Magnetoelectric nature of skyrmions in a chiral magnetic insulator Cu${}_2$OSeO${}_3$

Dielectric properties were investigated under various magnitudes and directions of magnetic field ($H$) for a chiral magnetic insulator Cu${}_2$OSeO${}_3$. We found that the skyrmion crystal induces electric polarization ($P$) along either in-plane or out-of-plane direction of the spin vortices depending on the applied $H$ direction. The observed $H$ dependence of $P$ in ferrimagnetic, helimagnetic, and skyrmion crystal state can be consistently described by the $d$-$p$ hybridization model, highlighting an important role of relativistic spin-orbit interaction in the magnetoelectric coupling in Cu${}_2$OSeO${}_3$. Our analysis suggests that each skyrmion quasiparticle can locally carry electric dipole or quadrupole, which implies that the dynamics of skyrmions are controllable by the external electric field.

Figure 1

(a) Crystal structure of Cu${}_2$OSeO${}_3$. (b) Symmetry elements compatible with the crystal lattice of Cu${}_2$OSeO${}_3$. (c) Magnetic skyrmion crystal formed within a plane normal to applied magnetic field ($H$). Background color represents the out-of-plane component of local magnetization vector ($m_z$). (d) Proper screw helical spin texture with a magnetic modulation vector $q$. Here, green dashed lines represent twofold rotation axes (2) or twofold rotation axes followed by time reversal ($2^{\prime }$), and a small black hexagon does a sixfold rotation axis (6) along the out-of-plane direction. (e) Schematic illustration of $d$-$p$ hybridization mechanism. (f)–(h) Magnetically induced electric polarization ($P$) under various directions of $H$ for Cu${}_2$OSeO${}_3$, predicted by the symmetry analysis (see text).

Figure 2

(a) [110] and (b) [001] components of $P$ simultaneously measured in $H$ rotating around the $\left[\overline{1}10\right]$ axis. (c) [001] component of $P$ under $H$ rotating around the [001] axis. Both measurements were performed at 2 K with $H=0.5$ T (i.e., collinear ferrimagnetic state). Dashed lines indicate the theoretically expected behaviors based on Eq. (1), and arrows denote the direction of $H$ rotation. In (d) and (e), the experimentally obtained relationships between the directions of $P$ and $M$ in the ferrimagnetic state, as well as the definition of $\theta$ and $\phi$ (the angle between the $H$ direction and the specific crystal axis), are summarized. Here the directions of $M$ and $P$ are indicated, while $\times$ denotes $P=0$.

Figure 3

Magnetic field dependence of (a) magnetization $M$, (b) ac magnetic susceptibility ${\chi }^{\prime }$, and (c) electric polarization $P$ for Cu${}_2$OSeO${}_3$ measured at 57 K with $H\parallel \left[001\right]$. The corresponding profiles for $H\parallel \left[110\right]$ and $H\parallel \left[111\right]$ are also indicated in (d)–(f) and (g)–(i), respectively. Letter symbols f, s, h, and h’ stand for ferrimagnetic, SkX, helimagnetic (single $q$ domain), and helimagnetic (multiple $q$ domains) states, respectively. Dashed lines indicate the theoretically expected behavior based on Eq. (1) for the single domain helical spin state. (j) Development of helical spin structure under applied $H$. (k) $P$-$M$ correspondence for the helical and SkX states in Cu${}_2$OSeO${}_3$, calculated based on the $d$-$p$ hybridization model (see text).

Figure 4

Calculated spatial distribution of (a) local magnetization vector $\vec{m}$, (b)–(d) local electric polarization vector $\vec{p}$, and (e)–(g) local electric charge $\rho$ for the skyrmion crystal state (see text). Magnetic field is along the out-of-plane direction, and the results for (b) and (e) $H\parallel \left[001\right]$, (c) and (f) $H\parallel \left[110\right]$, and (d) and (g) $H\parallel \left[111\right]$ are indicated. The background color represents relative value of $m_z$ for (a), $p_z$ for (b)–(d), and $\rho$ for (e)–(g), respectively. Here, $m_z$ and $p_z$ stand for the out-of-plane component of $\vec{m}$ and $\vec{p}$, respectively. The dashed hexagon indicates a magnetic unit cell of the skyrmion crystal, or a single skyrmion quasiparticle.