectric susceptibilities $\mathrm{Im}{\chi }_{aa}^{\mathrm{mm}}$, $\mathrm{Im}{\chi }_{cc}^{\mathrm{ee}}$, and $\mathrm{Im}{\chi }_{ca}^{\mathrm{em}}$ as functions of the frequency for the conical state at $H=518.4\mathrm{Oe}$ and the ferromagnetic state at $H=1382.4\mathrm{Oe}$ when the static magnetic field $\mathit{H}\parallel [010](\parallel \mathit{b})$ is applied.

Figure 4

Calculated frequency dependence of the absorption coefficients ${\alpha }_{+}$ and ${\alpha }_{-}$ for microwaves with $\mathrm{sgn}(\mathrm{Re}{k}^{\omega })=+1$ and $\mathrm{sgn}(\mathrm{Re}{k}^{\omega })=-1$, respectively, at several values of $H$ when ${\mathit{K}}^{\omega }\parallel \mathit{H}\parallel [010]$, ${\mathit{H}}^{\omega }\parallel [100]$, and ${\mathit{E}}^{\omega }\parallel [001]$ in the conical state (a) and the ferromagnetic state (b).

Figure 5

Calculated magnetic-field dependence of the resonance frequency and magnitude of the magnetochiral dichroism $\mathrm{\Delta }\alpha /{\alpha }_{-}$.