Magnetoelectric resonances and predicted microwave diode effect of the skyrmion crystal in a multiferroic chiral-lattice magnet

We theoretically discover that unique eigenmodes of skyrmion crystal (SkX) are not only magnetically active to an ac magnetic field (${\mathit{H}}^{\omega }$) but also electrically active to an ac electric field (${\mathit{E}}^{\omega }$) in a multiferroic chiral-lattice magnet Cu${}_2$OSeO${}_3$, which amplifies the dynamical magnetoelectric coupling between ${\mathit{E}}^{\omega }$ and the spin texture. The resulting intense interference between their electric and their magnetic activation processes can lead to an unprecedentedly large diode effect on the microwave, i.e., its absorption by SkX changes up to $\sim$20$\%$ when the incident direction is reversed. Our results demonstrate that the skyrmion could be a promising building block for microwave devices.

Figure 1

(a) Spin structure of Cu${}_2$OSeO${}_3$, composed of tetrahedra of four Cu${}^{2+}$ ions ($S=1/2$) with three-up and one-down spins. (b) Phase diagram of the spin model, (1). Here HL, SkX, and FM denote the helical, skyrmion-crystal, and ferromagnetic phases, respectively. (c) Spin structure in the SkX phase, which possesses a sixfold rotation axis, 6, and six twofold rotation axes, followed by time reversal, $2^{\prime }$. Arrows represent in-plane spin components. (d) Under $\mathit{H}\parallel \left[110\right]$, the system becomes polar along [001] and the emergence of ferroelectric polarization $\mathit{P}\parallel \left[001\right]$ is allowed. (e) Symmetry axes in a Cu${}_2$OSeO${}_3$ crystal, which belongs to the $P$2${}_1$3 space group: threefold rotation axes, 3, along $\langle 111\rangle$ and twofold screw axes, $2_1$, along $\langle 100\rangle$. (f) Real-space configurations of local electric polarizations ${\mathit{p}}_i$ in the skyrmion under $\mathit{H}\parallel \left[110\right]$.

Figure 2

(a) Imaginary parts of the dynamical magnetic susceptibilities, Im${\chi }_{yy}^{\mathrm{mm}}$ and Im${\chi }_{zz}^{\mathrm{mm}}$, as functions of the frequency $\omega$ at $g{\mu }_{\mathrm{B}}{\mu }_0{H}_z/J=3.75\times {10}^{-3}$. In Ref. 32, the strong (weak) resonance in Im${\chi }_{yy}^{\mathrm{mm}}$ was ascribed to the counterclockwise (clockwise) rotation mode, while the resonance in Im${\chi }_{zz}^{\mathrm{mm}}$ was ascribed to the breathing mode. (b) Imaginary parts of the dynamical dielectric susceptibilities, Im${\chi }_{zz}^{\mathrm{ee}}$ and Im${\chi }_{yy}^{\mathrm{ee}}$, as functions of $\omega$. The three above-mentioned magnetically active resonances can be seen in the spectra of dielectric susceptibilities at the same frequencies, indicating their simultaneous electric activities.

Figure 3

Configurations of the microwave ${\mathit{H}}^{\omega }$ and ${\mathit{E}}^{\omega }$ components, for which nonreciprocal directional dichroism is expected when $\mathit{P}\parallel \mathit{y}$ and $\mathit{M}\parallel \mathit{z}$ with $\mathit{P}\perp \mathit{M}$ and ${\mathit{K}}^{\omega }\parallel \mathit{P}\times \mathit{M}\parallel \mathit{x}$: (a) ${\mathit{E}}^{\omega }\parallel \mathit{z}$ and ${\mathit{H}}^{\omega }\parallel \mathit{y}$, and (b) ${\mathit{E}}^{\omega }\parallel \mathit{y}$ and ${\mathit{H}}^{\omega }\parallel \mathit{z}$. Cartesian coordinates $\mathit{x}\parallel \left[\overline{1}10\right]$, $\mathit{y}\parallel \left[001\right]$, and $\mathit{z}\parallel \left[110\right]$ are defined as shown, where the $z$ axis is parallel to $\mathit{H}$.

Figure 4

Calculated absorption coefficients, ${\alpha }_{+}\left(\omega \right)$ and ${\alpha }_{-}\left(\omega \right)$, for microwaves with sgn(Re$K^{\omega })=+1$ and sgn(Re$K^{\omega })=-1$, respectively, at several values of $H_z$ in the cases of (a) ${\mathit{H}}^{\omega }\parallel \mathit{y}$ and ${\mathit{E}}^{\omega }\parallel \mathit{z}$ and (b) ${\mathit{H}}^{\omega }\parallel \mathit{z}$ and ${\mathit{E}}^{\omega }\parallel \mathit{y}$.